# Full Text: Compositional Approaches to Linguistic Case for Cognitive Modeling

> Extracted from `cognitive_case_diagrams_v1_DAF_04-23-2026.pdf`

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## Page 1

Cognitive Diagrams: Reviewing Categorical Accounts of Linguistic Case
Daniel Ari Friedman
Active Inference Institute
daniel@activeinference.institute
ORCID: 0000-0001-6232-9096
DOI: 10.5281/zenodo.19695260
2026-04-22
2026-04-22
Contents
1
Abstract
6
2
Introduction: Bridging Formal Syntax and Situated Neuropragmatics
7
2.1
Case as Relational Algebra: From Surface Inflection to Categorical Morphism . . . . . . . . . . . . . . . .
7
2.2
Why Diagrams Earn Free Rides: Spatial Constraints as Inferential Engines
. . . . . . . . . . . . . . . . .
7
2.3
Six Converging Linguistic and Mathematical Traditions
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.3.1
Pillar 1: Case Systems, Alignment Typology, and Structural Admissibility . . . . . . . . . . . . . .
9
2.3.2
Pillar 2: Categorial and Type-Theoretic Grammar . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.3.3
Pillar 3: Categorical Compositional Distributional Semantics (DisCoCat)
. . . . . . . . . . . . . .
9
2.3.4
Pillar 4: Enriched Category Theory and Distributional Measures . . . . . . . . . . . . . . . . . . .
9
2.3.5
Pillar 5: Topos-Theoretic Bridges and Inter-Theoretic Transfer . . . . . . . . . . . . . . . . . . . .
9
2.3.6
Pillar 6: Biolinguistics, Neuropragmatics, and Oscillatory Interfaces
. . . . . . . . . . . . . . . . .
9
2.4
Active Inference Closes the Loop: Case Diagrams as the Structural Core of the Generative Model . . . . .
10
2.5
Roadmap: Navigating the Eleven Principal Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.6
What Is New in This Work
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
3
Research Questions and Manuscript Navigation
12
4
Case Systems: From Pāṇinian Kāraka to Cross-Linguistic Alignment Typology
13
4.1
Five Analytical Traditions That Shaped the Modern Theory of Case . . . . . . . . . . . . . . . . . . . . .
13
4.1.1
The Pāṇinian Kāraka Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
4.1.2
Jakobson’s Structural Features and the Prague School . . . . . . . . . . . . . . . . . . . . . . . . .
13
4.1.3
Fillmore’s Deep Case and Generative Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
4.1.4
Mel’čuk’s Meaning-Text Theory (MTT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
4.1.5
Dowty’s Proto-Roles and Graded Topologies
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
4.2
Alignment Typology: How Languages Group S, A, and P
. . . . . . . . . . . . . . . . . . . . . . . . . . .
14
4.2.1
The Three Core Argument Primitives: S, A, P
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
4.2.2
The Five Cross-Linguistic Alignment Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
5
Case Categories: Roles as Objects, Relations as Morphisms, Alignment as Functors
17
5.1
Eight Case Roles as Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
5.2
Accusative vs. Ergative as Structurally Non-Isomorphic Functors . . . . . . . . . . . . . . . . . . . . . . .
17
5.3
Graded Proto-Roles as [0, 1]-Weighted Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
5.4
Alignment Shifts as Natural Transformations: Functor Commutativity Encodes Grammar Agreement . . .
19
6
Categorial Grammar: Syntax as Algebraic Composition and Proof
22
6.1
Each Word Is Its Own Grammar Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
6.2
Lambek’s Residuation Law: Consuming and Producing Types in Linear Order
. . . . . . . . . . . . . . .
22
6.2.1
Pregroups Unlock Compact Closure: The Bridge to DisCoCat String Diagrams . . . . . . . . . . .
22
6.3
Cups, Caps, and Joyal–Street: How Wires Prove Grammaticality . . . . . . . . . . . . . . . . . . . . . . .
22

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7
Case Subscripts, Passivization, and the Curry–Howard Proof
27
7.1
Syntactic Derivation as Proof, Case Assignment as Type Inference
. . . . . . . . . . . . . . . . . . . . . .
27
7.2
Song’s Monadic Root Syntax: Sublexical Decomposition via Embedded Monads . . . . . . . . . . . . . . .
28
7.3
Passivization Is a Swap: Voice Alternation as Topological Wire Crossing . . . . . . . . . . . . . . . . . . .
28
8
Categorical Distributional Semantics (DisCoCat): Composing Vector Spaces via Type Derivations 30
8.1
The Formalism–Distribution Impasse and Why Montague Cannot Meet Harris
. . . . . . . . . . . . . . .
30
8.2
From Word2Vec to Transformer Attention: DisCoCat as the Algebraic Formalization of Distributional
Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
8.3
The Meaning Functor 𝐹∶Preg →FVect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
8.3.1
Pregroups and Vector Spaces Share a Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
8.3.2
Tensoring, Then Contracting: How “Alice Chases Bob” Becomes a Vector . . . . . . . . . . . . . .
31
8.3.3
DisCoCat Resolves “Dog Bites Man” vs “Man Bites Dog” . . . . . . . . . . . . . . . . . . . . . . .
31
8.4
Case-Typed Noun Spaces and Alignment as Natural Transformation . . . . . . . . . . . . . . . . . . . . .
32
9
Compact Closure: Snake Equation, Valency, and Four Complexity Metrics
33
9.1
The Snake Equation Powers Every Pregroup Contraction
. . . . . . . . . . . . . . . . . . . . . . . . . . .
33
9.1.1
The Snake Equation (Zigzag Identity) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
9.1.2
Four Metrics Quantify Derivational Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
10 Beyond the Sentence: State Wires Accumulate Semantic History Across Discourse
36
10.1 DisCoCirc State Wires Resolve Coreference and Role Shifts . . . . . . . . . . . . . . . . . . . . . . . . . .
36
10.2 Alice’s Role Trajectory: NOM→ACC→NOM Across Three Sentences . . . . . . . . . . . . . . . . . . . . .
36
10.3 lambeq Gen II Compiles DisCoCirc Discourse Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
10.4 No Barren Plateau for Local Observables
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
11 [0, 1]-Enriched Case Categories: Hom-Values as Distributional Proximity
42
11.1 Why Binary Morphisms Are Not Enough
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
11.2 Enriching Over ([0, 1], ⋅, 1): Identity, Sub-Multiplicative Composition, and Four Hom-Value Readings . . .
42
11.2.1 The Identity and Composition Axioms for [0, 1]-Enriched Case Categories . . . . . . . . . . . . . .
42
11.2.2 Four Linguistic Readings of the Hom-Value: Probability, Proto-Role, Similarity, Predictability
. .
42
11.2.3 When the Composition Inequality Fails: A Worked English NOM–ACC–DAT Example
. . . . . .
42
12 Magnitude and Magnitude Homology: Effective Role Count, Lawvere Similarity Spaces, and Lan-
guage as Enriched Category
45
12.1 Language as Enriched Category: Transformer Attention Weights Are Context-Dependent Hom-Values . .
46
12.2 Lawvere’s Insight: Case Categories Are Similarity Spaces
. . . . . . . . . . . . . . . . . . . . . . . . . . .
46
13 Topos-Theoretic Bridges: Transferring Results Across Case-Theoretic Frameworks
48
13.1 The Inter-Theoretic Translation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
13.2 Classifying Toposes: The Logical Shape of a Theory
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
13.2.1 A Topos Is a Self-Contained Logical Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
13.2.2 Morita Equivalence: Invariant Transfer Across Typologies . . . . . . . . . . . . . . . . . . . . . . .
48
13.3 A Chain of Morita Equivalences Connects the Four Case Theories
. . . . . . . . . . . . . . . . . . . . . .
48
13.3.1 Each Case Framework as a Geometric Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
13.3.2 The Bridge Programme: A Chain of Classifying Toposes for Case Theory
. . . . . . . . . . . . . .
48
13.4 Reconciling Classical DAIF with Intuitionistic Topos Logic via Sheaf Cohomology
. . . . . . . . . . . . .
49
13.5 Phillips’s Result: Language-of-Thought Properties as Universal Topos Constructions . . . . . . . . . . . .
49
13.6 Caramello’s Syntactic Learning Algorithm: Inducing Case Theories from Annotated Corpora
. . . . . . .
51
13.7 Morita Equivalence Diagrams Are Themselves Free-Ride Inferences . . . . . . . . . . . . . . . . . . . . . .
51
13.8 Python Implementation: Proxy Invariant Checks (implemented and tested) . . . . . . . . . . . . . . .
51
13.8.1 Extracting Geometric Theories from CaseCategory Instances . . . . . . . . . . . . . . . . . . . . . .
51
13.8.2
ClassifyingTopos Invariants and the Morita Equivalence Check . . . . . . . . . . . . . . . . . . . .
52
13.8.3 Concrete Morita Equivalence: A Two-Object Illustration . . . . . . . . . . . . . . . . . . . . . . . .
52
14 Active Inference as a Process Theory of Case
53
14.1 Static Categories Are Not Enough
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
14.2 Surprise Minimization Drives Case-Frame Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
14.2.1 Free Energy Bounds Surprisal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
14.2.2 The Five-Step Prior–Observation–Update–Prediction–Action Loop . . . . . . . . . . . . . . . . . .
54
2

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14.2.3 Case Diagrams as Instantiated Situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
14.2.4 Belief Dynamics Over Competing Case Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
15 Diagrams as Cognitively Privileged Representations: Free Rides, ERP Predictions, and Six-Strand
Synthesis
56
15.1 Why the Brain Prefers Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
15.2 P600 Signals and Garden-Path Reanalysis in Diagrammatic Models . . . . . . . . . . . . . . . . . . . . . .
56
15.3 Three Falsifiable ERP Predictions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
15.4 Integration: Six Strands Become One Generative Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
15.5 1197 Automated Tests Confirm the Formalism Is Executable . . . . . . . . . . . . . . . . . . . . . . . . . .
57
15.5.1 System Architecture and Categorical Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
15.5.2 Automated Test Suite and Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
16 Distributional Active Inference (DAIF): Convergence of Semantic Topologies and Reinforcement
Learning
59
16.1 Push-Forward Returns and the Distributional Bellman Operator
. . . . . . . . . . . . . . . . . . . . . . .
59
16.2 Quantile Temporal Difference and Implicit Quantile Networks . . . . . . . . . . . . . . . . . . . . . . . . .
60
16.3 Variational Message Passing and Bethe Free Energy
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
16.4 Policy Selection and Expected Free Energy
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
16.5 ERP Amplitude Profiles from Distributional Prediction Error . . . . . . . . . . . . . . . . . . . . . . . . .
64
16.6 Convergence Diagnostics and Distributional Metrics
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
16.6.1 Two Supporting Utilities Exposed by src/daif/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
16.6.2 Dimensional Analysis
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
16.7 Limitations and Neurobiological Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
16.8 CEREBRUM: Eight Cases as Functional Specializations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
16.8.1 Architecture and Design Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
16.8.2 Case Roles as Functional Specializations in CEREBRUM
. . . . . . . . . . . . . . . . . . . . . . .
69
17 Topological Quantum Neural Networks and ZX-Calculus: From Spin-Networks to Categorical Case
Diagrams
70
17.1 QNNs as Spin-Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
17.1.1 TQFT as the Forward Pass: Reshetikhin–Turaev Invariants Compute Network Amplitudes
. . . .
70
17.1.2 TQNNs Are Universal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
17.1.3 QRFs Select the Measurement Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
17.2 ZX-Calculus: Topological String Diagrams Where Graph Rewrites Are Quantum Proofs . . . . . . . . . .
71
17.2.1 String Diagrams for Quantum Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
17.2.2 Pregroup Cups and ZX Spiders Are Instances of the Same Compact-Closed Morphism . . . . . . .
71
17.3 One Diagram, Three Interpretations: TQNN, ZX, and DisCoCat Share a Monoidal Functor . . . . . . . .
71
17.3.1 A Common Language of Ribbon and Tensor Diagrams . . . . . . . . . . . . . . . . . . . . . . . . .
71
17.3.2 Generalized Flow Guarantees Causal Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
18 Quantum Meaning Spaces: Case Roles as Hilbert-Space Measurements
73
18.1 Case Probabilities via POVM: 𝑃(𝑐∣𝜌) = Tr(𝐸𝑐𝜌)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
18.2 Three Correspondences: Wires, Spiders, and Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
18.3 Sheaf Cohomology Governs Semantic Alignment
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
18.3.1 Contextuality, Entanglement, and Discord as Semantic Resources . . . . . . . . . . . . . . . . . . .
75
18.3.2 The TQNN/ZX Circuit as the Base Graph of a Quantum Semantic Sheaf . . . . . . . . . . . . . .
75
18.4 Case Assignment as Holographic Measurement
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
18.4.1 A Table of Correspondences: Classical Case Assignment Versus the Quantum Topological Model .
75
18.4.2 From Predictive Processing to Topological Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
18.4.3 Entanglement Strictly Exceeds Classical Semantic Capacity . . . . . . . . . . . . . . . . . . . . . .
76
19 Categorical Communication Protocols: Composing Agent Interactions via Typed Case Morphisms 77
19.1 A2A, MCP, ACP, ANP Are Missing Compositional Semantics . . . . . . . . . . . . . . . . . . . . . . . . .
77
19.2 Case Roles in Agent Protocols: NOM Requests, INS Executes, ACC Receives, DAT Benefits . . . . . . . .
77
19.3 Transformers Through Gavranović’s Lens: Attention as Parameterized Optics . . . . . . . . . . . . . . . .
78
19.4 Interpretability for Free: DisCoCat Diagrams Make Every Compositional Step Human-Readable
. . . . .
78
19.5 Multi-Turn Dialogue as a DisCoCirc Discourse Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
19.6 Multi-Agent Equilibria as Fixed Points of an Enriched Functor . . . . . . . . . . . . . . . . . . . . . . . .
79
19.7 DCST: Double-Categorical Morphisms for Sequential and Hierarchical Agent Interaction . . . . . . . . . .
79
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19.8 Five Properties of a Categorical Protocol
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
20 Prompt Injection as Categorical Type Violation: Detection and Defense
81
20.1 Injection Promotes ACC to NOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
20.2 Dependent Types, Monoidal Functors, and Multi-Turn Limits . . . . . . . . . . . . . . . . . . . . . . . . .
83
20.3 Four Defenses Against Prompt Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
20.4 The Attack Surface of an Active Inference Agent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
20.5 Quantum Key Distribution, Semantic Channels, and Functorial Encryption . . . . . . . . . . . . . . . . .
84
20.5.1 Quantum Key Distribution for Relational Semantics . . . . . . . . . . . . . . . . . . . . . . . . . .
84
20.5.2 Functorial Encryption and Diagram Obfuscation: Encrypting Compositional Meaning Itself . . . .
85
20.6 Three Epistemic Attack Vectors and Categorical Defenses . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
20.7 Present-Day Enforcement Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
20.8 Limitations and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
21 Conclusion: Elevating Language Models from Vectors to Enriched Category Frameworks
87
21.1 What This Paper Actually Did: Eleven Concrete Deliverables . . . . . . . . . . . . . . . . . . . . . . . . .
87
21.1.1 C1: Case Categories as a Formal Algebraic Framework . . . . . . . . . . . . . . . . . . . . . . . . .
87
21.1.2 C2: String Diagrams for Case Derivation Visualization . . . . . . . . . . . . . . . . . . . . . . . . .
87
21.1.3 C3: Case-Marked DisCoCat, the Distributional–Formal Synthesis, and Discourse Extension . . . .
87
21.1.4 C4: Enriched Cases, Categorical Magnitude, and Information Theory
. . . . . . . . . . . . . . . .
87
21.1.5 C5: Topos-Theoretic Transfer via Morita Equivalence
. . . . . . . . . . . . . . . . . . . . . . . . .
87
21.1.6 C6: Diagrams as Cognitively Privileged Representations . . . . . . . . . . . . . . . . . . . . . . . .
88
21.1.7 C7: Computational Verification and Test Suite (implemented and tested) . . . . . . . . . . . .
88
21.1.8 C8: Quantum Active Inference and Topological Semantic Flow (theoretical bridge)
. . . . . . .
88
21.1.9 C9: Cognitive Security and Case-Theoretic AI Safety (specification and proxy implementation) 88
21.1.10C10: Falsifiable Neurolinguistic Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
21.1.11C11: Categorical Communication Protocols for Multi-Agent AI . . . . . . . . . . . . . . . . . . . .
88
21.2 Eight Open Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
21.2.1 F1: Computational Experiments with DisCoCirc and lambeq . . . . . . . . . . . . . . . . . . . . .
89
21.2.2 F2: Topos-Theoretic Grammatical Induction from Corpora
. . . . . . . . . . . . . . . . . . . . . .
89
21.2.3 F3: Quantum Case Categories on Near-Term Hardware
. . . . . . . . . . . . . . . . . . . . . . . .
89
21.2.4 F4: Neural Predictive Processing and Electrophysiological Predictions . . . . . . . . . . . . . . . .
89
21.2.5 F5: Cross-Modal Case Structure in Embodied Cognition . . . . . . . . . . . . . . . . . . . . . . . .
89
21.2.6 F6: Enriched Category Learning from Distributional Data . . . . . . . . . . . . . . . . . . . . . . .
90
21.2.7 F7: Extending Distributional Active Inference for Linguistic Agents
. . . . . . . . . . . . . . . . .
90
21.2.8 F8: Synthesizing Biolinguistic Syntax with Neuropragmatic Inference via the ROSE Model
. . . .
90
21.3 Five Takeaways: One Argumentative Line per Formal Pillar . . . . . . . . . . . . . . . . . . . . . . . . . .
90
21.4 What the Paper Does Not Claim: Consolidated Limitations . . . . . . . . . . . . . . . . . . . . . . . . . .
91
21.5 Anticipated Objections and Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
21.6 What to Read Next, by Reader Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
21.7 Case Categories Are the Geometry of Meaning: A Unifying Coda . . . . . . . . . . . . . . . . . . . . . . .
92
22 Appendix A: Syntactic and Semantic Case Assignment Diagrams
93
22.1 Syntactic Trees and Pregroup Types: Eight Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
22.1.1 Ergative Clauses and the Alignment Functor
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
22.1.2 Passivisation as a Swap Morphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
22.1.3 Relative Clauses and Wire Threading
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
22.1.4 Complex Construction and the Complexity Metric . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
23 Appendix B: Notation Reference
95
23.1 Linguistic Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
23.2 Category Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
23.3 Enriched Categories and Magnitude
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
23.4 Distributional Semantics and LLMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
23.5 Distributional Active Inference (DAIF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
23.6 Active Inference and Cognitive Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
100
23.7 Quantum and Topological Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
101
23.8 Logical and Type-Theoretic Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
102
23.9 AI and Communication Protocols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
103
4

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23.10Diagrammatic Reasoning
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
104
23.11Notation Conventions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
104
24 Appendix C: Automated Test Suite Inventory
106
5

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1
Abstract
Commutative diagrams are cognitively privileged representations because a single diagram simultaneously encodes three
things a cognitive agent needs at once: the algebraic structure of a relational situation, the distributional semantics by
which language reports on that situation, and the inference process by which belief is updated when new evidence arrives;
the traditional linguistic category of case — who did what to whom — is the natural fulcrum on which all three layers
turn. We review formalized linguistic case systems as categories whose objects are case roles and whose morphisms are
grammatical relations, and we capture every cross-linguistic alignment type (nominative-accusative, ergative-absolutive,
tripartite, active-stative, fluid-S) as a structure-preserving functor between case categories. Sentences and discourses be-
come compact-closed string diagrams via DisCoCat and DisCoCirc, computable as executable witnesses of grammatical
and cognitive composition; enriching the case category with [0,1]-weighted hom-values supplies quantitative measures and
an explicit link to distributional proximity; a topos-theoretic bridge then ties the typological, type-logical, distributional-
semantic, enriched-categorical, and quantum layers together. Integrating this stack with Distributional Active Inference
yields first-principles, falsifiable predictions for neurophysiological event-related potentials during sentence comprehension,
and translating grammatical relations into Positive-Operator-Valued Measurements scales the same scaffolding to coherent
multi-agent discourse. In terms of Cognitive Security and AI safety, typological invariants motivate a protocol-level defense:
when multi-turn agent interactions are modeled as a fixed category of licensed morphisms with explicit role types and
wiring, prompt injection aligns with ill-typed role promotion and can be analyzed as a functorial type violation — an engi-
neering and specification target, not an automatic guarantee, on present-day LLM APIs. The resulting category-theoretic
scaffolding makes quantitative neurophysiological predictions, makes prompt injection statically decidable in principle
relative to that same fixed protocol (where the interaction grammar is enforced), and turns cross-linguistic typology into
a proof-by-functor rather than a taxonomy; the manuscript is accompanied by 1197 executable tests across sixty-four
test files at 95.96% line-and-branch coverage on src/ (from coverage.json) and 30 programmatically generated figures, so
every formal claim either runs or is honestly flagged as future work. The complete source code, test suite, manuscript,
and all figures are available open source on the GitHub repository and archived with DOI 10.5281/zenodo.19695260.
6

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2
Introduction: Bridging Formal Syntax and Situated Neuropragmatics
Central claim. We argue — and then make computationally precise across eleven contributions — that commutative
diagrams are cognitively privileged representations because a single diagram carries, without translation, three things
a language-using agent needs at once: the algebraic structure of a relational situation (who stands in which role),
the distributional semantics by which language reports on that situation (how likely is each assignment given the
sentence heard so far), and the inference process by which belief is revised when new evidence arrives (how should the
assignment be updated). Linguistic case is the natural pivot on which the three layers turn: case roles are the objects of
the relational algebra, case-marked words supply the distributional evidence, and case assignments are the latent variables
whose posterior is updated. Every subsequent pillar of the paper — from ancient 𝑘̄𝑎𝑟𝑎𝑘𝑎theory in section 4, through
DisCoCat and DisCoCirc across section 6–section 10, through distributional active inference in section 16, through POVM-
based quantum semantics in section 17, to categorical prompt-injection defenses in section 20 — is a witness to this single
central claim.
2.1
Case as Relational Algebra: From Surface Inflection to Categorical Morphism
Every natural language must solve the same fundamental problem: encoding who does what to whom, with what, where,
and why. Morphologically rich languages encode these situational relations overtly via distinct word modifications marking
the agent, patient, instrument, or location. Languages like Mandarin or English achieve the same end through strict word
order and adpositions. Beneath this surface variation lies a universal computational challenge: a cognitive agent must
dynamically assemble and assign structured roles to the participants of every encountered scenario, predicting relationships
and mapping them against the constraints of a conceptual space.
Today, this same universal computational challenge lies at the heart of the artificial intelligence alignment crisis. The
inability of contemporary Large Language Models (LLMs) to securely distinguish who is allowed to do what to whom
within a text block—allowing passive data to masquerade as commanding instructions—is the root cause of prompt
injection and agentic hijacking.
This universality suggests that linguistic case is not a morphosyntactic accident, but the reflection of a deeper, embodied
cognitive architecture. The lineage of this hypothesis is long: from Pāṇini’s ancient kāraka theory (which abstracted
nominal roles from root actions), through Jakobson’s structuralist feature matrices and Fillmore’s [1968] establishment of
a universal inventory of deep cases (Agent, Patient, Instrument), to Mel’čuk’s Meaning-Text Theory [1988], which formally
treats cases as relational networks mapping surface form to deep semantic structure. Modern typological work by Polinsky
and Preminger [2014], Blake [2001], and Haspelmath [2009] confirms that cross-linguistic variation is tightly bounded into
a small number of alignment types, related by systematic transformations.
This review shows that category theory supplies the mathematical language to formalize relational structure. Commutative
diagrams function as cognitively privileged representations. By constraining relational algebras to a two-dimensional graph,
these diagrams recruit spatial reasoning, providing boundary enforcement and free-ride inferences unavailable to linear
encodings.
2.2
Why Diagrams Earn Free Rides: Spatial Constraints as Inferential Engines
The cognitive science of diagrammatic reasoning provides a strong empirical foundation for this claim. To cast a problem
onto a two-dimensional visual topology is to recruit the nervous system’s innate, ancient capacity for spatial navigation.
Larkin and Simon [1987] demonstrated that diagrams can be computationally superior to sentential representations: their
planar constraint enables efficient perceptual grouping that would otherwise require painstaking, explicit logical deduction.
Shimojima [1996] refined this with the concept of free ride inferences—conclusions that “fall out” automatically from the
geometric constraints of a sketch without syntactic manipulation.
In the context of case theory, commutative diagrams offer exactly these embodied advantages. A commutative triangle
expressing that morphism 𝑔∘𝑓equals morphism ℎsimultaneously encodes:
1. The compositional structure of the relation (two steps factor through an intermediate case role).
2. The commutativity constraint (the direct and indirect visual routes yield the same mathematical result).
3. The full relational context (the entire shape of the scenario is apprehensible in a single perceptual glance).
This is exactly the kind of gestalt understanding that linear notation obscures. Figure 1 lays out a concrete instance. Let
𝑓∶NOM →INS be uses (𝑤= 0.9), 𝑔∶INS →ACC be applied_to (𝑤= 0.7), and ℎ∶NOM →ACC be acts_on with ℎ= 𝑔∘𝑓
and 𝑤(ℎ) = 0.63 = 𝑤(𝑔) ⋅𝑤(𝑓) as in Equation 3. The single morphism ℎis drawn twice in the diagram: once as the direct
arrow NOM →ACC and once as the composite path through INS; commutativity identifies those routes—they are not two
independent relations. A separate licensed arrow NOM →VOC (addresses, 𝑤= 0.85) lies outside that triangle. Dashed
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red arrows depict structurally prohibited maps (not elements of any Mor(𝒞)), including VOC →NOM and ACC →NOM,
for contrast with the solid licensed morphisms. As Giardino [2017] emphasizes, mathematical diagrams engage a hybrid
mode of reasoning, intimately wedding perceptual pattern-recognition with theoretical knowledge—a mode effortlessly
aligned with the predictive processing architecture of active inference [Namjoshi, 2026]. The intellectual lineage traces
through Peirce’s existential graphs: his spatial logic system proved that first-order reasoning can be conducted entirely
diagrammatically on a “sheet of assertion.” Our case diagrams extend this Peircean tradition into relational semantics:
inference proceeds not by algebraic churning, but by tracing lines and composing arrows—drawing the contours of thought
itself.
Figure 1: Commutativity in a small case category yields free-ride inference (Equation 3). Objects: NOM, ACC, INS,
VOC. Solid green arrows: licensed morphisms; edge labels use 𝑓, 𝑔, ℎon the commuting triangle and weights 𝑤∈[0, 1].
Dashed red: structurally prohibited VOC →NOM and ACC →NOM (not in Mor(𝒞)).
Generated by introduc-
tory_case_category() and render_case_category.
2.3
Six Converging Linguistic and Mathematical Traditions
The present synthesis draws on six research traditions, each contributing an essential formal ingredient:
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2.3.1
Pillar 1: Case Systems, Alignment Typology, and Structural Admissibility
The empirical foundation comes from linguistic typology. We formalize the case inventory and alignment systems cata-
logued by Polinsky and Preminger [2014], Claassen [2019], and Wu [2024] as categories with noun case roles as objects
and grammatical relations as morphisms. Crucially, the categorical framework makes explicit a distinction often implicit
in traditional accounts: structural admissibility—which morphisms between case roles are licensed (structurally well-
formed) versus prohibited (ill-formed). For example, the morphism NOM→ACC (“agent acts on patient”) is universally
licensed, while VOC→NOM (“addressee as agent”) is structurally inadmissible. This admissibility structure, visualized
through solid licensed edges, dashed prohibited edges, and absent transitions in the case category diagram (Figure 1),
encodes the combinatorial constraints of argument structure more transparently than any linear notation. Dowty’s [1991]
proto-role theory—which decomposes thematic roles into clusters of entailments rather than discrete atoms—motivates our
use of enriched (weighted) morphisms, where the morphism weight quantifies the degree of admissibility along a licensed
transition. Synthetic Case-Role Algebra (introduced in section 4) extends this line by formalizing proto-roles in a
[0, 1]-enriched monoidal setting, enabling context-sensitive algebraic manipulation of case-role semantics without ad hoc
feature bundles.
2.3.2
Pillar 2: Categorial and Type-Theoretic Grammar
Lambek’s [1958] syntactic calculus reformulates grammatical combination as algebraic cancellation in a pregroup, yielding
derivations that can be visualized as Joyal and Street’s [1991] string diagrams. Song [2022a] extends this with monadic
semantics for root syntax, showing that category-theoretic structure reaches down to the sublexical level.
2.3.3
Pillar 3: Categorical Compositional Distributional Semantics (DisCoCat)
Coecke, Sadrzadeh, and Clark’s [2010] DisCoCat framework composes distributional word meanings according to syn-
tactic structure using monoidal categories and string diagrams.
We lift grammatical pregroup reductions directly to
strong monoidal functors, mapping linguistic structures to inner-product vector spaces independent of basis. This frame-
work bridges the two great traditions of linguistic meaning—formal (truth-conditional, compositional) and distributional
(context-dependent, statistical)—by using the algebra of compact closed categories to compose vector representations
according to type-logical derivations. The resulting categorical semantics provides the algebraic formalization of the dis-
tributional programme that modern large language models (from Word2Vec [Mikolov et al., 2013] through transformer
architectures [Vaswani et al., 2017, Devlin et al., 2019]) implement empirically, making it possible to analyse both sym-
bolic and neural approaches to language within a single mathematical framework. DisCoCirc [de Felice and Coecke, 2020,
de Felice et al., 2022] and the lambeq QNLP stack [Lorenz et al., 2021, Quantinuum, 2025] extend this to discourse and
circuit compilation; section 10 gives the discourse-level and Gen II details.
2.3.4
Pillar 4: Enriched Category Theory and Distributional Measures
Bradley et al.’s [2021] enrichment over [0, 1] equips hom-sets with distributional proximity measures, supplying the bridge
from syntax to statistical semantics. Categorical magnitude (scalar “effective size’ ’ of an enriched category) is the
Leinster-style invariant that summarises how much relational structure a case system encodes; Bradley’s [2021; 2024]
operadic and information-theoretic work complements that picture. Leinster and Shulman [2021] extend magnitude to
magnitude homology, a graded homological invariant that detects higher-dimensional differences between case systems
indistinguishable by scalar magnitude alone.
2.3.5
Pillar 5: Topos-Theoretic Bridges and Inter-Theoretic Transfer
Caramello’s [2016; 2021; 2023] bridge technique uses classifying toposes as transfer points between mathematical theories:
when two formalizations are Morita equivalent (or linked by an explicit bridge), topos-level invariants proved in one
setting carry over without separate proof. Phillips [2024] strengthens this by showing that the Language of Thought
admits universal constructions in a topos, grounding the systematicity of case assignment in the deepest structures of
categorical logic.
2.3.6
Pillar 6: Biolinguistics, Neuropragmatics, and Oscillatory Interfaces
The cognitive neuroscience of language is caught between two paradigms: the formal, algebra-driven constraints of biolin-
guistics, which models syntax via the order-free, non-associative topological boundaries of the MERGE operator [Murphy,
2026], and the context-driven, highly associative dynamics of neuropragmatics, which models intention and dialogue
via Bayesian inference [Gutierrez Cisneros et al., 2026]. To bridge this gap—translating rigid mathematical geometry into
flexible social cognition—we adopt Murphy’s ROSE architecture [Murphy, 2023] (Representation, Operation, Structure,
9

## Page 10

Encoding): slow oscillatory phase codes (delta/theta) protect the strict algebraic hierarchy of syntax (a “mesoscopic pro-
tectorate”), while fast gamma rhythms bind semantic and pragmatic probabilities to these structures. Such bioelectric
handoffs provide the mechanistic interface where literal, formal structures are exported into the Default Mode Network
and Theory of Mind hubs for situated speech-act evaluation.
The algebraic frameworks in Pillars 2–5 (DisCoCat, enriched categories) map the strict wiring of syntax to vector spaces.
The categorical formulation sets topological boundaries within which context and distributed meanings update. This
sketches a mechanistic interface between syntax and situated inference.
2.4
Active Inference Closes the Loop: Case Diagrams as the Structural Core of the Gen-
erative Model
The unifying framework is active inference [Namjoshi, 2026], which casts cognition as approximate Bayesian inference
under a generative model. Case-marked commutative diagrams serve as the structural core of such generative models:
each diagram specifies a pattern of expected relational dependencies, and the agent’s task in understanding (or producing)
a sentence is to minimize surprise relative to this diagrammatic prior. The CEREBRUM architecture [Friedman and
Active Inference Institute, 2024]—Case-Enabled Reasoning Engine with Bayesian Representations for Unified Modeling—
provides a computational instantiation of this framework, with case roles as functional specializations of model components
in an active inference cycle.
The situation semantics of Barwise and Perry [1983] already conceptualized meaning as structured situations with typed
constituents—an ontology that maps naturally onto the objects and morphisms of our case categories. Active inference
adds dynamics: the agent actively samples evidence to confirm or update its case assignments, using diagrammatic
structure to guide exploration.
2.5
Roadmap: Navigating the Eleven Principal Contributions
The remainder of the paper develops this synthesis in order: section 4–section 5 (typology and functorial alignment);
section 6–section 7 (pregroup diagrams and case-marked types); section 8 and subsection 8.3 (DisCoCat); section 9
(compact closure and diagram complexity); section 10 (discourse and QNLP); section 11–section 12; section 13; section 14,
section 15, and section 16 (active inference, diagrammatic ERP predictions, and DAIF); quantum extensions (section 17,
section 18); applications (section 19, section 20). section 3 maps questions RQ1–RQ11 to sections and contributions
C1–C11. Computational verification is summarized in section 15 with test inventory in section 24.
Each subsequent section is self-contained yet cumulative: the reader may enter at any pillar and follow the cross-references
forward to the synthesis. The manuscript articulates how DisCoCat functors from pregroup grammars to FinHilb interface
with topos-theoretic invariants where Morita equivalence holds. This sketches connections between variational free energy
minimization, ZX-style rewrites, and secure protocol enforcement (see C5, C8, C9 for evidence classes).
2.6
What Is New in This Work
The individual ingredients we build with have independent prior art. What is genuinely new here is the integration and
its computational witness. Building on the prior art most readers will know, our distinct contributions are:
• Extending Coecke–Sadrzadeh–Clark DisCoCat [2010; 2017]: we add to the pregroup / FinHilb functorial
pipeline explicit case-marked wires, making grammatical role a first-class typed structure on every wire rather
than an emergent property of word-order composition (section 7). The three-sentence DisCoCirc discourse we ship
(section 10) tracks case-role trajectories — Alice NOM→ACC→NOM — that single-sentence DisCoCat does not
address.
• Building on Akgül et al. Distributional Active Inference [2026]: we instantiate DAIF on a case-theoretic
generative model rather than a generic MDP, obtaining first-principles derivations of the N400 and P600 ERP
amplitudes from a first-order expansion of Δ𝐹(subsection 16.5) — a derivation, not an empirical ansatz.
• Extending Friedman CEREBRUM [2024]: we add a distributional layer (the full src/daif/ subpackage) to the
eight-case functional-specialisation architecture, replacing point beliefs with return-distribution posteriors through-
out the inference cycle (subsection 16.8).
• Building on Friston active inference [2017]: we preserve the canonical three-term 𝐺(𝜋) as the 𝛽risk = 0 spe-
cialisation of a four-term decomposition (Equation 27) that exposes the risk-sensitive variance term needed for
distributional RL, and we supply a formal collapse identity back to the canonical form so the canonical form is
recovered as a special case rather than discarded.
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## Page 11

• Juxtaposing contemporary QNLP (lambeq, Quantinuum) [2021]: our quantum layer is a specification —
POVM-based case assignment with honestly-disclosed classical-mixture limits on 𝜌— sitting alongside, not in place
of, hardware QNLP; the section 16 distributional pipeline runs entirely on classical numerics and every section 17 /
section 18 claim is explicitly status-labelled as either implemented or literature-bridge.
• Building on present-day LLM agent-safety practice: we recast prompt injection in a type-theoretic reformu-
lation (section 20) in which multi-turn role promotion becomes an ill-typed morphism in a fixed protocol category,
complementing existing content filters with a concrete decidability story — engineering target, not automatic guar-
antee on today’s APIs.
The entire synthesis is backed by a public test suite (1197 tests across 64 files at 95.96% line-and-branch coverage on src/
(from coverage.json)) and 224 tests specifically covering the DAIF distributional layer, so every formalism in the paper
either executes or is honestly flagged as future work in subsection 16.7.
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3
Research Questions and Manuscript Navigation
This paper addresses eleven core research questions, each developed in specific sections and yielding a principal contribution
(C1–C11, as enumerated in section 21). The table below provides a navigational overview; the reading guide that follows
explains the logical dependencies between questions.
Table 1: Core research questions mapped to corresponding manuscript sections and resulting contributions.
#
Research Question
Addressing
Sections
C
RQ1
Can linguistic case systems—across all major typological alignment patterns—be formalized
within a single algebraic framework, with roles as objects and grammatical relations as
morphisms?
section 4,
section 5
C1
RQ2
Do string diagrams derived from pregroup type reductions provide a computationally and
cognitively superior representation of case derivations compared to linear notation?
section 6,
section 7,
section 22
C2
RQ3
Can the DisCoCat meaning functor be extended with case-typed noun spaces to unify formal
and distributional semantics—and does this unification generalize to discourse via DisCoCirc?
section 8,
subsec-
tion 8.3,
section 9,
section 10
C3
RQ4
Does equipping case categories with [0, 1]-enriched hom-values yield a principled bridge
between symbolic grammar and statistical semantics, and can categorical magnitude serve as
a complexity invariant?
section 11,
section 12
C4
RQ5
Can classifying toposes and Morita equivalence (or explicit bridge toposes) scaffold transfer of
topos-expressible invariants between distinct formalizations of case theory (typological,
type-logical, distributional, enriched), when equivalence is exhibited?
section 13
C5
RQ6
Are commutative diagrams cognitively privileged representations—recruiting spatial reasoning,
free-ride inference, and predictive processing—beyond being merely convenient notation?
section 2,
section 14
C6
RQ7
Can the entire categorical framework be computationally verified with real (non-mock)
automated tests at ≥90% coverage, confirming that the abstractions are executable?
section 15,
section 16
C7
RQ8
Do categorical string diagrams extend naturally into quantum generalizations via TQNNs,
ZX-calculus, and sheaf-theoretic quantum semantics—and does quantum entanglement
provide genuine advantages for semantic communication?
section 17,
section 18
C8
RQ9
Can prompt injection attacks be formulated mathematically as illicit cross-category
morphisms (type violations), yielding a decidable static check relative to a fixed
protocol / interaction category (compile-time where that grammar is enforced)?
section 20
C9
RQ10
Does integrating enriched category theory with active inference generate falsifiable
neurolinguistic predictions (P600/N400 amplitudes scaling with enriched morphism weights)?
section 14,
section 16
C10
RQ11
Can grammatical case roles (NOM, ACC, INS) rigorously type-check the tensor payloads of
multi-agent AI framework protocols (A2A, MCP, ANP) to prevent agentic hijacking?
section 19
C11
Reading guide. RQ1–RQ5 build the mathematical framework cumulatively: each layer depends on the
preceding one, from case categories (RQ1) through string diagrams (RQ2), distributional semantics (RQ3), en-
riched structure (RQ4), to topos-theoretic transfer (RQ5). RQ6 provides the cognitive science meta-argument
that threads through the entire paper, grounding diagrams in spatial reasoning and predictive processing. RQ7
supplies computational validation of the framework via automated testing. RQ8–RQ11 extend the framework
into four application domains: quantum semantics (RQ8), cognitive security (RQ9), falsifiable neurolinguistic
predictions (RQ10), and multi-agent AI protocols (RQ11). The conclusion (section 21) revisits each question
with a summary of results and identifies eight open directions (F1–F8).
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4
Case Systems:
From Pāṇinian Kāraka to Cross-Linguistic Alignment
Typology
Where we are in the argument. section 2 stated the central claim — that commutative diagrams are cognitively
privileged because they encode algebra, distribution, and inference at once, with case as the pivot. Before we formalise
case as a category in section 5, this chapter surveys the linguistic input the formalism must cover: five analytical traditions
(Pāṇinian 𝑘̄𝑎𝑟𝑎𝑘𝑎, Fillmorean deep case, Jakobsonian features, Mel’čukian dependency, Haspelmathian typology) and the
five alignment typologies (nominative-accusative, ergative-absolutive, tripartite, active-stative, fluid-S) on which cross-
linguistic variation converges.
4.1
Five Analytical Traditions That Shaped the Modern Theory of Case
4.1.1
The Pāṇinian Kāraka Framework
The formal study of grammatical case originates with Pāṇini (circa 4th century BCE), whose Aṣṭādhyāyī —a grammar of
approximately 4,000 rules that predates Euclid—formalized the kāraka theory. This framework first classified semantic
roles—agent, patient, instrument, and others—as deep relational functions linking verbs to their arguments, abstracting
away from surface morphosyntactic inflections. Etymologically meaning “that which brings about” an action, the kāraka
system establishes a rigorous mapping from conceptual predication to phonological realization [2021; 1987].
4.1.2
Jakobson’s Structural Features and the Prague School
This profound semantic foundation lay predominantly dormant until structural linguists resurrected it in the mid-20th
century. Roman Jakobson’s Morphologic Inquiry into Slavic Declension (1958) decomposed grammatical cases into binary
distinctive features (e.g., [±directional], [±peripheral]) [1958]. By treating case oppositions analogously to phonological
features, Jakobson shifted the field from Pāṇini’s holistic roles to a componential analysis that exposes deep relational
hierarchies and markedness.
Concurrently, Louis Hjelmslev’s La catégorie des cas [1935] reinforced this functionalist
tradition by formalizing case as a purely relational category within a dynamic semiotic network.
To make Jakobson’s componential analysis tangible, consider the six-case singular paradigm of the Russian masculine
inanimate noun stol “table”: NOM stol / ACC stol / GEN stolá / DAT stolú / INS stolóm / PREP (LOC) stolé. The same
root surfaces in six morphological guises whose contrasts Jakobson decomposes along the binary features [±directional]
(the dative singles out goal-directed cases) and [±peripheral] (the instrumental and locative push out from the core
argument cases). Serbian/BCS prijatelj “friend” runs a parallel paradigm but adds an overt vocative prijatelju! “friend!”,
giving a seven-way contrast that exercises the entire eight-role inventory we adopt in Table 4 (with the ablative absorbed
into Slavic genitive-of-source and the prepositional/locative). These overt morphological contrasts will reappear in section 5
as concrete witnesses for the alignment functor and in section 11 as a calibration source for [0, 1]-enriched hom-values.
Slavic languages — with their dense nominal morphology, productive derivation, and unambiguous case suffixes — are also
a natural stress-test candidate for the Distributional Active Inference framework developed in section 16; the limitations
discussion in subsection 16.7 returns to this point explicitly, noting that a Russian or Serbian/BCS sentence would deliver
an information-theoretically sharper drop in posterior entropy than the German example currently shipped.
4.1.3
Fillmore’s Deep Case and Generative Roots
Charles Fillmore’s seminal “The Case for Case” (1968) built directly upon this structuralist lineage, explicitly translating
these concepts into the burgeoning generative linguistics framework. Fillmore proposed deep cases (e.g., Agentive, Objec-
tive, Dative) as universal semantic primitives underlying surface syntax. Crucially, he argued that verbs assign underlying
case frames which subsequently generate surface structures [1968]. This evolution transformed Pāṇini’s kāraka into deep
relational networks, permanently prioritizing universal semantics over language-specific morphology.
4.1.4
Mel’čuk’s Meaning-Text Theory (MTT)
A distinct but parallel formalization emerged with Aleksandr Žolkovskij and Igor Mel’čuk’s Meaning-Text Theory (MTT)
[Žolkovskij and Mel’čuk, 1965, Mel’čuk, 1981, 1988]. Developed initially in Moscow, MTT models natural language as a
rigorous, many-to-many correspondence between meanings (deep semantic representations) and texts (surface-phonological
representations). The transition from meaning to text unfolds across a multi-level synthesis process: Deep Semantics, Deep-
Syntactic, Surface-Syntactic, Morphological, and Phonological. At the deep-syntactic level, meaning is represented via
dependency trees linking lexical units through dependency relations. The nodes are populated by actants—semantic roles
closely aligning with thematic grids but strictly language-specific in their surface realization.
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Crucially, MTT establishes that grammatical case is not directly semantic, but rather a final surface-morphological phe-
nomenon governed by syntactic linearization and dependency constraints. Lexical functions map actants to surface struc-
tures, explicitly demonstrating that morphosyntactic inflection serves merely as the end-stage formal realization of deep
semantic relations. By monotonically mapping meaning to text via hierarchical dependency graphs, MTT provided an
exhaustive synthesis that presages our modern categorical formalizations mapping conceptual structures to syntactic types.
4.1.5
Dowty’s Proto-Roles and Graded Topologies
Where MTT models the mapping from deep semantic roles to surface morphology via hierarchical dependency graphs,
Dowty’s contribution is to decompose the deep roles themselves into continuous, gradient primitives. Fillmore’s deep
cases serve as the direct precursors to modern thematic role theory. Dowty [1991] refined this paradigm by decomposing
thematic roles into clusters of sentential entailments, yielding two proto-roles: the Proto-Agent (characterized by voli-
tional involvement and causation) and the Proto-Patient (characterized by incremental themes and causal affectedness).
This decomposition proves critical for our categorical formalization: it replaces discrete, named roles with a graded struc-
tural topology where role assignment is a matter of degree rather than kind. Consequently, morphisms in our cognitive
case diagrams carry continuous weights 𝑤∈[0, 1] representing the degree to which a noun phrase satisfies proto-role
entailments—a design choice that yields the enriched category structure developed formally in section 11.
4.2
Alignment Typology: How Languages Group S, A, and P
Contemporary typological work reveals that the world’s languages realize case systems according to a small number of
alignment types—systematic patterns governing how the core arguments of transitive and intransitive clauses are grouped
[Polinsky and Preminger, 2014, Blake, 2001, Haspelmath, 2009].
4.2.1
The Three Core Argument Primitives: S, A, P
The cross-linguistic comparison rests on three primitives:
Table 2: The three core argument primitives used in cross-linguistic case typology.
Symbol
Role
Definition
S
Sole argument of intransitive
“The child sleeps”
A
Agent-like argument of transitive
“The child broke the vase”
P
Patient-like argument of transitive
“The child broke the vase”
4.2.2
The Five Cross-Linguistic Alignment Types
The key insight from typological research is that languages differ in how they group these three roles for purposes of case
marking, agreement, and other grammatical processes:
Table 3: Five alignment types and their grouping of the three core argument primitives.
Alignment
Grouping
Exemplar Languages
Nominative–Accusative
S = A ≠P
English, Latin, Finnish, Russian
Ergative–Absolutive
S = P ≠A
Basque, Dyirbal, Georgian (partly)
Active–Stative
S splits by agentivity
Lakhota, Guaraní, Eastern Pomo
Tripartite
S ≠A ≠P
Nez Perce, some Australian languages
Fluid-S
S marking varies by context
Bats (NE Caucasian), Acehnese
Slavic case morphology as a stress-test for the formalism. Russian (six cases) and Serbian/BCS (seven cases
including a productive vocative) sit firmly in the nominative-accusative column of Table 3, yet their nominal paradigms
exercise the full eight-role inventory of Table 4 in ways that English’s collapsed paradigm hides. Two animacy-conditioned
syncretisms in Russian masculine singular illustrate the point precisely:
• Inanimate masculine — NOM = ACC syncretism. stol “table” surfaces identically in subject and direct-object
position (NOM stol / ACC stol; Stol stoit “the table stands” vs. Vižu stol “I see the table”). The surface-realisation
functor 𝑀∶𝒞grammatical →𝒞morphological identifies {NOM, ACC} for this declension class — a partial kernel that
coexists with full distinguishability for feminine and neuter paradigms.
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• Animate masculine — ACC = GEN syncretism. brat “brother” takes ACC = GEN brata (Vižu brata “I see the
brother”, same form as the genitive bez brata “without the brother”). The morphology overtly licenses the canonical
formalism’s claim that the surface functor neutralises distinct grammatical objects on a paradigm-by-paradigm basis,
not globally.
Serbian/BCS replicates the same animacy split (čovek / čoveka parallel to brat / brata) and adds a productive vocative
(prijatelju! “friend!”, bože! “God!”) that English lacks entirely. These are exactly the empirical witnesses the alignment
functor of section 5 predicts: where English shows a single inflectionless noun, Slavic morphology exposes a non-trivial
kernel structure on a per-declension-class basis, supplying ready-made cross-linguistic targets for the categorical apparatus
we develop next.
Fluid-S and Context-Dependent Functors. In Bats (Nakh-Daghestanian), the intransitive subject of a single verb
surfaces in different cases strictly depending on the speaker’s internal construal of agentive volition. For example, the verb
fall assigns an absolutive S when the action is accidental (The child-ABS fell), but assigns an ergative S when the action
is volitional (The child-ERG fell [on purpose]).
Categorically, we model Fluid-S as a context-dependent functor 𝐹𝜃∶𝒰→ℒparameterized by a continuous volition
feature 𝜃∈[0, 1]. Figure 2 visualizes the resulting volition landscape: case categorization boundaries shift dynamically as
a direct function of the agent’s internal construal, satisfying naturality only up to a probabilistic reparameterization of 𝜃.
Synthetic Case-Role Algebra. We introduce Synthetic Case-Role Algebra as a novel, computational upgrade to the
Dowty-style proto-role framework. Where Dowty modeled proto-roles as static clusters of entailments [1991], we formalize
them as objects in a [0, 1]-enriched monoidal category with tensor product over role compositions. This advancement
enables purely algebraic manipulation of semantic roles: composition, weighting, and transformation of roles proceed
through functorial operations representing complex event structures such as causativization, serial verb constructions, and
argument-structure alternations. Crucially, this enriched structure demonstrates that “case assignment” is not a discrete
binary choice but a vector-valued expectation in continuous case space—providing the mathematical bridge between the
symbolic traditions of formal grammar and the statistical representations of modern neural language models (section 8).
Claassen [2019] surveys the explanatory frameworks proposed for alignment diversity, arguing that no single factor (pro-
cessing efficiency, disambiguation, discourse pragmatics) suffices—a conclusion that motivates our multi-dimensional cate-
gorical formalization. Wu [2024] offers a detailed case study of Amis (Austronesian), demonstrating how verb classification,
case marking, and grammatical relations interact in a language that defies simple alignment classification.
Beyond the three core arguments, languages distinguish a rich inventory of oblique cases. Our formalization follows the
CEREBRUM framework [Friedman and Active Inference Institute, 2024] in adopting eight fundamental cases (Table 4):
Table 4: Eight fundamental case roles adopted from the CEREBRUM framework [Friedman and Active Inference Institute,
2024].
Case
Abbreviation
Semantic Core
Syntactic Prototype
Nominative
NOM
Agent / experiencer
Intransitive subject,
transitive agent
Accusative
ACC
Patient / theme
Direct object, incremental
theme
Genitive
GEN
Possessor / source
Possessive modifier,
partitive
Dative
DAT
Recipient / goal
Indirect object,
beneficiary
Instrumental
INS
Instrument / means
Adverbial of means
Locative
LOC
Location / context
Spatial/temporal ground
Ablative
ABL
Origin / cause
Source of motion, causal
adjunct
Vocative
VOC
Addressee
Direct address
While historically used merely to diagram sentences, this exact eight-role inventory is what enables the Categorical AI
Protocol introduced in section 19. By rigidly mapping artificial agent capabilities to corresponding grammatical cases—
e.g., treating an API as strictly INS, passive data strictly as ACC, and system context rigidly as LOC—the cognitive case
diagram enforces computational boundary constraints that formally constrain prompt injection attacks through explicit
role typing (section 20).
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Figure 2: Fluid-S alignment is a continuous mapping, not a binary switch. The context-dependent functor 𝐹𝜃∶𝒰→ℒ
is rendered as a 2D decision surface with volitional control 𝜃∈[0, 1] on the x-axis and proto-agentivity [Dowty, 1991]
on the y-axis. Color intensity represents 𝑃(ERG ∣𝜃, agentivity), computed via a logistic boundary. Nakh-Daghestanian
verb exemplars (Bats language) are overlaid at their typologically attested coordinates: low-volition actions (sneeze,
accidental fall) cluster in the ABS region; high-volition actions (jump, fight) occupy the ERG region. The dashed curve
marks the functor decision boundary where 𝐹𝜃(𝑆) transitions from ABS to ERG. Generated programmatically from
src.visualization.fluid_s_plots.plot_fluid_s_volition_landscape().
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5
Case Categories: Roles as Objects, Relations as Morphisms, Alignment
as Functors
Where we are in the argument. section 4 surveyed the cross-linguistic input data (five traditions, five alignment types).
This chapter converts that data into the first formal object of the framework: a category whose objects are case roles,
whose morphisms are grammatical relations, and whose alignment typologies are structure-preserving functors between
categories — the layer on which every subsequent pillar (syntax, semantics, enrichment, cognitive inference) is built.
5.1
Eight Case Roles as Objects
We define a case category 𝒞as a small category where:
• Objects are case roles (NOM, ACC, GEN, DAT, INS, LOC, ABL, VOC)
• Morphisms are grammatical relations between roles (e.g., “transitive action”: NOM →ACC)
• Identity morphisms represent the reflexive relation of each case role to itself
• Composition models the transitivity of grammatical dependencies
This formalization is implemented in our CaseCategory class, which uses set-based object tracking and list-based morphism
storage as the underlying representation. Each object carries its role enum and optional morphosyntactic features; each
morphism carries a relation label and an enriched weight 𝑤∈[0, 1]. Figure 3 shows the full eight-case standard category.
5.2
Accusative vs. Ergative as Structurally Non-Isomorphic Functors
An alignment functor 𝐹∶𝒰→ℒformally maps a universal case category 𝒰to a language-specific category ℒby
systematically collapsing objects that the target language treats as equivalent. For example, in an accusative language, the
functor forcibly merges S and A into a single NOM role while preserving P as a distinct ACC role: 𝐹(S) = 𝐹(A) = NOM,
𝐹(P) = ACC.
Mathematically, this functor guarantees three properties:
• Surjective on objects: Every target case role is the explicit image of a universal role.
• Structure-preserving: Grammatical relations in 𝒰strictly map to relations in ℒ.
• Diagnostic kernels: The kernel of 𝐹(the set of objects mapping to the identical target) uniquely identifies the
language’s alignment typology.
Thus, the alignment functor formalizes linguistic neutralization with mathematical precision: two semantically distinct
roles receive identical morphological treatment precisely because the functor collapses them into a single object in the
target category.
Explicit functor construction.
Let 𝒰be the universal three-role category with objects {𝑆, 𝐴, 𝑃} and morphisms
𝑓∶𝐴→𝑃(transitive action), 𝑔∶𝑆→𝑆(intransitive). The accusative functor 𝐹acc ∶𝒰→ℒacc and ergative functor
𝐹erg ∶𝒰→ℒerg act on objects as:
𝐹acc(𝑆) = 𝐹acc(𝐴) = NOM,
𝐹acc(𝑃) = ACC
(1)
𝐹erg(𝑆) = 𝐹erg(𝑃) = ABS,
𝐹erg(𝐴) = ERG
(2)
On morphisms, each functor strictly preserves the transitive relation: 𝐹acc(𝑓) = 𝑓′ ∶NOM →ACC and 𝐹erg(𝑓) =
𝑓″ ∶ERG →ABS.
Crucially, the kernel of 𝐹acc—the exact set {(𝑋, 𝑌) ∣𝐹acc(𝑋) = 𝐹acc(𝑌)}—resolves to {(𝑆, 𝐴)},
formally encoding the syntactic identification of the intransitive subject with the transitive agent. Conversely, the kernel
of 𝐹erg resolves to {(𝑆, 𝑃)}. This topological kernel structure successfully supplies a highly compact, computable algebraic
fingerprint for every known alignment type.
Figure 4 shows three alignment systems rendered from our CaseCategory implementation.
5.3
Graded Proto-Roles as [0, 1]-Weighted Morphisms
Following Dowty [1991], we equip morphisms with weights in [0, 1] that encode the degree of proto-role satisfaction. A
morphism 𝑓∶NOM →ACC with weight 𝑤= 0.9 indicates a strong transitive action (clear agent acting on clear patient),
while 𝑤= 0.4 might indicate an experiencer construction (“The child fears the dark”) where the nominative argument
only weakly satisfies Proto-Agent entailments.
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Figure 3: Eight case roles form a directed graph with weighted grammatical morphisms. The standard linguistic case
category 𝒞with objects NOM, ACC, GEN, DAT, INS, LOC, ABL, VOC and directed morphisms encoding grammatical
relations. Edge labels identify relation types (acts_on: NOM→ACC, possesses: GEN→NOM, received_by: ACC→DAT,
located_at: NOM→LOC); edge weights 𝑤∈[0, 1] reflect proto-role satisfaction per Dowty’s [1991] decomposition. The
enriched structure over ([0, 1], ⋅, 1) ensures multiplicative weight attenuation under composition (Equation 3). Generated
programmatically from the CaseCategory class.
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Figure 4: Functor kernels uniquely fingerprint each alignment typology. Three alignment systems realized as functors
from 𝒰= {𝑆, 𝐴, 𝑃}: Nominative–Accusative merges {𝑆, 𝐴} →NOM (kernel {(𝑆, 𝐴)}, Equation 1); Ergative–
Absolutive merges {𝑆, 𝑃} →ABS (kernel {(𝑆, 𝑃)}, Equation 2); Tripartite is injective (kernel ∅). Color-coded nodes
reveal neutralization patterns: shared colors indicate functor identification of roles. Generated programmatically from the
CaseCategory implementation.
Slavic quirky case as overt evidence for sub-unit morphism weights. Russian and Serbian/BCS supply a sharper,
morphologically overt witness for 𝑤< 1: a class of verbs that systematically refuses the canonical NOM →ACC arrow and
instead routes its object through GEN, DAT, or INS. In Russian, boyat’sja “to fear” governs the genitive (boyus’ sobaki “I
fear the dog-GEN”, not *sobaku-ACC); pomogat’ “to help” governs the dative (pomogayu drugu “I help the friend-DAT”);
upravlyat’ “to manage / steer” governs the instrumental (upravlyaet mašinoj “drives the car-INS”). Serbian/BCS shows
the same pattern: čestitati “to congratulate” assigns DAT (čestitam prijatelju “I congratulate the friend-DAT”); bojati se
“to fear” assigns GEN (bojim se psa “I fear the dog-GEN”). Each such verb supplies an enriched morphism whose target is
not ACC, equivalently a NOM →ACC arrow whose Dowtian weight has been reduced to near zero — and the reduction
is visible in the suffix on the noun, not merely hypothesised from semantics. These quirky-case lexemes are the cleanest
empirical anchor for the graded [0, 1]-enrichment we develop in section 11.
Composition of enriched morphisms multiplies weights:
𝑤(𝑔∘𝑓) = 𝑤(𝑔) ⋅𝑤(𝑓)
(3)
This multiplicative composition reflects the intuition that grammatical dependencies attenuate as they chain through
intermediate roles. Figure 5 illustrates the categorical composition of two morphisms through an intermediate case role.
The resulting structure is a category enriched over ([0, 1], ⋅, 1)—a connection we develop fully in section 11.
5.4
Alignment Shifts as Natural Transformations:
Functor Commutativity Encodes
Grammar Agreement
Having established that alignment systems are functors 𝐹, 𝐺∶𝒰→ℒfrom a universal case category to language-specific
categories, a natural question arises: how do different alignment systems relate to each other? The categorical answer
is a natural transformation 𝛼∶𝐹⇒𝐺—a systematic family of morphisms 𝛼𝐴∶𝐹(𝐴) →𝐺(𝐴) for each case role 𝐴,
satisfying the naturality condition:
𝐺(𝑓) ∘𝛼𝐴= 𝛼𝐵∘𝐹(𝑓)
for every morphism 𝑓∶𝐴→𝐵
(4)
This naturality constraint is the formal expression of grammatical coherence: transforming one alignment into another and
then applying a grammatical relation yields the same result as first applying the relation and then transforming—ensuring
that alignment comparison respects the relational fabric of the grammar.
Worked example. Consider the accusative functor 𝐹(which maps S and A to NOM, P to ACC) alongside the tripartite
functor 𝐺(which maps explicitly to S →ABS, A →ERG, P →ACC). The identity natural transformation id𝐹∶
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Figure 5:
Morphism composition attenuates weights multiplicatively through intermediate case roles.
Morphism
𝑓∶NOM →ACC (acts_on) and 𝑔∶ACC →DAT (received_by) compose to ℎ= 𝑔∘𝑓∶NOM →DAT per Equation 3;
in the enriched category weights multiply (e.g., 𝑤𝑓= 0.9, 𝑤𝑔= 0.7 ⇒𝑤(𝑔∘𝑓) = 0.63). The commutative triangle
encodes that DAT assignment factors through ACC—the multiplicative attenuation reflects the typological observation
that subject–recipient relations are weaker than the constituent subject–object and object–recipient links. Generated
programmatically from the CaseCategory class.
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𝐹⇒𝐹provides components (id𝐹)𝐴= id𝐹(𝐴) over every role 𝐴, trivially satisfying the naturality condition. We construct
the vertical composition 𝛽∘𝛼of two natural transformations 𝛼∶𝐹⇒𝐺and 𝛽∶𝐺⇒𝐻purely componentwise:
(𝛽∘𝛼)𝐴= 𝛽𝐴∘𝛼𝐴.
Our NaturalTransformation class implements these operations, with ComponentMorphism objects encoding each 𝛼𝐴, and
compose_transformations() implementing vertical composition. The IdentityNaturalTransformation constructor automat-
ically generates identity components for every object in an AlignmentFunctor’s domain. This machinery provides the formal
infrastructure for comparing alignment types not merely by listing their neutralization patterns but by characterizing the
structural mappings between them—e.g., the natural transformation from accusative to tripartite alignment is injective
(no two roles merge in the target), while the transformation from tripartite to ergative is non-injective (S and P merge
into ABS).
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6
Categorial Grammar: Syntax as Algebraic Composition and Proof
Where we are in the argument. section 5 built the case category — a static snapshot of licensed role-to-role morphisms.
This chapter equips the framework with a compositional syntax on top of that static layer: Lambek pregroup types let
each word carry its own combinatory potential, and cup/cap contractions reduce a parallel tensor of word-boxes to the
single sentence type 𝑠, so a sentence becomes a proof of well-typedness rendered graphically as a string diagram.
6.1
Each Word Is Its Own Grammar Rule
Traditional generative grammar computes sentence structure using top-down phrase-structure rules that recursively com-
bine constituents. Categorial grammar inverts this perspective: rather than specifying abstract construction rules, it
assigns each lexical item a dedicated type that encodes its combinatory potential. For example, a transitive verb like
“chases” is not loosely defined as “a word requiring a subject and object,” but specifically assigned the algebraic type
(𝑛\𝑠)/𝑛—an active function that categorically demands a noun to its right (the object) and a noun to its left (the subject)
to yield a complete sentence.
6.2
Lambek’s Residuation Law: Consuming and Producing Types in Linear Order
Lambek [1958] laid the algebraic foundations by introducing a residuated structure on syntactic types. He defined two bi-
nary operations—the left residual \ and the right residual /—derived from the multiplicative connective ⊗(concatenation).
The fundamental axiom governing this system is the residuation law:
𝐴⊗𝐵≤𝐶
⟺
𝐴≤𝐶/𝐵
⟺
𝐵≤𝐴\𝐶
(5)
This law brilliantly captures the fundamental duality of syntax: a verb of type (𝑛\𝑠)/𝑛actively consumes available
noun arguments to produce a well-formed sentence. The resulting Lambek calculus operates as a non-commutative
intuitionistic linear logic—non-commutative because strict word order dictates meaning, and linear because the derivation
logically consumes each lexical resource exactly once.
6.2.1
Pregroups Unlock Compact Closure: The Bridge to DisCoCat String Diagrams
Lambek [2004] later simplified the framework to pregroup grammars, where each type 𝑎has a left adjoint 𝑎𝑙and a
right adjoint 𝑎𝑟satisfying:
𝑎𝑙⋅𝑎≤1 ≤𝑎⋅𝑎𝑙
𝑎⋅𝑎𝑟≤1 ≤𝑎𝑟⋅𝑎
(6)
Coecke, Sadrzadeh, and Clark revolutionized this formal structuralism in 2010 with the DisCoCat (Distributional Com-
positional Categorical) model [2010]. DisCoCat mathematically unifies categorial grammar with distributional semantics
by defining strong monoidal functors mapping pregroup grammars directly into vector spaces. As Duneau emphasizes,
this approach “allows the meaning of a sentence to be computed as a function of both the distributional meaning of the
words involved, as well as its grammatical form” [2021]. By proving that pregroups and finite-dimensional vector spaces
both behave as rigid monoidal categories, DisCoCat allows us to compute whole-sentence vector meanings linearly from
constituent word vectors.
In this system, grammaticality reduces to algebraic verification: checking whether a sequence of types maps to the
sentence type 𝑠via contraction (𝑎𝑙⋅𝑎→1) and expansion (1 →𝑎⋅𝑎𝑙) operations.
This reformulation unlocks the
DisCoCat framework (section 8) because pregroups function as compact closed categories, natively supporting a
computationally sound diagrammatic calculus. When a derivation is ill-typed, contractions fail to close: there is no valid
reduction to 𝑠. section 20 develops the security reading of that failure mode—typed interaction boundaries and prompt
injection as illicit role reassignment.
6.3
Cups, Caps, and Joyal–Street: How Wires Prove Grammaticality
The pivotal connection between categorial grammar and visualization comes from Joyal and Street [1991], who proved
that morphisms in monoidal categories can be faithfully represented as string diagrams—planar graphs where:
• Wires represent types (objects of the category)
• Boxes represent words or operations (morphisms)
• Vertical composition represents sequential application
• Horizontal juxtaposition represents the tensor product (concatenation)
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• Cups and caps represent the contraction/expansion of a pregroup
Figure 7 surveys three syntactic structures side by side (progression from intransitive to passive); Figure 6 shows the
same transitive sentence rendered by our native matplotlib pipeline—a direct computational output confirming word-box
layout, wire routing, and cup-contraction geometry independently of DisCoPy; and Figure 8 demonstrates that type-logical
structure remains invariant across diverse surface word orders (SVO, free, SOV).
Figure 6: Native matplotlib rendering cross-validates DisCoPy pregroup geometry. String diagram for “Alice chases Bob”
generated via src.visualization.string_diagrams.render_discocat_sentence() without the DisCoPy library. Three Word
boxes carry pregroup types: 𝑛(Alice, blue = NOM), 𝑛𝑟⊗𝑠⊗𝑛𝑙(chases, neutral), and 𝑛(Bob, red = ACC). Dashed
arcs trace cup connections from noun wires to the verb’s adjoint slots, rendering 𝜀∶𝑛𝑟⊗𝑛→1 and 𝜀∶𝑛𝑙⊗𝑛→1 as
paired ligatures. This confirms that our visualization pipeline correctly reconstructs pregroup string-diagram geometry
using only case-role metadata from src.case_systems.case_category, cross-validating the DisCoPy canonical output in
Figure 9.
The Latin point in Figure 8 generalises directly to highly inflected modern Slavic. Russian Sobaka kusaet čeloveka “the
dog bites the man” and its scrambled counterpart Čeloveka kusaet sobaka differ in linear order but share the reduction
𝑛NOM ⊗(𝑛𝑟⊗𝑠⊗𝑛𝑙) ⊗𝑛ACC →𝑠, with the case suffixes (-a NOM.SG.F vs -a ACC.SG.M.ANIM after stem hardening)
supplying the role information that English encodes positionally. Serbian/BCS exhibits the same robustness: Pas ujeda
čoveka / Čoveka ujeda pas / Pas čoveka ujeda (and three further permutations) all type-check to 𝑠because the morphological
case marking on the noun wires uniquely identifies which is NOM and which is ACC. In Slavic languages the type calculus
is therefore not just consistent with the surface order — the surface order is, to a first approximation, immaterial, with
case morphology carrying the entire wiring diagram explicitly.
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Figure 7: Valency determines diagram topology: Cup count tracks verb argument structure. A 1×3 DisCoPy pregroup
grammar visualization of increasing argument complexity.
Panel (a) intransitive “Bob runs” (NOM subject, 1 Cup
contraction, type 𝑛⊗(𝑛𝑟⊗𝑠) →𝑠); Panel (b) transitive “Alice chases Bob” (NOM + ACC, 2 Cup contractions, type
𝑛⊗(𝑛𝑟⊗𝑠⊗𝑛𝑙) ⊗𝑛→𝑠); Panel (c) passive “Bob is chased by Alice” (patient promoted to NOM, former agent demoted
to an oblique by-phrase; full transitive-shape verb type 𝑛𝑟⊗𝑠⊗𝑛𝑙with 2 Cup contractions — which noun enters each cup
is reversed relative to panel (b)). The role reassignment is visible as a topological swap of which noun lands in the left
(subject) cup and which in the right (oblique) cup; cup count itself encodes argument-slot count, not syntactic prominence.
Complexity metric 𝜅(𝐷) per Equation 12.
Figure 8: Type-logical structure is invariant under surface word-order permutation.
Pregroup derivations for “SUBJ
chases OBJ” across three typologically diverse languages: English (SVO), Latin (inflected, free order), and Japanese
(agglutinative, SOV). Each renders an identical type reduction 𝑛⊗(𝑛𝑟⊗𝑠⊗𝑛𝑙) ⊗𝑛→𝑠, confirming the central claim of
categorial grammar: syntactic universals reside in the type algebra, not in linear word order.
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This result—the soundness and completeness of string-diagrammatic reasoning—formally guarantees that any topological
conclusion drawn visually from the diagram is algebraically valid: the diagram is not a heuristic but a rigorous proof
instrument.
This profound visual transparency instantiates exactly the “free ride” phenomenon identified by Shimojima [1996]: a
single diagram simultaneously exposes the syntactic derivation, the deep argument structure, and the compositional flow
of semantic meaning—entirely eliminating the need for sequential, explicit inference steps.
Concrete derivation in DisCoPy. The type reduction for “Alice chases Bob” is computationally verified using the
DisCoPy library’s discopy.rigid module:
from discopy.rigid import Ty, Box, Cup, Id
n, s = Ty('n'), Ty('s')
alice = Box('Alice', Ty(), n)
chases = Box('chases', Ty(), n.r @ s @ n.l)
bob
= Box('Bob',
Ty(), n)
words = alice @ chases @ bob
# n ⊗(n.r ⊗s ⊗n.l) ⊗n
cups
= Cup(n, n.r) @ Id(s) @ Cup(n.l, n)
diagram = words >> cups
# reduces to s
assert diagram.cod == s
# type-checks: sentence type
The two Cup operations successfully contract 𝑛with 𝑛𝑟(resolving the subject–verb link) and 𝑛𝑙with 𝑛(resolving the
verb–object link). This topological reduction collapses the five-wire tensor product into a single, valid sentence wire 𝑠.
Consequently, the assertion diagram.cod == s constitutes a direct, machine-processable proof of grammaticality.
From pregroup types to graded types. The standard pregroup derivation produces a rigid, binary grammaticality
judgment: the final codomain of the fully contracted tensor product either matches the sentence type 𝑠(grammatical)
or fails (ill-formed). However, we can grade this binary verdict by replacing the underlying Boolean algebra {0, 1} with
the continuous unit interval [0, 1]. This substitution yields a graded type theory where type judgments carry continuous
confidence weights rather than truth values.
Asudeh and Giorgolo [2020] developed this approach using a monadic
semantics that wraps base types inside a computational effect tracking epistemic uncertainty—an idea whose categorical
content is precisely a change of enrichment base.
Mapped into our framework, this operation corresponds to the [0, 1]-enrichment detailed in section 11: the enriched scalar
weight on any morphism 𝑓∶𝐴→𝐵quantifies the confidence that the categorical case assignment 𝐴→𝐵remains well-
typed. From there, categorical magnitude aggregates these sparse, local confidence scores into a global complexity measure
evaluating the entire syntactic derivation. The progression from rigid pregroup strings, through graded types, into fully
enriched case categories operates mathematically as a cascade of base-change functors (Bool ↪[0, 1] ↪R≥0)—where
each successive level grants greater representational nuance while demanding higher computational complexity.
For readers encountering pregroup reduction for the first time, Figure 10 unpacks the same construction pedagogically:
the four panels walk from the raw word types (panel 1), through the parallel tensor that juxtaposes the boxes (panel 2),
through the application of the two cup contractions 𝜀𝑛that cancel each argument wire against its adjoint (panel 3), to
the normal form in which only the sentence wire 𝑠survives (panel 4). Each panel is coloured by case role (NOM = blue,
ACC = red, verb output in indigo) so the identity of each wire is unambiguous.
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Figure 9: Direct DisCoPy machine output confirms pregroup type reduction to sentence wire 𝑠. Nine Word boxes carry
complex noun-phrase types (e.g., 𝑛⊗𝑛𝑙for adjectives mapping an adjacent noun to a modified noun) and core verb types
(𝑛𝑟⊗𝑠⊗𝑛𝑙for the transitive “intercepts”). Eight Cup contractions sequentially resolve the argument scopes from the
deepest noun phrases outward, ultimately canceling all local adjoint pairs and reducing the extended initial tensor product
to the single sentence wire 𝑠. Unlike the schematic progression in Figure 7, this is the direct computational output of a multi-
word discourse component, strictly confirming diagram.cod == s. By the Curry–Howard correspondence, this diagram
simultaneously constitutes a syntactic derivation, a proof of well-typedness, and (under the meaning functor of section 8)
a compositional semantic computation. Our extension decorates 𝑛-typed wires with functorial states 𝑆∶Ent →Case,
tracking role assignments across discourse boundaries via DisCoCirc entity wires.
Figure 10:
Pregroup reduction unpacked in four panels.
Panel 1:
raw pregroup types for Alice (𝑛), chases
(𝑛𝑟⊗𝑠⊗𝑛𝑙), Bob (𝑛).
Panel 2:
parallel composition ⊗exposes all five dangling wires.
Panel 3:
the two
cups 𝜀𝑛cancel Alice’s 𝑛against the verb’s 𝑛𝑟and the verb’s 𝑛𝑙against Bob’s 𝑛, leaving only the central 𝑠wire.
Panel 4:
normal form — the entire sentence reduces to a single 𝑠-typed wire.
Generated programmatically from
src.visualization.category_unpacking.render_pregroup_reduction_unpacking().
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7
Case Subscripts, Passivization, and the Curry–Howard Proof
Where we are in the argument. section 6 installed pregroup syntax — cups, caps, and compact closure — as the
algebraic backbone of composition. This chapter decorates that backbone with case information: every noun wire picks
up a case subscript (𝑛NOM, 𝑛ACC, …) so grammatical role becomes a first-class typed structure on wires, and passivization
becomes a wire-level swap together with case relabeling (Equation 7/Equation 8) rather than an exception to the type
system — the Curry–Howard image is a proof rewrite of that rearrangement.
Case-marked noun phrases receive compound types that encode both their grammatical role and their combinatory po-
tential. For example, in a nominative–accusative language, the type assignment proceeds as follows (Table 5):
Table 5: Case-marked pregroup type assignments for a standard nominative-accusative transitive clause.
Expression
Type
Gloss
“Alice” (subject)
𝑛NOM
Noun, nominative-marked
“the ball” (object)
𝑛ACC
Noun, accusative-marked
“kicks”
(𝑛NOM\𝑠)/𝑛ACC
Seeks ACC right, NOM left
“to Bob” (recipient)
(𝑠\𝑠)/𝑛DAT
Modifies sentence via DAT argument
The specific subscripts structurally refine the base noun type 𝑛with mandatory continuous case features, mathematically
guaranteeing that the transitive verb “kicks” exclusively selects a nominative-marked subject and an accusative-marked
object. Any feature mismatch automatically blocks the algebraic derivation, accurately modeling native ungrammaticality.
Alignment and natural transformations. Cross-linguistic alignment is modeled functorially in section 5: alignment
types are structure-preserving functors from a universal role category to a language-specific case category, compared
via natural transformations when one asks how two alignments cohere on the same underlying argument structure. At
the level of case-marked pregroup types (this section), that functorial picture suggests treating agreement checks as
naturality-style coherence constraints: the morphosyntactic alignment chosen for arguments must remain compatible with
the verb’s combinatorial requirements so that reductions still factor through the intended case-typed slots. We do not
claim a full adjunction between identity and alignment functors in the pregroup category here; the precise categorical
status of alignment is anchored in section 5, while the typed reductions below supply the concrete syntax.
7.1
Syntactic Derivation as Proof, Case Assignment as Type Inference
A deep structural parallel underlies the Lambek calculus: derivations of syntactic types correspond to proofs in intuition-
istic logic (via the Curry–Howard isomorphism), and therefore to programs in a typed lambda calculus, as summarized in
Table 6.
Table 6: The Curry–Howard isomorphism connecting syntactic, logical, and computational domains.
Syntactic Domain
Logical Domain
Computational Domain
Types (𝐴/𝐵, 𝐴\𝐵)
Propositions
Types
Derivations
Proofs
Programs (𝜆-terms)
Cut elimination
Proof normalization
𝛽-reduction
Commutativity of cuts
Confluence of rewriting
Church–Rosser property
This foundational correspondence dictates two critical consequences for our cognitive framework:
1. Semantic compositionality strictly follows syntactic well-formedness: A successfully typed derivation au-
tomatically guarantees a well-formed geometric meaning representation (specifically, the exact 𝜆-term isolated and
extracted directly from the formal proof).
2. Case assignment reduces to type inference: Dynamically determining the correct case for a noun phrase
in context is computationally equivalent to inferring the type of an unknown variable in a 𝜆-expression—thereby
grounding case assignment in the well-understood algorithmic landscape of type inference (Hindley-Milner and its
extensions).
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7.2
Song’s Monadic Root Syntax: Sublexical Decomposition via Embedded Monads
Song [2022a; 2022b] significantly extends this categorial framework by deploying a novel monadic semantics for root
syntax. He successfully models the sublexical decomposition of complex verb roots by embedding specialized monads
directly into the base syntactic category. This computational approach captures the deep intuition that a seemingly simple
verb like “break” harbors layered internal structure—specifically, an active causative layer tightly coupled to a passive
result-state layer—which dynamically dictates subsequent case assignment. The formal monad elegantly encapsulates
this tangled lexical complexity within one streamlined categorical construction, empowering the type-logical grammar to
process lexical features and syntactic composition simultaneously and uniformly.
This monadic architecture seamlessly interfaces with modern graded type theory. For instance, Asudeh and Giorgolo [2020]
previously developed a monadic semantics tracking evidentiality, deploying graded computational effects to quantify the
shifting epistemic real-world status of various propositions. Our framework extends this exact pattern into the domain
of morphological case, where the continuous “grade” attached to any specific morphism formally encodes the statistical
strength and validity of that local semantic role assignment.
7.3
Passivization Is a Swap: Voice Alternation as Topological Wire Crossing
Within categorical linguistics, passivization emerges as a uniquely revealing topological operation. Traditionally de-
scribed as a syntactic rule that promotes the patient to subject position while (optionally) demoting the agent to an
oblique role, passivization in our framework ceases to be an ad-hoc transformation; instead, it combines (i) a Swap
morphism 𝜎𝐴,𝐵∶𝐴⊗𝐵→𝐵⊗𝐴that crosses the noun wires feeding into the verb’s pregroup derivation—so which noun
meets the verb’s left vs. right adjoint slots reverses relative to the active diagram—and (ii) updated case features on
those wires: the promoted subject is 𝑛NOM, not a carried-over 𝑛ACC.
In a pregroup grammar, the active transitive “Alice chases Bob” has the type reduction:
𝑛NOM ⋅(𝑛𝑟⋅𝑠⋅𝑛𝑙) ⋅𝑛ACC →𝑠
(7)
For “Bob is chased by Alice,” Bob is the grammatical subject (𝑛NOM) and Alice appears in the oblique by-phrase (𝑛OBL),
with surface order subject–verb–oblique:
𝑛NOM ⋅(𝑛𝑟⋅𝑠⋅𝑛𝑙) ⋅𝑛OBL →𝑠
(8)
This is not the naive swap of subscripts in (7) (which would incorrectly leave the promoted patient typed as accusative).
The DisCoPy Swap primitive makes the wire crossing explicit; the case-marked types in (8) match the figure and standard
promotion/demotion. The generated passive diagram diverges from its active counterpart through that crossing together
with the reassignment of which noun occupies each cup. This transforms abstract syntactic voice alternation into a highly
visible, instantly readable topological feature embedded directly within the string diagram.
This diagrammatic transparency exemplifies Shimojima’s [1996] free-ride inference capability. By inspecting the visual
topology, a cognitive agent immediately verifies that passivization preserves the verb’s inherent argument structure while
rearranging its surface realization—a structural fact that typically requires chaining inference steps to algebraically estab-
lish within standard linear notation (Figure 11).
Relative to Equation 12, the passive “Bob is chased by Alice” diagram in Figure 11 has the same cup count as the transitive
active and therefore nearly identical 𝜅(𝐷); the syntactic reassignment is visible in which cup each noun enters, not in the
total count of contractions.
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Figure 11: Passivization as role reassignment in the DisCoPy pregroup diagram for “Bob is chased by Alice” with type
𝑛NOM ⊗(𝑛𝑟⊗𝑠⊗𝑛𝑙) ⊗𝑛OBL →𝑠. Bob’s 𝑛wire contracts into the verb’s left adjoint 𝑛𝑟via the left Cup (𝑛–𝑛𝑟) —
promoting the semantic patient to grammatical subject — while Alice’s 𝑛wire contracts into the right adjoint 𝑛𝑙via the
right Cup (𝑛𝑙–𝑛), now demoted to the oblique by-phrase. Compare with Figure 12 (active “Alice chases Bob”), where
Alice occupies the left subject slot and Bob the right object slot: the reassignment of which noun lands in each Cup is
what makes voice alternation a topological swap 𝜎𝑛,𝑛∶𝑛⊗𝑛→𝑛⊗𝑛rather than a lexical substitution. Cup count is
unchanged (two, matching the transitive active), so 𝜅(𝐷) tracks argument slots rather than surface case marking; per
Equation 12 only the box count distinguishes the two diagrams (the passive inserts the “is chased by” box in place of the
active “chases” box).
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8
Categorical Distributional Semantics (DisCoCat):
Composing Vector
Spaces via Type Derivations
Where we are in the argument. section 6–section 7 gave the framework a syntactic apparatus: pregroup derivations
with case-typed wires. This chapter supplies the matching semantic one — the Coecke–Sadrzadeh–Clark meaning functor
𝐹∶Preg →FinHilb that sends each word to a tensor in a vector space and each pregroup contraction to an inner-product
composition, so the same syntactic diagram simultaneously proves well-typedness and computes the composed meaning.
8.1
The Formalism–Distribution Impasse and Why Montague Cannot Meet Harris
The study of linguistic meaning has historically been fractured between two traditions offering complementary strengths
and weaknesses:
1. Formal semantics, following Montague [1973], strictly models meaning compositionally: the algebraic meaning of
any complex expression functions as a direct product of its parts and their syntactic combination. This tradition
excels at capturing rigid logical structure (quantification, negation, modality) but severely struggles with fluid lexical
meaning, typically treating content words as opaque, unanalyzed primitives.
2. Distributional semantics models meaning empirically: a word’s meaning organically emerges from its distribution
across contexts, typically encoded computationally as a dense vector in high-dimensional space. This distributional
hypothesis—famously summarized as “You shall know a word by the company it keeps” [Firth, 1957]—traces directly
to J.R. Firth and Zellig Harris [1954], whose algebraic analysis proved that linguistic elements occurring in similar
environments inherently share semantic properties. While this tradition beautifully captures graded similarity and
geometric analogy, it lacks rigorous compositional structure. Under pure distribution, “dog bites man” and “man
bites dog” collapse into identical vector representations.
This theoretical tension represents one of the deepest impasses in modern cognitive science. Turney and Pantel’s [2010]
survey of vector space models proved that distributional methods capture fine-grained semantic distinctions—synonymy,
antonymy, hypernymy, analogy—but isolate these victories to the word level. Baroni and Lenci [2010] built a tensor-
based Distributional Memory framework partially addressing this compositionality by structuring co-occurrence data into
a three-way tensor over (word, relation, word) triples, yet this approach lacked a principled type-logical backbone. Lenci
[2018] surveyed this landscape and identified the central open problem: how to fuse the algebraic compositionality of
formal semantics with the empirical grounding of distributional vector models.
The DisCoCat (Distributional Compositional Categorical) framework [Coecke et al., 2010] resolves this long-standing
tension. It deploys category theory to compose distributional meanings directly according to syntactic structure, yielding
a semantic framework that is simultaneously compositional (via categorial grammar), distributional (via corpus-derived
vector spaces), and algebraically principled (via rigid monoidal category theory).
8.2
From Word2Vec to Transformer Attention: DisCoCat as the Algebraic Formalization
of Distributional Semantics
The distributional programme has undergone a dramatic computational intensification in the era of large language models
(LLMs). The trajectory from classical co-occurrence matrices through static word embeddings to contextual transformers
can be understood as a progressive enrichment of the distributional hypothesis itself:
1. Static embeddings:
Mikolov et al.’s [2013] Word2Vec (skip-gram and CBOW) demonstrated that targeted
prediction-based training on local context windows naturally generates word vectors exhibiting striking algebraic
regularity—yielding the famous “king −man + woman
𝑎𝑝𝑝𝑟𝑜𝑥queen” geometric analogy. Pennington et al.’s [2014] GloVe successfully incorporated global co-occurrence
statistics via log-bilinear regression, yielding dense vectors whose inner products cleanly approximate underlying
pointwise mutual information. Both models instantiate the distributional hypothesis in its classical Firthian form:
contextual use strictly determines meaning, permanently encoding it as geometric position within a learned vector
space.
2. Contextual embeddings: BERT [Devlin et al., 2019] and GPT [Radford et al., 2018] dramatically push be-
yond static type-level representations into dynamic token-level contextualized embeddings, where the identical word
dynamically receives entirely different coordinate vectors in varying contexts. In terms of the formal vs. distribu-
tional dichotomy, this jump represents a critical advance: contextualized embeddings successfully (though implicitly)
capture compositional structure by continuously conditioning word representations upon their full sentential envi-
ronment. This addresses polysemy and complex constructional effects that static embeddings routinely conflate.
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3. Transformer architecture: The transformer [Vaswani et al., 2017] implements distributional composition via
multi-head self-attention, where each head computes a weighted combination of input token representations. The
attention weights 𝛼𝑖𝑗= softmax(𝑄𝑖𝐾𝑇
𝑗/√𝑑𝑘) operate analogously to the enriched hom-values detailed in section 11:
they encode the degree of contextual relevance between tokens 𝑖and 𝑗within a tuned representational subspace—
yielding a graded, learned distributional mapping.
The mapping back to our categorical framework is direct: DisCoCat supplies the algebraic formalization of what
modern LLMs learn empirically. A transformer builds sentence representations by attending to syntactically and
semantically relevant tokens through learned weight matrices. DisCoCat achieves the same goal by composing word vectors
through type-logical derivations within a compact closed category. The central functor 𝐹∶Preg →FVect defining
DisCoCat is therefore the principled version of the composition that neural attention learns from data. Gavranović’s
[2024] categorical learning programme confirms this perspective rigorously, proving that neural attention heads operate
as parameterized optics—categorical constructions composing functorially, mirroring DisCoCat derivations.
For case theory specifically, the transformer analogy is illuminating: each attention head in a transformer can be understood
as learning a particular relational role—attending to subjects, objects, modifiers, or other grammatical functions. This is
precisely the role that case marking plays in natural language: structuring who-does-what-to-whom. The case-typed noun
spaces of our enriched DisCoCat model (𝑁NOM, 𝑁ACC, 𝑁DAT, …) correspond to the role-specific representational subspaces
that different attention heads learn to inhabit.
8.3
The Meaning Functor 𝐹∶Preg →FVect
8.3.1
Pregroups and Vector Spaces Share a Category
DisCoCat’s central observation is that pregroup grammars and vector spaces share a common abstract structure: both
are compact closed categories. This means there exists a meaning functor:
𝐹∶Preg →FVect
(9)
from the pregroup grammar category (where objects are types and morphisms are grammatical reductions) to the category
of finite-dimensional vector spaces (where objects are vector spaces and morphisms are linear maps).
Under this functor:
• Noun types 𝑛map to a vector space 𝑁(the noun space)
• Sentence types 𝑠map to a vector space 𝑆(the sentence space)
• A transitive verb of type 𝑛𝑟⋅𝑠⋅𝑛𝑙maps to a tensor in 𝑁⊗𝑆⊗𝑁
• Pregroup contractions (cups/caps) map to the standard inner product and its dual
8.3.2
Tensoring, Then Contracting: How “Alice Chases Bob” Becomes a Vector
The compositional meaning of a sentence is computed by tensoring the word meanings and then contracting the result
along the indices determined by the syntactic derivation. For “Alice chases Bob”:
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
Alice chases Bob = (⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
Alice ⊗⃡⃡⃡⃡⃡⃡⃡⃡⃡⃡⃡⃡⃡⃡⃡⃡⃡⃡⃡⃡
chases ⊗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
Bob) ∘(𝜀𝑁⊗1𝑆⊗𝜀𝑁)
(10)
where 𝜀𝑁∶𝑁⊗𝑁→ℝis the compact closure map (inner product).
This computation has a direct diagrammatic
representation as a string diagram—the same Joyal–Street [Joyal and Street, 1991] formalism that governs the syntax.
8.3.3
DisCoCat Resolves “Dog Bites Man” vs “Man Bites Dog”
Grefenstette and Sadrzadeh [2015] demonstrated that DisCoCat models can outperform purely distributional baselines
on disambiguation and sentence similarity tasks. The key advantage is that compositional structure resolves ambiguities
that bag-of-words models cannot: “dog bites man” and “man bites dog” receive different sentence vectors because the
syntactic structure assigns different roles to the nouns.
Attention as Cup contraction. The claim that DisCoCat provides the “algebraic formalization” of transformer-based
LLMs can be made precise. In a transformer’s self-attention mechanism, the query–key inner product softmax(𝑄𝐾⊤/
√
𝑑)𝑉
selects which word vectors interact—effectively implementing a soft version of the Cup contraction. The pregroup Cup
𝜀∶𝑛𝑟⊗𝑛→𝐼contracts two noun wires into a scalar; the attention inner product 𝑞𝑖⋅𝑘𝑗contracts two contextualized
vectors into an attention weight, which then mixes the value vectors. The difference is that the DisCoCat Cup is binary
(either the types match or they don’t) while the attention Cup is graded (producing a real-valued weight). This is precisely
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the [0, 1]-enrichment of section 11 applied to the type-reduction level: each attention head learns an enriched Cup that
contracts types with learned confidence weights rather than categorical yes/no type-matching. Multi-head attention then
corresponds to the tensor product of ℎindependent enriched contraction maps, each attending to a different substring
of the relational structure—analogous to how DisCoCat composes multiple Cup contractions for multi-argument verbs
(Figure 12).
Figure 12: Cup contractions compute sentence meaning by contracting word tensors along syntactically determined indices.
The DisCoCat meaning functor 𝐹∶Preg →FVect applied to “Alice chases Bob.” Left panel: pre-contraction tensor
product 𝑛⊗(𝑛𝑟⊗𝑠⊗𝑛𝑙) ⊗𝑛, showing three Word boxes with pregroup types. Right panel: fully contracted diagram
where two Cup contractions (𝜀∶𝑛𝑟⊗𝑛→1 and 𝜀∶𝑛𝑙⊗𝑛→1) reduce the five-wire product to sentence wire 𝑠. Under 𝐹,
noun types map to 𝑁, the sentence type to 𝑆, and the verb’s type to ⃡⃡⃡⃡⃡⃡⃡⃡⃡⃡⃡⃡⃡⃡⃡⃡⃡⃡⃡⃡
chases ∈𝑁⊗𝑆⊗𝑁. The Cup contractions become
inner products 𝜀𝑁∶𝑁⊗𝑁→ℝ, computing ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
Alice chases Bob ∈𝑆.
8.4
Case-Typed Noun Spaces and Alignment as Natural Transformation
The present contribution lies in showing how case marking enriches the DisCoCat framework with explicit role structure.
In a case-marked DisCoCat model:
1. Typed noun spaces: Instead of a single noun space 𝑁, we define case-specific spaces 𝑁NOM, 𝑁ACC, 𝑁DAT, …
Morphisms between these spaces encode case-changing operations (passivization, dative shift, etc.).
2. Case-constrained composition: The meaning functor 𝐹maps case-typed pregroup derivations to tensor contrac-
tions that respect case constraints. A verb seeking a nominative subject and accusative object contracts only with
vectors from the appropriate spaces.
3. Alignment as Natural Transformation: Cross-linguistic alignment differences correspond to different meaning
functors. However, we go further by modeling case alignment as natural transformations between functors. An
accusative-language transformation 𝜈acc ∶ℐ→𝐹acc identifies S and A arguments; an ergative transformation 𝜈erg
identifies S and P. (We write 𝜈here so 𝜂remains reserved for the compact-closure cap in pregroup diagrams.) This
categorification allows us to model “grammar” as the requirement that these transformations commute with the
DisCoCirc entity wires, ensuring role persistence across sentences.
The monotonic complexity–valency relationship (Equation 12 in section 9) is visible in the cup count of Figure 13; the
diagram also exemplifies Shimojima’s [1996] free-ride inference over argument structure and compositional flow.
The following sections extend this model to discourse and quantum computation.
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Figure 13: Ditransitive pregroup diagram for “Alice gives Bob a book”: four word boxes (Alice, gives, Bob, book) with
three Cup contractions reducing the five-argument tensor to sentence type 𝑠, exhibiting higher valency (𝜅(𝐷)) than a
transitive clause. Generated by render_discopy_ditransitive() in src/visualization/discopy_diagrams.py.
9
Compact Closure: Snake Equation, Valency, and Four Complexity Met-
rics
Where we are in the argument. section 6–section 8 made each sentence a string diagram that simultaneously expresses
syntax and semantics. This chapter zooms in on the axiom that makes the machine run — the compact-closure snake
equation (𝜀𝑛⊗1𝑛)∘(1𝑛⊗𝜂𝑛) = 1𝑛— and uses its consequences to define four graded complexity metrics (word count, cup
count, cap count, depth) that turn “how hard is this sentence to parse?” from a qualitative question into a computable
one, feeding directly into the enriched hom-values of section 11.
9.1
The Snake Equation Powers Every Pregroup Contraction
9.1.1
The Snake Equation (Zigzag Identity)
The essential algebraic engine driving DisCoCat is the strict compact closure of the governing pregroup category. For
every lexical type 𝑛, the corresponding adjunction maps—𝜂𝑛∶1 →𝑛⊗𝑛𝑟(the cap expansion) and 𝜀𝑛∶𝑛𝑟⊗𝑛→1 (the
cup contraction)—must mathematically satisfy the fundamental snake equation (also known as the topological zigzag
identity):
(𝜀𝑛⊗1𝑛) ∘(1𝑛⊗𝜂𝑛) = 1𝑛
(11)
Translated into intuitive string-diagrammatic terms, a cup geometrically composed with a cap bridging adjacent wires
immediately “straightens out” into a solid identity wire—a topological zigzag seamlessly canceling into a continuous
straight line. This foundational axiom is not merely a geometric curiosity: it physically functions as the engine powering
every valid pregroup type reduction. Every recorded grammatical contraction (e.g., a noun canceling against an active
verb argument slot) constitutes a direct instance of the cup map 𝜀, and every grammatical expansion instantiates the cap
map 𝜂. The continuous snake equation algebraically guarantees that these interwoven contractions and expansions remain
globally well-behaved—ensuring they can be freely inserted and removed without ever altering the final semantic meaning
of the derivation.
Cognitively, the snake equation provides an immediate visual proof of coherence. A cognitive agent inspecting a string
diagram can verify well-formedness by confirming that all zigzags cancel—a purely spatial operation requiring no sequential
algebraic computation, and thus instantiating Shimojima’s [1996] free-ride inference in its most direct form (Figure 14).
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Figure 14: The snake equation is the algebraic engine powering every pregroup type reduction. The compact closure axiom
(Equation 11) rendered by DisCoPy as a three-panel equality: left panel shows the right-adjoint zigzag (𝜀𝑛⊗1𝑛)∘(1𝑛⊗𝜂𝑛)
with the 𝑥𝑟intermediary; middle panel is the identity wire 1𝑛that the zigzag equals; right panel shows the mirror-
image left-adjoint zigzag using the 𝑥𝑙adjoint. Verified computationally via diagram.normal_form() == Id(Ty('x')). Each
pregroup contraction (noun canceling with verb argument) is an instance of 𝜀; the snake equation guarantees that cup-
cap pair insertions and removals leave derivations invariant (cf. section 6).
Generated by render_discopy_snake() in
src/visualization/discopy_diagrams.py.
Figure 15 unpacks the same axiom into three explicit panels: the left-hand zigzag carrying its 𝜂𝑛and 𝜀𝑛labels, the identity
wire 1𝑛it equals, and an algebraic recap of both compact-closure equations with cap and cup type signatures.
Figure 15: Snake equation unpacked. Panel 1: the zigzag (𝜀𝑛⊗1𝑛) ∘(1𝑛⊗𝜂𝑛) with 𝜂at the top (cap, 1 →𝑛⊗𝑛𝑟)
and 𝜀at the bottom (cup, 𝑛𝑟⊗𝑛→1). Panel 2: the identity wire 1𝑛it equals. Panel 3: axiom recap showing both
zigzag equations straighten to the identity, with explicit type signatures for 𝜂and 𝜀. Generated programmatically from
src.visualization.category_unpacking.render_snake_equation_unpacking().
9.1.2
Four Metrics Quantify Derivational Complexity
The algebraic properties of pregroup diagrams support quantitative analysis of derivational complexity. Our complex-
ity_metrics module implements four complementary measures using the DisCoPy library:
1. Box count: The number of lexical entries (Word boxes) in the diagram, corresponding to the sentence’s word count
from the type-logical perspective. A transitive sentence has 3 boxes (subject, verb, object); a ditransitive sentence
has 4 or more.
2. Cup/Cap count: The number of contraction and expansion operations. Cups (denoted 𝜀) count argument consump-
tion; caps (denoted 𝜂) count argument introduction. The cup count directly reflects verb valency: an intransitive
verb requires 1 cup, a transitive verb 2, and a ditransitive verb 3.
3. Normal form: A diagram is in normal form if no further simplifications (zigzag cancellations, box reordering)
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## Page 35

are possible. The compute_normal_form() operation computes this canonical representative of the diagram’s equiva-
lence class. Normal form preservation under algebraic manipulation provides a correctness check for compositional
operations.
4. Syntactic complexity score: A composite metric defined as:
complexity(𝐷) = 𝑤𝑤⋅|𝐷|words + 𝑤𝑐⋅|𝐷|cup + 𝑤𝑎⋅|𝐷|cap + 𝑤𝑑⋅depth(𝐷)
(12)
where |𝐷|words is the lexical box count (Cup/Cap plumbing excluded), |𝐷|cup and |𝐷|cap are contraction and expansion
counts, and depth(𝐷) is the longest input-to-output path measured in boxes. The weights 𝑤𝑤, 𝑤𝑐, 𝑤𝑎, 𝑤𝑑are keyword
arguments to syntactic_complexity_score(); the defaults 𝑤𝑤=1.0, 𝑤𝑐=0.5, 𝑤𝑎=0.25, 𝑤𝑑=0.1 encode the intuition that
lexical entries contribute most, contractions moderately, expansions least, and depth a small residual penalty. Passing
𝑤𝑤=𝑤𝑐=𝑤𝑎=𝑤𝑑=1.0 recovers an unweighted total-structure count.
Deeper diagrams encode more complex syntactic
derivations—a ditransitive sentence like “Alice gave Bob a book” (depth 7) is structurally more complex than a simple
intransitive “Alice runs” (depth 3).
The compare_diagrams() function applies these metrics across a collection of diagrams, producing tabular comparisons
suitable for cross-linguistic analysis. Figure 16 visualizes these metrics across sentence types of increasing valency, demon-
strating the monotonic relationship between argument structure and derivational complexity.
Figure 16: Verb valency dominates diagram complexity: categorical complexity score 𝜅(𝐷) (Equation 12) plotted for ten
sentence types ranging from intransitive to adjunct-heavy constructions, with natural-language exemplar sentences overlaid
at each bar. Generated by render_complexity_comparison() in src/visualization/complexity_plots.py (orchestrated by
scripts/generate_diagrams.py).
Key finding. Adverb and adjective modifiers increase box count but do not always add core argument Cups (modifier
types 𝑛⋅𝑛𝑙contribute one box and one cup each, but some adverbs add a box without an argument Cup), so 𝜅can grow
more slowly with adjunct stacking than with valency increases. Cup count |𝐷|cup remains a reliable proxy for verb valency.
In code, cup counts are obtained by iterating DisCoPy diagram boxes and testing for Cup instances (see compare_diagrams()
in the project).
These metrics connect naturally to the enriched framework of section 11: diagram depth serves as a syntactic complexity
proxy that can be incorporated into the enriched hom-value, providing a principled bridge between type-logical deriva-
tion cost and distributional semantic distance. Discourse-level persistence of entity wires (DisCoCirc) is developed in
section 10. For multi-agent security, when tracking networks isolate an adversarial identity wire (a prompt injection)
covertly attempting to merge with an ongoing command wire across communication boundaries, the circuit topology can
flag the type violation before execution (section 20).
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10
Beyond the Sentence:
State Wires Accumulate Semantic History
Across Discourse
Where we are in the argument. section 8–section 9 formalised individual sentences as compact-closed string diagrams
and their complexity as a four-metric summary. This chapter extends composition across sentence boundaries: de Felice–
Coecke DisCoCirc adds persistent entity wires that carry an entity’s case-role history across a discourse, so that a three-
sentence discourse traces Alice’s NOM →ACC →NOM trajectory and Bob’s ACC →NOM trajectory as first-class
diagrammatic data rather than coreference annotation bolted on after the fact.
10.1
DisCoCirc State Wires Resolve Coreference and Role Shifts
De Felice and Coecke [2020] address this limitation by formulating DisCoCirc (Distributional Compositional Circuits).
DisCoCirc extends the base categorical framework, empowering it to dynamically handle discourse-level semantic structure.
Specifically, DisCoCirc introduces continuous state wires that actively persist across isolated sentence boundaries, continu-
ously encoding the dynamically evolving states of distinct discourse entities (e.g., characters, objects, shifting topics). For
example, a multi-sentence sequence like “Alice chased Bob. He was terrified.” is explicitly represented as an integrated
geometric circuit where:
• Alice and Bob are wires that persist across both sentences.
• The pronoun “He” is resolved by actively connecting its topological wire directly to Bob’s wire.
• The shifting emotional state “terrified” seamlessly updates the cumulative state information carried exclusively by
Bob’s persistent wire.
De Felice et al. [2022] subsequently extended this approach, developing a full-fledged, multi-layered circuit model built
to systematically resolve ambiguity, tangled coreference, and overarching discourse coherence—all locked within the exact
same continuous categorical formalism.
A CCG-based pipeline for generating discourse circuits from syntactic parse
trees has demonstrated that DisCoCirc can scale to text, dynamically composing sentence-level diagrams along shared
entity wires via an iterative process of coreference resolution and wire merging [2021]. This pipeline approach has since
been extended to a comparative framework for evaluating compositional AI model architectures [2025], confirming that
the DisCoCirc wire-merging strategy generalizes across model families. Complementary work on DiscoSG (Discourse
Scene Graphs) extends this approach to multi-sentence image captions, parsing text into scene graphs that capture cross-
sentence coreference relations. Figure 17 illustrates a multi-sentence discourse diagram where entity wires persist across
sentence boundaries. For case theory, DisCoCirc is significant because it shows how case-marked argument structure
composes across discourse: the nominative subject of one sentence can become the accusative object of the next, and this
transformation is tracked as a morphism in the discourse category.
10.2
Alice’s Role Trajectory: NOM→ACC→NOM Across Three Sentences
The power of DisCoCirc for case theory becomes particularly vivid in multi-sentence discourses where the same entity
occupies different case roles across sentences. Consider the three-sentence discourse:
“Alice chases Bob. Bob fears Alice. She smiles.”
In this discourse, Alice undergoes a complete cycle of case role reversals:
1. Sentence 1: Alice is NOM (Proto-Agent, the one chasing) and Bob is ACC (Proto-Patient, the one chased).
2. Sentence 2: Bob is now NOM (the one fearing) and Alice is ACC (the one feared)—a role reversal where Alice
moves from agent to patient.
3. Sentence 3: “She” resolves anaphorically to Alice, who returns to NOM as the agent of smiling.
This NOM →ACC →NOM trajectory for Alice across three sentences is precisely the kind of dynamic case assignment
that static single-sentence analyses cannot capture. The categorical representation as a triple tensor product 𝑠⊗𝑠⊗𝑠
(Figure 19) encodes each sentence as an independent pregroup derivation while preserving the entity identity that links
them.
This role reversal essentially requires applying the topological Swap operation (introduced for passivization in
section 7) dynamically across the discourse boundary. In a full DisCoCirc implementation, Alice’s entity wire would carry
accumulated semantic state—the meaning of “She” in sentence 3 inherits the enriched state of an Alice who has first
chased and then been feared, not merely the bare lexical entry for “Alice.”
The trajectory is executable: the src.diagrams.string_diagram.Discourse class implements exactly this entity-wire-with-
role-history bookkeeping. The following snippet, taken verbatim from the public API exercised by the test suite, builds
the three-sentence Alice/Bob discourse and reads back the role history that drives Figure 19:
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Figure 17: DisCoCirc type structure encodes discourse coherence as a tensor product of sentence types. DisCoPy pregroup
grammar for “Alice chases Bob. Bob runs.” showing inter-sentential composition. Sentence 1 (left subdiagram): Alice (𝑛)
and Bob (𝑛) contract into “chases” (𝑛𝑟⊗𝑠⊗𝑛𝑙) via two Cups, yielding sentence type 𝑠. Sentence 2 (right subdiagram):
Bob (𝑛) contracts into “runs” (𝑛𝑟⊗𝑠) via one Cup, yielding 𝑠. The joint discourse type 𝑠⊗𝑠encodes inter-sentential
coherence as a tensor product, the computational primitive that DisCoCirc state wires exploit to propagate Bob’s semantic
state from ACC (sentence 1) to NOM (sentence 2). See Figure 18 for wire-colour confirmation of the role transition, and
Figure 19 for the three-sentence DisCoPy tensor layout (role reversal; anaphora is developed in prose and in the Discourse
API below, not as a separate pronoun box in the DisCoPy figure).
from src.diagrams.string_diagram import Sentence, Discourse
discourse = Discourse()
discourse.add_sentence(Sentence.transitive("Alice", "chases", "Bob"))
discourse.add_sentence(Sentence.transitive("Bob", "fears", "Alice"))
discourse.add_sentence(Sentence.intransitive("Alice", "smiles"))
# Persistent entity wires (DisCoCirc state-wire reading)
assert discourse.entities == {"Alice", "Bob"}
assert discourse.num_sentences == 3
# Role history per entity reproduces the NOM →ACC →NOM trajectory
assert [r.name for r in discourse.role_history["Alice"]] == ["NOM", "ACC", "NOM"]
assert [r.name for r in discourse.role_history["Bob"]]
== ["ACC", "NOM"]
# `role_reversal_entities` flags every entity whose case role
# changes across the discourse — the formal analogue of the
# trajectory pictured in \autoref{fig:three-sentence-discourse}.
assert set(discourse.role_reversal_entities()) == {"Alice", "Bob"}
Slavic discourse supplies a morphologically overt analogue of the persistent state wire: in Serbian/BCS, second-position
clitics (Wackernagel position) such as je AUX.3SG, mu DAT.3SG.M, ga ACC.3SG.M form a fixed-order cluster that
carries the case role of the discourse referent forward in time. Marko ga je video “Marko him.ACC AUX saw” and Dao
mu ga je “He gave him.DAT it.ACC AUX” make Bob’s wire-and-case-history visible in the surface string in a way that
English pronouns (which collapse case morphology onto a tiny three-form I/me/my paradigm) cannot. Russian, which
lacks the BCS clitic cluster, instead encodes the same persistence directly in the suffix on the noun or full pronoun (ego
ACC, emu DAT, im INS), so role reversals like Alice’s NOM→ACC→NOM trajectory in Alisa gonyaet Boba. Bob boitsja
Alisy. Ona ulybaetsja are reconstructible from the morphology alone, with no positional ambiguity. These are the cleanest
natural-language witnesses for the entity-wire-with-role-history bookkeeping that the Discourse API formalises.
The
point
of
including
the
snippet
in
prose
is
reproducibility,
not
novelty:
every
assertion
above
is
ex-
ercised
in
tests/test_diagrams_string_diagram.py,
so
a
reader
who
clones
the
repository
can
re-derive
the
discourse-level role trajectories with no additional setup beyond
uv
sync.
The same
Discourse object is what
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## Page 38

Figure 18: Wire colour provides independent visual confirmation of the ACC→NOM role transition that Figure 17
encodes algebraically.
Native matplotlib DisCoCirc diagram for the same two-sentence discourse, generated via
src.visualization.string_diagrams.render_discocirc_discourse() without the DisCoPy library. Vertical grey wires:
persistent entity wires for Alice and Bob, confirming that identity is tracked across the sentence boundary. Colour-coded
role discs: Bob’s wire is marked red (ACC) at sentence 1 (chased object) and blue (NOM) at sentence 2 (running agent).
The ACC→NOM colour change is directly readable from the diagram without algebraic computation — Shimojima’s free-
ride inference in action. This independent rendering cross-validates that src.diagrams.string_diagram.Discourse correctly
resolves entity identity and role reassignment using only case-role metadata.
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src.visualization.string_diagrams.render_discocirc_discourse() consumes to produce Figure 18 — closing the loop
between symbolic case assignment, computational state-wire bookkeeping, and the colour-coded role transitions visible
in the diagram.
Figure 19: Dynamic case role reversal tracks entity identity across three-sentence discourse. DisCoPy rendering with
lexical heads matching the discourse above: “Alice chases Bob. Bob fears Alice. Alice smiles.” (sentence 3 uses the subject
node Alice—same referent as anaphoric She in the running text; DisCoPy boxes are lexical, not pronominal). Sentence 1:
Alice NOM, Bob ACC. Sentence 2: Bob NOM, Alice ACC—a complete agent–patient reversal. Sentence 3: Alice NOM
as agent of smiles. Alice’s trajectory NOM→ACC→NOM is tracked by the triple tensor 𝑠⊗𝑠⊗𝑠. In a full DisCoCirc
implementation, Alice’s entity wire carries accumulated semantic state—“She” in the gloss inherits the enriched state of
an Alice who has both chased and been feared. This dynamic case assignment across discourse boundaries is what lambeq
Gen II [Krawchuk et al., 2025b] compiles into parameterized quantum circuits.
The DisCoPy rendering encodes entity persistence algebraically (each sentence contributes a tensor factor 𝑠𝑖and the entity
wires are shared by construction), but the reader must infer the shared-entity structure from context. Figure 20 makes the
structure visual: each entity is given its own colour (Alice indigo, Bob amber), the three sentence panels sit side-by-side
with cups coloured by the entity each one contracts, and the bottom role-history ribbon explicitly plots Alice’s NOM →
ACC →NOM trajectory and Bob’s ACC →NOM trajectory with the case-role palette used throughout the paper. It is
the same object as Figure 19 — re-presented with the Frobenius-spider entity-identity bookkeeping that DisCoCirc posits,
made graphically explicit.
10.3
lambeq Gen II Compiles DisCoCirc Discourse Diagrams
The categorical structure of DisCoCat maps naturally onto quantum circuits: the tensor product structure of FVect is
identical to the tensor product structure of Qubit, the category of qubit systems. This observation underlies the QNLP
(Quantum Natural Language Processing) program [Meichanetzidis et al., 2020], which implements DisCoCat models as
parameterized quantum circuits.
The lambeq library [Lorenz et al., 2021] provides a practical pipeline:
1. Parse a sentence into a pregroup derivation (via the neural CCG parser Bobcat or rule-based parsers)
2. Convert the derivation into a string diagram
3. Translate the diagram into a parameterized quantum circuit (or a classical tensor network)
4. Train the parameters on NLP tasks (classification, similarity, question answering)
Kartsaklis et al. [2021] demonstrate that this pipeline achieves competitive performance on question-answering tasks,
confirming that the categorical structure captures genuine linguistic regularities even when instantiated on noisy near-
term quantum hardware.
lambeq Gen II (released May 21, 2025 by Quantinuum) marks a significant advance by incorporating full DisCoCirc
support as its core mathematical foundation, enabling the framework to scale beyond single-sentence semantics to discourse-
level NLP [Krawchuk et al., 2025b, Quantinuum, 2025]. With over 50,000 downloads, lambeq Gen II achieves language
neutrality, improved trainability, and compositional interpretability for explainable AI on quantum hardware. The new
DisCoCircReader API automatically compiles long texts and multi-page documents into discourse-level quantum circuits,
with entity wires tracking semantic state persistence across sentence boundaries—closing the gap between sentence-level
DisCoCat and discourse-level case role tracking. This is directly relevant to the case role reversal phenomena discussed in
Figure 19: lambeq Gen II can, in principle, compile such multi-sentence case-dynamic discourses into trainable quantum
circuits.
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Figure 20:
DisCoCirc entity-persistence unpacking for Alice chases Bob.
Bob fears Alice.
Alice smiles.
Three
sentence panels (top) plus a role-history ribbon (bottom).
Each entity has a dedicated colour (Alice indigo, Bob
amber) and the cup contractions are coloured by the entity each one consumes.
The ribbon shows Alice travers-
ing NOM→ACC→NOM and Bob traversing ACC→NOM (with an “absent” marker for sentence 3), exhibiting
graphically the Frobenius-spider entity persistence that motivates DisCoCirc.
Generated programmatically from
src.visualization.category_unpacking.render_discocirc_entity_persistence().
Foundational work by Meyer and Lewis integrates DisCoCat with density matrices for modeling dynamic meaning and
lexical ambiguity in text, treating semantic states as mixed quantum states—providing an alternative to pure-state vector
models that naturally accommodates ambiguity and partial information [2020].
Recent work on string diagram rewriting by Bonchi et al. [2022] provides the theoretical foundation for diagram
simplification, showing that string diagram rewrite systems modulo Frobenius structure can be interpreted as double-
pushout hypergraph rewriting—ensuring that the algebraic simplifications applied during normal form computation are
provably sound. De Huybrecht [2024] extends DisCoCat with subcategorization for light verb constructions, demonstrating
that the categorical framework accommodates sublexical compositional structure—a development that connects naturally
to the monadic root syntax of Song [2022a] discussed in section 6.
For our case-theoretic framework, QNLP offers a concrete computational substrate: case categories could be implemented
as quantum circuits where case roles correspond to quantum registers and grammatical relations correspond to parame-
terized gates. This connection between linguistic case structure and quantum information processing—mediated entirely
by the shared categorical formalism—illustrates the power of the diagrammatic approach.
10.4
No Barren Plateau for Local Observables
A central challenge for practical QNLP on near-term quantum hardware is the trainability of parameterized quantum
circuits (PQCs): the vanishing gradient problem, or barren plateau, makes gradient-based optimization exponentially hard
as circuit width and depth scale.
Two recent results (2024) directly resolve this obstacle for linguistically motivated
circuits:
1. Rad et al. [2024] introduce reduced-domain parameter initialization: rather than sampling all parameters uniformly
from [0, 2𝜋), one initializes the circuit in a small-angle domain close to the identity. For circuits of the depth typical
of DisCoCirc discourse diagrams (compiled from multi-sentence texts via coreference resolution), this initialization
provably yields polynomial rather than exponential gradient decay—keeping optimization tractable as discourse
length grows.
2. Letcher et al. [2024] derive tight, assumption-free lower bounds on the variance of cost function gradients for
PQCs with local observables (e.g., Pauli operators restricted to a few-qubit subsystem). Their key finding is that,
for POVMs restricted to local observables—exactly the structure of the case-role measurement operators 𝐸𝑐of
Equation 38 (formalized in section 18)—no barren plateau effect occurs. This provides a theoretical guarantee that
case-role classification circuits implemented via lambeq remain optimizable regardless of total circuit size, so long as
the readout observable is local.
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## Page 41

Together, these results underpin the practical feasibility of the F1 and F3 research directions of section 21: scaling
case-marked DisCoCat/DisCoCirc models to corpora-scale quantum hardware without exponential gradient overhead.
The geometric structure of lambeq’s IQP and Sim4 ansätze, combined with these initialization and observable choices,
provides a principled recipe for quantum case category training on near-term devices.
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## Page 42

11
[0, 1]-Enriched Case Categories: Hom-Values as Distributional Proxim-
ity
Where we are in the argument. section 4–section 10 have been working in a category whose morphisms are Boolean
— a relation either exists between two case roles or it does not. This chapter upgrades that to [0, 1]-enrichment so every
pair of roles carries a graded hom-value 𝒞(𝐴, 𝐵) ∈[0, 1] — a distributional proximity that doubles as the precision 𝑤𝑓
used throughout section 14 / section 16 for prediction-error weighting, and that will feed into the magnitude invariant of
section 12 and the topos-theoretic bridge of section 13.
11.1
Why Binary Morphisms Are Not Enough
The categories established in section 4—modeling case roles as objects and grammatical relations as morphisms—capture
the qualitative topology of case systems: which roles exist and how they connect.
However, actual linguistic data is
fundamentally quantitative: certain grammatical relations are more probable than others, proto-role assignments vary in
strength, and distributional similarity is a matter of degree.
To accommodate this quantitative dimension without sacrificing algebraic structure, we advance from ordinary categories
to enriched categories. In an enriched category, hom-sets carry additional measurable structure rather than functioning
as mere discrete sets of morphisms. The framework of Bradley et al. [2021] supplies the key formal construction.
11.2
Enriching Over ([0, 1], ⋅, 1): Identity, Sub-Multiplicative Composition, and Four Hom-
Value Readings
11.2.1
The Identity and Composition Axioms for [0, 1]-Enriched Case Categories
A category 𝒞enriched over the unit interval ([0, 1], ⋅, 1) assigns to every pair of objects 𝐴, 𝐵a hom-value 𝒞(𝐴, 𝐵) ∈[0, 1]
satisfying:
𝒞(𝐴, 𝐴) = 1
(Identity)
(13)
𝒞(𝐴, 𝐶) ≥𝒞(𝐴, 𝐵) ⋅𝒞(𝐵, 𝐶)
(Composition)
(14)
The identity axiom dictates that every linguistic expression remains maximally related to itself. The composition inequality
imposes a strict logical boundary, demanding that distributional relatedness inherently composes sub-multiplicatively: if
expression 𝐴proves 80% related to 𝐵, and 𝐵remains 70% related to 𝐶, the overarching algebraic structure formally
guarantees that 𝐴must be at least 0.8 × 0.7 = 56% related to 𝐶.
11.2.2
Four Linguistic Readings of the Hom-Value: Probability, Proto-Role, Similarity, Predictability
Bradley et al. [2021] originally interpret these numerical hom-values strictly as empirical conditional probabilities measured
within a massive distributional model: 𝒞(𝐴, 𝐵) = 𝑃(context ∣𝐴and 𝐵co-occur). Within our customized case-theoretic
application, we actively broaden this interpretation to capture deep grammatical phenomena (Table 7):
Table 7: Four linguistic interpretations of hom-values in [0, 1]-enriched case categories.
Hom-value interpretation
Domain
Example
Conditional probability
Corpus statistics
𝑃(ACC role ∣transitive verb context)
Proto-role strength
Semantic typology
Degree of Proto-Agent satisfaction
Distributional similarity
Vector semantics
Cosine similarity of case-role
embeddings
Morphological predictability
Morpholexicology
Reliability of case-marking paradigm
11.2.3
When the Composition Inequality Fails: A Worked English NOM–ACC–DAT Example
Architectural note — two decoupled number systems.
The enriched hom-value 𝒞(𝐴, 𝐵) ∈[0, 1] defined
here is not the same scalar as the morphism weight 𝑤𝑓
∈[0, 1] that appears on every Morphism in the Case-
Category of section 4.
The CaseCategory assigns 𝑤𝑓
= 1.0 to every structurally present morphism (it encodes
admissibility — a binary question dressed as a real to support multiplicative composition along chains, per Eq.
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## Page 43

3), whereas the
EnrichedCategory assigns a graded distributional proximity between every pair of roles includ-
ing non-adjacent ones.
The two number systems serve different purposes and are intentionally decoupled in the
implementation:
src/case_systems/case_category.py::standard_case_category ships unit-weight morphisms,
while
src/enriched_cat/enriched.py::standard_enriched_category ships the full 8 × 8 proximity matrix 𝑍. When a prediction-
error formula in section 14 or section 16 references a “weight” 𝑤𝑓= 𝒞(𝐴, 𝐵), the referent is the enriched hom-value —
never the morphism weight in the case category.
Our EnrichedCategory class implements this structure directly. The constructor takes a list of CaseRole objects and a
NumPy proximity matrix encoding hom-values:
from src.enriched_cat.enriched import EnrichedCategory
from src.case_systems.case_category import CaseRole
import numpy as np
roles = [CaseRole.NOM, CaseRole.ACC, CaseRole.DAT]
proximity = np.array([
[1.00, 0.85, 0.30],
# NOM: high with ACC, low with DAT
[0.85, 1.00, 0.45],
# ACC: high with NOM, moderate with DAT
[0.30, 0.45, 1.00],
# DAT: low with NOM, moderate with ACC
])
cat = EnrichedCategory(
name="English Case Proximity",
roles=roles,
proximity_matrix=proximity,
)
# Verify composition inequality: 0.30 >= 0.85 * 0.45 = 0.3825?
# This fails! English NOM-DAT is too distant relative to the chain.
assert not cat.check_composition_inequality(CaseRole.NOM, CaseRole.ACC, CaseRole.DAT)
The composition inequality violation here is linguistically meaningful: it tells us that the NOM→ACC→DAT chain
overestimates the direct NOM→DAT relatedness, reflecting the typological fact that subject–recipient identity (e.g., in
benefactive constructions) is more restricted than the product of agent–patient and patient–recipient proximities. Figure 21
shows the pairwise hom-values 𝒞(𝐴, 𝐵) ∈[0, 1] between case roles as an annotated heatmap of the proximity matrix.
Slavic syncretism as an empirical anchor for high hom-values.
The fourth row of Table 7 — morphological
predictability — is most cleanly calibrated against Slavic case morphology. Where a paradigm collapses two morphological
cells into one form, the morphological-predictability hom-value approaches its upper bound:
• Russian masculine animate ACC = GEN (e.g. brata, čeloveka) ⇒𝒞(ACC, GEN) ≈1 for that declension class.
• Russian neuter NOM = ACC (e.g. okno “window” identical in subject and direct-object position) ⇒
𝒞(NOM, ACC) ≈1 for that paradigm.
• Serbian/BCS dative–locative singular merger in many declensions (e.g. gradu serves both prema gradu “toward-
DAT the city” and u gradu “in-LOC the city”) ⇒𝒞(DAT, LOC) ≈1.
These are not modelling choices: they are observed identifications of morphological cells, supplying directly-measurable
upper bounds on the enriched hom-values for the corresponding role pairs. They co-exist with low hom-values for the
same role pairs under the distributional or proto-role readings of Table 7 — a useful reminder that the four interpretations
are projections of a richer multi-channel proximity, not competing definitions.
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## Page 44

Figure 21: Core and peripheral argument complexes emerge from enriched distributional hom-values. Annotated heatmap
of the 8 × 8 matrix with entries 𝒞(𝐴𝑖, 𝐴𝑗) ∈[0, 1] (YlOrRd scale; numeric labels in each cell). High-proximity blocks
reveal the core argument complex (NOM–ACC–DAT: transitive and transfer morphisms) and the peripheral com-
plex (LOC–INS–ABL: spatial and instrumental relations). GEN shows elevated proximity to both blocks via posses-
sive modification; VOC shows high proximity to NOM (0.70, reflecting nominative-vocative morphological syncretism
in many IE languages) but low proximity to the spatial-instrumental periphery (INS=0.20, LOC=0.15, ABL=0.15),
reflecting its pragmatic rather than referential function.
These hom-values assemble the similarity matrix 𝑍used
for categorical magnitude in section 12 (Equation 15), with axioms Equation 13–Equation 14.
Generated program-
matically from src.visualization.enriched_diagrams.render_enriched_heatmap() applied to EnrichedCategory data (see
scripts/generate_diagrams.py).
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12
Magnitude and Magnitude Homology: Effective Role Count, Lawvere
Similarity Spaces, and Language as Enriched Category
Where we are in the argument.
section 11 introduced [0, 1]-enrichment so that the case category carries a full
distributional-proximity matrix. This chapter extracts the principal quantitative invariant of that matrix — Leinster’s
magnitude |𝒞| = ∑𝑖,𝑗(𝑍−1)𝑖𝑗— which gives a single-number answer to “how many effectively distinct case roles does the
language use?” (|𝒞| ≈2.50 for the standard eight-case system), and which will serve as the N400 magnitude-change proxy
in section 15 and as a cognitive-security invariant in section 20.
A central mathematical invariant unique to enriched categories is their magnitude—a numerical quantity capturing the
“effective size” of the category by discounting distributional overlap between objects.
For an enriched category with 𝑛objects, let 𝑍be the 𝑛× 𝑛similarity matrix where 𝑍𝑖𝑗= 𝒞(𝑖, 𝑗).
The categorical
magnitude is:
|𝒞| = ∑
𝑖,𝑗
(𝑍−1)𝑖𝑗
(15)
assuming 𝑍is invertible. This magnitude metric connects to information theory:
• For a discrete category (no non-trivial relationships), |𝒞| = 𝑛(the number of objects)
• For a highly connected category, |𝒞| < 𝑛(objects are “redundant”)
• Magnitude connects to the diversity measures studied in ecology (species diversity), graph theory (effective graph
resistance), and information geometry (Fisher information)
Worked example. Consider the minimal 3-case category with objects {𝑆, 𝐴, 𝑃} and hom-values 𝒞(𝑆, 𝐴) = 0.85 (S and
A share agentive contexts), 𝒞(𝐴, 𝑃) = 0.70 (transitive co-occurrence), 𝒞(𝑆, 𝑃) = 0.40 (weak S–P overlap). The similarity
matrix 𝑍is:
𝑍= ⎛
⎜
⎝
1.00
0.85
0.40
0.85
1.00
0.70
0.40
0.70
1.00
⎞
⎟
⎠
,
𝑍−1 ≈⎛
⎜
⎝
4.93
−5.51
1.88
−5.51
8.12
−3.48
1.88
−3.48
2.68
⎞
⎟
⎠
(16)
The magnitude is |𝒞| = ∑𝑖,𝑗(𝑍−1)𝑖𝑗≈4.93 −5.51 + 1.88 −5.51 + 8.12 −3.48 + 1.88 −3.48 + 2.68 ≈1.52—substantially
less than the cardinality 3 of the role inventory. The deficit 3 −1.52 = 1.48 is substantial—approximately 49% of the
cardinality—reflecting that S and A share dense agentive contexts (𝑤= 0.85) and A and P share transitive co-occurrence
contexts (𝑤= 0.70), making the proto-role system considerably redundant. By contrast, an accusative alignment—which
merges S and A into a single NOM role—would yield a 2-object category with magnitude exactly 2.0, and the deficit
3 −2.0 = 1.0 quantifies the information lost by neutralization.
Scaling to full categories. Our EnrichedCategory implementation computes magnitude for any case category. For the
standard 8-case English category with empirically calibrated distributional proximity values, the magnitude is approxi-
mately 2.50—the deficit 8−2.50 = 5.50 reflects substantial distributional overlap: NOM and ACC share transitive contexts
(𝑤= 0.85), DAT and ACC overlap in double-object constructions (𝑤= 0.55), GEN and NOM co-occur in possessive con-
structions (𝑤= 0.60), and even peripheral cases such as VOC overlap with NOM (𝑤= 0.70). Only 2.50 of the 8 case roles
encode genuinely independent relational distinctions. This magnitude differential provides a quantitative formalization
of Silverstein’s [1976] case hierarchy: languages with more alignment-based neutralization (lower magnitude) have less
relational discriminability, while richer case inventories (higher magnitude) make finer-grained relational distinctions.
Bradley [2021] established a link connecting categorical magnitude to classical information entropy via topological operad
derivations.
Her result proves that Shannon entropy acts as the unique algebraic derivation of a specific topological
operad—a categorical structure governing the composition of enriched categories. This supplies theoretical justification
for magnitude as a measurable geometric invariant quantifying linguistic complexity: the magnitude of any case category
quantifies how much irreducible “information” that case system encodes regarding relational meaning.
Leinster and Shulman [2021] further develop magnitude homology, which categorifies magnitude from a scalar invariant
to a graded homological invariant—detecting not just the “effective number” of objects but the higher-dimensional “holes”
in the distributional landscape. For case categories, magnitude homology can distinguish between two systems with the
same magnitude but different topological structure: a category where NOM–ACC–DAT form a tight cluster and all other
cases are isolated looks identical in magnitude to one where the clustering is evenly distributed, but their magnitude
homology groups differ, revealing that the former has a 1-dimensional “hole” (a missing transitive link) that the latter
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fills. This finer invariant provides a richer classification of case systems than magnitude alone. Bradley and Vigneaux
[2025] realize this programme on natural language by building categories of texts enriched from language-model next-
token probabilities and computing magnitude and magnitude homology for associated metric spaces of texts—a concrete
large-scale application of Leinster–Shulman theory beyond finite toy examples.
However, that LM-enriched construction exposes a vulnerability relevant to our cognitive synthesis: if magnitude homol-
ogy computations are ported into non-classical environments, such as lambeq Gen II’s Parameterized Quantum Circuits
(section 17), the inherent environmental quantum noise (decoherence) acts as a non-trivial topological perturbation. Un-
less error-correction syndromes explicitly preserve the sequence of homology groups, the invariant may technically fail
to commute.
Thus, comparing the 1-dimensional “holes” of classical case-frames against their quantum algorithmic
analogues must be approached with mathematical caution, respecting the shear forces introduced by measurement non-
commutativity.
12.1
Language as Enriched Category:
Transformer Attention Weights Are Context-
Dependent Hom-Values
Bradley’s [2024; 2025] broader program treats natural language itself as an enriched category, where:
• Objects are expressions (words, phrases, sentences)
• Hom-values encode distributional co-occurrence probabilities
• Composition models transitivity of distributional relatedness
This “language as enriched category” perspective has profound implications for case theory:
1. Case roles emerge from distributional structure: Rather than being imposed a priori, case distinctions arise
from clusters of high hom-values in the enriched language category. Nouns that frequently appear in agent contexts
cluster together, forming the “nominative” region of the category.
2. Alignment types correspond to enriched structure: Different languages partition the enriched category dif-
ferently, and these partitions correspond to the alignment types (accusative, ergative, etc.) discussed in section 4.
3. Language models are enriched functors: A neural language model (such as a transformer) can be viewed
as an enriched functor from the syntactic category to the semantic category, mapping type-logical derivations to
distributional meaning representations while preserving the enriched structure.
The deep connection to modern distributional semantics is this: static embeddings operationalize hom-values as cosine
similarity in a learned space, while contextualized transformers [Devlin et al., 2019, Vaswani et al., 2017] compute dynamic
hom-values from sentential context—a move from a fixed enriched category to a parameterized one. The attention-as-
enriched-cup analogy (developed in section 8) carries the same intuition here: layer-wise weights grade how strongly tokens
couple, alongside Bradley et al.’s [2021] probabilistic reading of hom-objects.
12.2
Lawvere’s Insight: Case Categories Are Similarity Spaces
The [0, 1]-enrichment connects to a deep tradition in categorical algebra. Lawvere showed that metric spaces are categories
enriched over ([0, ∞], +, 0): the hom-value is the distance between points, the identity axiom says 𝑑(𝑥, 𝑥) = 0, and the
composition inequality is the triangle inequality 𝑑(𝑥, 𝑧) ≤𝑑(𝑥, 𝑦) + 𝑑(𝑦, 𝑧). Our [0, 1]-enrichment is the multiplicative
analogue: hom-values are similarities rather than distances, and the composition inequality is sub-multiplicative rather
than sub-additive. The inequality direction reverses between the two settings because the monoidal structure reverses:
in the additive metric setting we want small composites (triangle inequality is an upper bound on distance), whereas for
multiplicative similarity we want large composites (so the natural inequality is the lower bound 𝒞(𝐴, 𝐶) ≥𝒞(𝐴, 𝐵)⋅𝒞(𝐵, 𝐶),
equivalently the upper-bound form used in the section 23 notation appendix).
When is magnitude defined? Leinster’s magnitude requires the similarity matrix 𝑍to be invertible. For the standard
eight-case category (Table 4) 𝑍has condition number ≈9.5 and magnitude is well-defined (|𝐶| ≈2.50, deficit 5.50).
For degenerate categories where roles are distributional clones (e.g. 𝑍has repeated rows), 𝑍becomes singular; the
implementation in src/enriched_cat/enriched.py falls back to a Moore–Penrose pseudo-inverse with a logged warning,
which yields a best-least-squares approximate magnitude but not the exact Leinster magnitude (which is simply undefined
in that case).
Distributional-semantics embedding into [0, 1]. A word-embedding model instantiates the enriched category via
the explicit map 𝒞(𝑤, 𝑤′) = (cos(𝑣𝑤, 𝑣𝑤′) + 1)/2 ∈[0, 1], where 𝑣𝑤, 𝑣𝑤′ are the model’s vector representations and cos
is cosine similarity. This is the concrete choice that allows a pretrained transformer or word2vec model to be read as a
[0, 1]-enriched functor.
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This Lawvere-style perspective unifies our case categories with the geometry of distributional semantics: case roles are
points in a “similarity space,” and morphisms between them are paths weighted by distributional proximity. The mag-
nitude of this space then quantifies the “effective dimensionality” of the case system—how many independent relational
distinctions the language makes.
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13
Topos-Theoretic Bridges: Transferring Results Across Case-Theoretic
Frameworks
Where we are in the argument. section 4–section 12 have erected four formal layers — typological categories (section 5),
type-logical pregroup syntax (section 6–section 7), distributional semantics (section 8–section 10), and enriched magnitude
(section 11–section 12). This chapter supplies the meta-framework that lets results proven in one layer transfer to the
others: classifying toposes and Morita equivalence (with the honest limitation that the repository checks necessary-but-
not-sufficient invariant matching, not full topos equivalence).
13.1
The Inter-Theoretic Translation Problem
The preceding sections constructed four formally distinct perspectives on case: typological, type-logical, distributional, and
enriched. A central methodological question arises: when can structural results proved in one framework be carried over
to another without starting from scratch? Caramello’s [2016] topos-theoretic bridge technique provides this methodology
for properties that are invariants of a shared classifying topos, once Morita equivalence (or a suitable bridge topos)
is in hand. We adopt this methodology as a research programme for case theory: establishing full Morita equivalences for
the specific case-theoretic formalizations developed here remains largely open (see the implementation note later in this
section), but the framework precisely identifies what must be proved and what would transfer as a result.
13.2
Classifying Toposes: The Logical Shape of a Theory
13.2.1
A Topos Is a Self-Contained Logical Universe
A topos is a category possessing the structural richness of a generalized “universe of sets”: products, exponentials, and a
subobject classifier (a “truth-value object”) that supports internal first-order reasoning. The most familiar example is the
category Set of ordinary sets; other important instances include presheaf categories [𝒞op, Set] and sheaf categories over
topological spaces. Intuitively, a topos provides a self-contained logical universe within which mathematical reasoning can
proceed—and different toposes encode different logical constraints.
13.2.2
Morita Equivalence: Invariant Transfer Across Typologies
Every geometric theory 𝕋(axiomatized by sequents with finite conjunctions, arbitrary disjunctions, and existential quan-
tification) generates a unique classifying topos ℰ𝕋: a canonical topos such that, for any Grothendieck topos ℱ, models
of 𝕋in ℱcorrespond naturally to geometric morphisms ℱ→ℰ𝕋. The classifying topos encodes the theory’s “logical
shape” independently of any particular model.
Caramello’s insight [2016; 2021] is that formally different theories can share the same classifying topos up to geometric
equivalence—a relation termed Morita equivalence. When ℰ𝕋1 ≃ℰ𝕋2, any property expressible as an invariant of the
shared topos transfers automatically from 𝕋1 to 𝕋2 without re-proof.
13.3
A Chain of Morita Equivalences Connects the Four Case Theories
13.3.1
Each Case Framework as a Geometric Theory
We formalize each case-theoretic framework as a geometric theory:
1. Typological case theory 𝕋typ: Objects are case roles, morphisms are grammatical relations, axioms specify
alignment constraints (e.g., “S = A” in accusative alignment).
2. Type-logical case theory 𝕋log: Objects are syntactic types, morphisms are Lambek calculus derivations, axioms
specify well-formedness conditions (pregroup contractions).
3. Distributional case theory 𝕋dist: Objects are vector spaces, morphisms are linear maps, axioms specify the
composition law for distributional meaning (DisCoCat).
4. Enriched case theory 𝕋enr: Objects are expressions, morphisms carry [0, 1]-valued weights, axioms specify the
identity and composition inequalities.
13.3.2
The Bridge Programme: A Chain of Classifying Toposes for Case Theory
The research programme we pursue is that these four perspectives admit a topos-theoretic alignment: formally distinct
case theories should be related by bridge toposes and, where Morita equivalence can be established, invariants proved in
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one formulation transfer without re-proof. Equation (17) sketches the target picture—a chain of classifying toposes linked
by intermediate geometric theories—not a single theorem asserted for all details in this manuscript.
ℰ𝕋typ ←ℰ𝕋bridge →ℰ𝕋log ←ℰ𝕋bridge′ →ℰ𝕋dist
(17)
Intermediate toposes would be supplied by geometric theories simultaneously interpretable in both flanking frameworks.
Were such Morita equivalences established, one would expect:
• Syntactic theorems port to semantics: Commutativity results in the type-logical setting would align with
compositionality statements in the distributional setting.
• Typological universals constrain distributional models: Alignment types (accusative, ergative) would impose
structural constraints on the vector-space data of DisCoCat-style models.
• Enriched structure enriches all frameworks: [0, 1]-valued weights could be pulled back to probabilistic readings
of typological, logical, and distributional constructions.
Implementation status.
The Python topos module does not implement general classifying toposes or full Morita
equivalence proofs. It constructs finite invariant profiles (sort counts, symbol arities, axiom tallies) from CaseCategory
instances and uses those profiles for check_morita_equivalence() and guarded bridge_transfer(). The equivalence check
verifies necessary but not sufficient conditions: matching arity spectra and compatible axiom counts. A positive result
means equivalence is consistent with the evidence; it does not constitute a proof.
This proxy is a concrete, testable
approximation of the diagram in (17), not a replacement for topos-level equivalence. Full classifying topos construction
and Morita equivalence proof for specific case theory pairs remains an open research target (see F2).
Figure 22 visualizes the alignment functor between the Accusative and Ergative category instantiations, showing how the
same eight case roles are connected by different morphism structures under different alignment types.
13.4
Reconciling Classical DAIF with Intuitionistic Topos Logic via Sheaf Cohomology
An unresolved tension arises when binding topos-theoretic invariants of case structures with the Distributional Active
Inference (DAIF) layer (section 16). DAIF relies on Quantile Temporal Difference (TD) learning, which assumes Markovian
updates over classical probability distributions.
Conversely, the internal logic of a classifying topos is generally intuitionistic—yielding non-distributive lattices of truth
values where the classical law of excluded middle fails. Forcing classical Markovian density tracking onto non-distributive
topological spaces risks mathematical collapse: the case alignments (modeled as geometric properties) may fail to compose
covariantly under quantum NLP (lambeq Gen II) decoherence.
To resolve this, we propose that classical DAIF distributions cannot be mapped directly into the topos. Instead, a sheaf-
theoretic bridge is required: probability densities over case assignments must be treated as sections of a probability
sheaf. Local discrepancies in case assignment (e.g., during pragmatic garden-path discourse) resolve globally via sheaf
cohomology.
This ensures that while local parses resolve classically via quantile Huber loss, their global composition
respects the intuitionistic structure of the geometric case theory.
13.5
Phillips’s Result: Language-of-Thought Properties as Universal Topos Constructions
Phillips [2024] provides a striking application of topos-theoretic methods to cognitive science. He shows that the Language
of Thought (LoT) hypothesis—the claim that cognition operates over structured, combinatorial representations with
language-like properties—can be formalized categorically, and that the resulting structure is universal in the topos-theoretic
sense.
Specifically, Phillips demonstrates that:
1. LoT properties (discrete constituents, role-filler independence, systematicity) arise as universal constructions in
a topos—categorical products, fiber bundles, and presheaves.
2. Every topos supports an internal first-order logic, explaining how LoT-like logical capacities can emerge in systems
(biological or artificial) whose internal architecture forms a topos.
3. The “shape” of cognitive representations is fundamentally topological, captured by presheaves and fiber bundles
rather than by point-set structures.
Applied to case theory, Phillips’s result is significant: because the Language of Thought is topos-universal, and our case
categories are definable within any topos (as small categories governed by first-order axioms), every cognitive system
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Figure 22: The alignment functor preserves role inventory but restructures morphism grouping. Functor 𝐹∶𝒞acc →𝒞erg
mapping the Accusative case category (source, blue panel) to the Ergative (target, amber panel). Five nodes appear in
each panel — the three semantic primitives S, A, P plus the alignment-specific case labels (NOM, ACC in the accusative;
ERG, ABS in the ergative); purple dashed functor arrows show the object-level mapping 𝐹(role) (from AlignmentFunc-
tor.object_map). The key structural difference: Accusative groups {𝑆, 𝐴} →NOM (kernel {(𝑆, 𝐴)}, cf. Equation 1), while
Ergative groups {𝑆, 𝑃} →ABS (kernel {(𝑆, 𝑃)}, cf. Equation 2). This diagram provides one link in the alignment functor
structure that would, if full Morita equivalence were established, enable inter-theoretic bridge transfer (see F2 for the proof
programme). Generated programmatically from src.visualization.functor_diagrams.render_functor_diagram() with the
canonical accusative_to_ergative_functor() (scripts/generate_category_figures.py); panel layout is fixed for publica-
tion while cross-panel arrows follow the functor’s object map.
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with LoT-like architecture has the structural capacity to represent case assignments. This grounds the claim that case
structure is a universal feature of higher cognition in the mathematical framework of topos theory rather than in typological
observation alone.
13.6
Caramello’s Syntactic Learning Algorithm: Inducing Case Theories from Annotated
Corpora
Caramello [2023] extends the bridge technique to a learning theory: she shows that classifying toposes can be used to
learn the theory of a mathematical structure from finite data (a finite set of models). The learning algorithm constructs
a classifying topos from the observed data and then extracts the axioms of the underlying theory.
Applied to case systems, this suggests a principled approach to grammatical induction: given a corpus annotated with
case labels, one could construct the classifying topos of the implicit case theory and read off its axioms—recovering the
alignment type, the morphism structure, and the enriched weights from data alone. The procedure operates in four phases:
1. Extraction: Parse a Universal Dependencies treebank for a target language, collecting all case-labeled dependency
arcs. Each arc (𝑟1, rel, 𝑟2) instantiates a morphism 𝑟1 →𝑟2 in the implicit case category.
2. Saturation: Close the extracted morphism set under composition, identity, and the enriched weight constraints of
section 11. Compute the empirical hom-values as normalized co-occurrence frequencies.
3. Classification: Construct the classifying topos ℰ𝕋from the saturated theory—the canonical topos whose models
are exactly the case-assignment patterns attested in the corpus. The topos-theoretic invariants (sort count, arity
distribution, axiom count) combined with categorical magnitude and magnitude homology (section 12)—computed
from the [0, 1]-enriched hom-values of section 11—provide a multi-dimensional fingerprint of the language’s case
system.
4. Identification: Compare the fingerprint against the Morita equivalence classes of known alignment types. If the
classifying topos matches an existing class, the language’s alignment type is identified; if not, the procedure has
discovered a novel alignment pattern.
This topos-theoretic learning procedure would be provably correct (recovering the true theory in the limit) and maximally
general (not presupposing any particular alignment type). The scalar magnitude invariant from section 12 enters at step
3 as a summary of the learned category’s “effective size,” while magnitude homology provides a finer-grained topological
signature.
13.7
Morita Equivalence Diagrams Are Themselves Free-Ride Inferences
The bridge technique has a natural diagrammatic interpretation. Morita equivalence between theories is witnessed by
functorial translations—diagrams in the 2-category of toposes that commute up to natural isomorphism. These diagrams
serve the same cognitive function as the commutative diagrams of section 2: they make the transfer of structure visible,
allowing a researcher to verify at a glance that a result proved in one framework genuinely applies in another.
Manders [2008] observed that even in classical mathematics, diagrams serve not merely as illustrations but as inferential
instruments whose spatial properties encode proof-relevant information.
The topos-theoretic bridge diagrams extend
this observation to the meta-theoretical level: the commutative diagram expressing Morita equivalence is itself a “free
ride” inference, automatically transferring any topos-invariant property from one theory to another without requiring a
case-by-case verification.
Illustrative transfer (conditional on a bridge). The transitive pregroup derivation 𝑛⋅(𝑛𝑟⋅𝑠⋅𝑛𝑙) ⋅𝑛→𝑠yields
the same sentence type whether contractions are grouped left-to-right or right-to-left. That type-logical commutativity
is mirrored in DisCoCat by functoriality: the sentence vector for a fixed derivation is well-defined. A full Morita story
would package such facts as invariants of a shared classifying topos; here we use the example only to show what kind of
statement the bridge programme is meant to align across 𝕋log, 𝕋dist, and enriched formulations—not as a claim that every
step of (17) is already proved for our case theories.
13.8
Python Implementation: Proxy Invariant Checks (implemented and tested)
The topos-theoretic narrative above is paired with a topos Python module that implements finite geometric theories ex-
tracted from CaseCategory, invariant-profile comparison (check_morita_equivalence), and guarded bridge transfer
when profiles match—not a full classifying-topos construction inside the runtime.
13.8.1
Extracting Geometric Theories from CaseCategory Instances
The build_typological_theory() function constructs a geometric theory 𝕋typ from a CaseCategory by extracting:
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• Sorts: the case roles (objects of the category)
• Function symbols: the morphisms with their source/target pairs
• Axioms: identity morphism existence, composition closure, and alignment constraints
For the standard 8-case category, this yields a theory with 8 sorts and approximately 15 function symbols. The minimal
3-case category produces a theory with 3 sorts and 5 function symbols. Our build_enriched_theory() function further
annotates the geometric theory with [0, 1]-valued hom weights from the enriched structure of section 11.
13.8.2
ClassifyingTopos Invariants and the Morita Equivalence Check
The ClassifyingTopos class computes topological invariants—number of sorts, function arity distribution, and axiom
count—that characterize the “logical shape” of a theory. Two theories are Morita equivalent when their classifying
toposes share the same invariant profile:
T_std = build_typological_theory(standard_case_category())
T_min = build_typological_theory(minimal_case_category())
equivalent = check_morita_equivalence(T_std, T_min)
# False: 8-sort and 3-sort theories have different invariants
13.8.3
Concrete Morita Equivalence: A Two-Object Illustration
To build intuition, we work out a minimal example. Consider two presentations of the same relational structure — a
binary agent-patient dependency — formalized as geometric theories over the two-case system {NOM, ACC}:
Theory 𝕋syn (syntactic presentation): Sorts {𝑛, 𝑎}; one binary relation symbol acts_on(𝑛, 𝑎); one axiom asserting
that every element of sort 𝑛participates in at least one acts_on instance.
Theory 𝕋sem (semantic presentation): Sorts {+agent, +patient}; one binary relation symbol transfers_force; the
same participation axiom expressed using the semantic sort names.
Both theories present the same classifying topos: the category of sheaves over a two-node directed graph with a sin-
gle edge. Their invariant profiles match — two sorts, one function symbol of arity (1, 1), one existential axiom — so
check_morita_equivalence returns True:
T_syn = GeometricTheory("syn", TheoryType.TYPOLOGICAL)
T_syn.add_sort("nom"); T_syn.add_sort("acc")
T_syn.add_relation("acts_on", arity=("nom", "acc"))
T_syn.add_axiom(Axiom("participation", antecedent="nom(x)", consequent="∃y.acts_on(x,y)"))
T_sem = GeometricTheory("sem", TheoryType.TYPE_LOGICAL)
T_sem.add_sort("+agent"); T_sem.add_sort("+patient")
T_sem.add_relation("transfers_force", arity=("+agent", "+patient"))
T_sem.add_axiom(Axiom("participation", antecedent="+agent(x)", consequent="∃y.transfers_force(x,y)"))
check_morita_equivalence(T_syn, T_sem)
# True — same classifying topos
The Morita equivalence here licenses one concrete transfer: any result proved about acts_on morphisms in 𝕋syn — such as
transitivity conditions derivable from the participation axiom — applies verbatim to transfers_force morphisms in 𝕋sem,
without re-proof. In the full linguistic setting, this corresponds to transferring structural theorems about nominative-
accusative case from a surface-form typological theory to a semantic proto-role theory, provided the two share the same
classifying topos. The code proxy in topos_theory.py detects invariant equivalence; verifying the full Grothendieck-site
equivalence for richer theories remains a programme for future work (subsection 21.2).
The bridge_transfer() function implements the transfer mechanism: given two Morita-equivalent theories, it constructs
the functorial translation that carries properties (alignment constraints, composition laws) from one to the other. The
transfer is blocked when Morita equivalence fails, preventing unsound cross-theoretic reasoning—a computational enforce-
ment of the mathematical constraint.
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14
Active Inference as a Process Theory of Case
Where we are in the argument.
section 4–section 13 have given the framework a structural theory — objects,
morphisms, functors, enriched weights, and topos-theoretic bridges. This chapter supplies the missing process theory:
active inference recasts case assignment as variational Bayesian inference over a generative model whose state variable is
the case-role posterior, whose precision parameters are the enriched weights of section 11, and whose free-energy descent
turns “parsing a sentence” into a sequence of belief updates — the dynamics that section 15 exploits to derive ERP
predictions.
14.1
Static Categories Are Not Enough
The preceding sections constructed a mathematical infrastructure for analyzing case systems—categorical, type-logical,
distributional, enriched, and topos-theoretic. Yet these frameworks remain static: they describe the structure of case
grammar without explaining how a cognitive agent deploys that structure during real-time comprehension and production.
Bridging this gap requires a dynamic process theory of case-marked relational reasoning. Active inference [Namjoshi, 2026,
Friston et al., 2017] provides exactly this missing dynamic computational layer.
14.2
Surprise Minimization Drives Case-Frame Inference
14.2.1
Free Energy Bounds Surprisal
Active inference is the primary process theory derived from the free energy principle (FEP): every self-organizing system
maintains its structural integrity by minimizing the surprisal (negative log-probability) of its sensory observations under
an internal generative model of its environment [2010]. The system executes this minimization through two complementary
strategies:
1. Perceptual inference: Update internal beliefs to better predict current observations (reduce prediction error)
2. Active inference: Act on the environment to bring observations in line with predictions (reduce expected prediction
error)
Recent extensions of active inference to linguistics and cognitive science have modeled language comprehension and
production as forms of sequential Bayesian inference. As Donnarumma, Frosolone, and Pezzulo (2023) demonstrate in their
integration of large language models and active inference for modelling eye movements in reading, linguistic processing
constitutes “inference over a hierarchical generative model, facilitating predictions and inferences at various levels of
granularity, from syllables to sentences” [2023]. Similarly, Friston et al. (2021) have demonstrated how communication
emerges between synthetic subjects: “linguistic outcomes (specifically, the spoken word)… are selected to minimise the free
energy given current beliefs” via “high-order interactions among abstract (discrete) states in deep (hierarchical) models”
[2021; 2020].
Both strategies minimize the same mathematical quantity—variational free energy—and both draw from a single generative
model encoding the system’s prior expectations about the relational structure of its world.
Critically, recent neurolinguistic evidence directly supports this prediction-error account.
Li and Futrell [2023; 2024]
decompose surprisal into two orthogonal components: heuristic surprise (“shallow surprisal”), which tracks the N400 brain
potential and reflects lexical-associative prediction error, and a discrepancy signal (“deep surprisal”), which tracks the P600
and reflects structural reanalysis when the true parse diverges from the initially inferred structure. This decomposition
maps directly onto our enriched case framework: the N400 corresponds to distributional prediction error within a case-role
subspace (semantic mismatch), while the P600 corresponds to structural prediction error between case-diagram topologies
(morphosyntactic reanalysis requiring a change in the generative model’s case assignments). The formal equations in
section 16 (Equation 31–Equation 32) instantiate exactly this dual decomposition.
14.2.1.1
Generative Models of Relational Structure
Under this paradigm, language understanding manifests as
active inference over relational structure: the listener maintains a generative model anticipating who-does-what-to-whom,
and each incoming word supplies evidence that updates this model. Morphological case marking provides high-precision
evidence—for example, a nominative suffix predicts that the noun phrase functions as the agent, reducing uncertainty
about the relational structure of the unfolding event.
14.2.1.2
S-HAI: The Case Diagram as the Abstract “Schema” Level
These relational generative models find
their formal articulation in recent advances such as Schema-Based Hierarchical Active Inference (S-HAI) [Maele
et al., 2026]. Unifying predictive processing with schema theory, S-HAI employs a dual-level POMDP structure to model
rapid generalization across environments. In the linguistic domain, the “Level 2” model encodes abstract, hidden relational
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## Page 54

goals—which corresponds exactly to the case diagram structure we describe here. The “Level 1” model encodes concrete
sensorimotor navigation—for linguistics, this maps to the sequential parsing of surface word forms.
Just as S-HAI explains sudden “zero-shot” behavioral remapping in novel environments by preserving the high-level schema
mapping while updating the “grounding likelihoods” to new observables, a case frame enables an agent to rapidly generalize
the relational structure of a complex sentence regardless of novel vocabulary pairings. The abstract string diagram is the
schema; case inflection is the grounding likelihood.
14.2.2
The Five-Step Prior–Observation–Update–Prediction–Action Loop
The process unfolds as follows:
1. Prior: The listener has a prior belief about the relational structure (a “case diagram” encoding expected roles and
their connections)
2. Observation: Each word provides sensory evidence—its form, its case marking, its distributional properties
3. Update: The listener updates the case diagram to accommodate the evidence, using approximate Bayesian inference
(typically variational message passing)
4. Prediction: The updated diagram generates predictions about upcoming words (case-marked NPs, verb valency
patterns)
5. Action: In production, the speaker selects words and case markers that minimize expected free energy—choosing
expressions that are informative, contextually appropriate, and syntactically well-formed
14.2.3
Case Diagrams as Instantiated Situations
This dynamic active inference perspective connects naturally to situation semantics (Barwise and Perry [1983]), which
treats linguistic meaning as structured situations—specific configurations of individuals, relations typed by arity, and
spatiotemporal locations, grounded in an ecological realism where meanings are recurring relational patterns that organisms
attune to. Translated into our categorical framework, a situation is an instantiated case diagram: a specific assignment
of entities to roles with particular morphisms activated; the situation type is the structural case category itself, the
abstract pattern that any particular situation instantiates; and information flow between situations is a functorial
mapping between case categories. Where classical situation semantics left the dynamics implicit, active inference supplies
the computational engine: the agent moves through situations in real time, updating its case diagram with each incoming
word and using the updated diagram to predict which situation will arise next.
14.2.4
Belief Dynamics Over Competing Case Frames
Figure 23 shows a minimal scalar-belief simulation:
the agent holds a CaseDiagramBelief over alternative align-
ment frames (NOM–ACC vs. ERG–ABS). As syntactic evidence arrives, variational free energy and entropy track
the discrete update loop that section 16 extends to full return distributions in DAIF. Generated programmatically
from src/visualization/active_inference_plots.plot_alignment_frame_belief_dynamics() (belief trajectory from sequen-
tial_belief_update() in src/cognitive/belief_updating.py).
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Figure 23: Variational free energy drives convergence to the correct case frame during belief updating. The agent begins
with a uniform prior over possible case frames (NOM–ACC vs. ERG–ABS, 𝐻[𝑞] = log 2). Top: stacked 𝑃(frame) over
evidence steps (including the prior column). Middle: entropy 𝐻[𝑞] in nats, with the steepest drop annotated at the
most informative step. Bottom: variational free energy 𝐹[𝑞] after each update (per-step curve) and its running minimum
min𝜏≤𝑡𝐹[𝑞𝜏] (non-increasing envelope), with dashed vertical markers at each evidence index (same convention as word
arrivals in Figure 24). Likelihoods are synthetic categorical draws consistent with a TQNN-evaluated diagram. Generated
programmatically from src/visualization/active_inference_plots.plot_alignment_frame_belief_dynamics() (PNG from
scripts/generate_cognitive_figures.py).
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15
Diagrams as Cognitively Privileged Representations: Free Rides, ERP
Predictions, and Six-Strand Synthesis
Where we are in the argument. section 14 supplied the process theory (active inference over a case-role posterior).
This chapter cashes that theory out in two ways: first with three falsifiable ERP predictions (precision-weighted P600
amplitude, magnitude-change N400 proxy, garden-path reanalysis cost) that empirically test the framework, and second
with an automated test inventory demonstrating that every formal claim is backed by executable code — the “six-strand
synthesis” that brings typology, type logic, distributional semantics, enriched structure, topos theory, and biolinguistic
interfacing into a single generative model.
15.1
Why the Brain Prefers Diagrams
We can now substantially strengthen our initial architectural claim from section 2: formal commutative diagrams do
not merely provide a convenient pedagogical representation illustrating abstract case structure—they mathematically
constitute the exact native computational format through which biological cognitive agents actively maintain and query
their internal generative models representing relational environmental structure.
This structural claim draws support from converging empirical evidence:
1. Computational advantage (Larkin & Simon, [1987]): Diagrams enable search, recognition, and inference opera-
tions that are computationally prohibitive in sentential format. A commutative case diagram allows the agent to
verify consistency (does the direct path equal the composed path?) by simple spatial inspection.
2. Free ride inferences (Shimojima, [1996]): Properties of the diagram that are perceptually available but would
require explicit computation in a sentential format.
In a case diagram, transitivity of grammatical relations is
visible—the existence of a path from NOM to DAT through ACC is spatially apparent.
3. Hybrid reasoning (Giardino, [2017]): Mathematical diagrams engage a mode of reasoning that combines perceptual
pattern recognition with background theoretical knowledge. Case diagrams similarly engage both the perceptual
system (spatial layout) and linguistic knowledge (case constraints, verb valency).
4. Peirce’s existential graphs: Peirce’s graphical logic system demonstrated that first-order logic can be conducted
entirely diagrammatically, without algebraic symbols.
Our case diagrams extend this tradition: the relational
structure of a sentence is represented graphically, and inference proceeds by diagram manipulation (adding/removing
nodes, composing morphisms).
15.2
P600 Signals and Garden-Path Reanalysis in Diagrammatic Models
The standard predictive processing framework—for which active inference operates as the most formally mathematically
developed version—provides a remarkably natural, mechanistically precise account detailing exactly how biological agents
actively deploy these diagrammatic representations cognitively during real-time processing:
1. Top-down structural predictions: The currently active internal case diagram continuously generates precise,
high-precision predictions anticipating incoming expected sensory input (e.g., the system computationally predicts
“a nominative-marked noun phrase must immediately appear because the parsed transitive verb structurally demands
an active agent”).
2. Bottom-up prediction errors: Incoming sensory words that physically violate the model’s top-down diagrammatic
predictions instantly generate massive, measurable prediction errors (e.g., encountering a structurally unexpected
morphological case marker directly triggers a quantifiable P600 event-related neural potential measurable in the
biological brain).
3. Belief updating: The diagram is updated to accommodate the prediction error, potentially restructuring the
assignment of entities to case roles (garden-path reanalysis)
4. Precision weighting: The enriched weights on morphisms serve as precision parameters that control the relative
influence of prior expectations and incoming evidence. A high-weight morphism generates strong predictions that
are costly to override; a low-weight morphism generates weak predictions that are easily overridden.
15.3
Three Falsifiable ERP Predictions
The predictive processing account generates quantitative, falsifiable predictions about neural responses to case-marking
violations. In the active inference framework, a case-assignment violation triggers a prediction error whose amplitude
scales with the precision of the violated expectation—which is precisely the enriched hom-value of the violated morphism:
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PE(𝑓) ∝𝑤𝑓⋅|𝜇predicted −𝜇observed|
(18)
where 𝑤𝑓= 𝒞(𝐴, 𝐵) is the enriched morphism weight (acting as a precision on prediction error) for 𝑓∶𝐴→𝐵and 𝜇are
the expected vs. observed case features. This yields three concrete electrophysiological predictions:
1. P600 amplitude scales with morphism weight: A case violation on a high-weight morphism (NOM→ACC in a
transitive clause, 𝑤= 0.85 per 𝒞(NOM, ACC) in the standard enriched category) should elicit a larger P600 than a
violation on a low-weight morphism (NOM→INS in an experiencer construction, 𝑤= 0.35 per 𝒞(NOM, INS)). The
ratio of P600 amplitudes should approximate the ratio of enriched weights (0.85/0.35 ≈2.4).
2. N400 reflects distributional expectation: Semantic case violations—where the case-marked NP satisfies the
morphological case but not the distributional proto-role requirements (e.g., an inanimate NOM in an agentive
construction)—should elicit N400 effects proportional to the absolute change in categorical magnitude induced by
the violation, ∣|𝒞after| −|𝒞before|∣(section 11). The N400 is thus time-locked to the transition between pre-violation
and post-violation diagrams, not to any static property of either category alone; the n400_amplitude_proxy() function
in src/cognitive/reanalysis.py computes precisely this quantity.
3. Garden-path reanalysis costs track magnitude: The processing cost of reanalyzing a garden-path sentence’s
case structure should correlate with the change in categorical magnitude between the initial and revised case diagrams,
since magnitude quantifies how much relational information the agent’s generative model encodes.
15.4
Integration: Six Strands Become One Generative Model
The full picture emerges when we combine the five formal layers of section 2 (Pillars 1–5) with the sixth strand (Pillar
6: biolinguistics and oscillatory interfacing via ROSE), all within the active inference framework:
1. Case categories (section 4) provide the objects and morphisms of the generative model—the vocabulary of roles
and relations
2. Categorial grammar (section 6) provides the composition rules—how roles combine to form structured derivations
3. DisCoCat / DisCoCirc (section 8, section 9, section 10) provides the semantic functor and discourse extension—
mapping syntactic structure to distributional meaning
4. Enriched structure (section 11) provides the precision parameters—graded weights that control inference
5. Topos-theoretic bridges (section 13) provide transfer theorems—ensuring consistency across formalizations
6. Biolinguistic and neurocomputational interface (section 2, section 14): ROSE-style cross-frequency coupling
links hierarchical syntax (MERGE-level constraints) to the associative dynamics of comprehension—supplying the
neural-timescale bridge under which the formal diagram is deployed in real time.
The active inference agent uses this combined structure as a single, integrated generative model.
Each scenario it
encounters—a sentence heard, a scene observed, an action planned—is interpreted by instantiating a case diagram from
structure (1), parsing the input using rules (2), computing meaning via the semantic functor (3), weighting confidence
using enriched structure (4), transferring results across representational formats using bridge techniques (5), and routing
syntactic structure through the oscillatory interface (6) during online comprehension and production.
This is total cognitive scenario understanding: the agent doesn’t just parse a sentence or assign case labels—it constructs
a complete, internally consistent, generic, strongly typed, dynamically updating model of the relational structure of the
situation, and uses that model to predict, explain, and act.
15.5
1197 Automated Tests Confirm the Formalism Is Executable
The framework developed in this paper is computationally verified through an implementation and test suite that exercises
every categorical construction discussed above.
15.5.1
System Architecture and Categorical Core
The categorical core (CaseCategory,
EnrichedCategory,
AlignmentFunctor,
NaturalTransformation) is implemented in
Python with set-based object tracking and list-based morphism storage, enforcing categorical axioms at construction
time. 9 first-level packages under src/ structure the domain code; six further module groups extend the core: FluidS-
Functor (context-dependent alignment parameterized by volition), CaseDiagramBelief and active inference computations
(variational free energy, prediction error, belief updating), the src/daif/ subpackage (7 modules, 25 symbols—full dis-
tributional RL inference: push-forward returns, quantile TD, VMP, Bethe FE, EIG, ERP profiles, policy selection, and
metrics), CasePOVM and quantum case assignment (POVM-based probability via Equation 38 in section 18), Ditransi-
tiveSentence (three-argument verb support), and CaseFrameValidator (cognitive security via type-violation detection).
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The visualization layer produces all manuscript figures programmatically, ensuring exact correspondence between formal
claims and visual evidence. The DisCoPy integration library (installed version 1.2.2) provides an independent validation
path: pregroup types (Ty), lexical entries (Word), cup contractions (Cup), cap expansions (Cap), type permutations (Swap),
normal form computation (normal_form()), and circuit depth analysis (depth()) are exercised against the same categorical
structures described in section 6 and section 8.
15.5.2
Automated Test Suite and Verification
The implementation is validated by 1197 automated tests across 64 test files.
95.96% line-and-branch coverage on
src/ (from coverage.json) The configuration enforces ≥90% coverage on src/.
Every test uses real mathematical
computations—no mocks or fakes. The per-category inventory (counts, modules, and DAIF file breakdown) is listed
in section 24.
This computational verification demonstrates that the category-theoretic framework is a working computational architec-
ture: the categorical abstractions compile, execute, and produce verifiable results, bridging the gap between formal theory
and implemented system.
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16
Distributional Active Inference (DAIF): Convergence of Semantic
Topologies and Reinforcement Learning
The claim of this section, precisely. Three mathematical objects that arose independently in three fields — Firth’s
distributional-semantics vectors [1957] in linguistics, Bellemare–Dabney–Munos return distributions [2017] in reinforce-
ment learning, and Friston variational posteriors [2017] in active inference — are the same structural object seen from
three angles. Each assigns to a pair of states a [0, 1]-valued hom-value (a similarity, a probability mass on a support
atom, a posterior weight); each does so because the underlying framework had to replace an inadequate scalar summary
(co-occurrence count, expected return, MAP estimate) with a full distribution in order to be expressive; and each com-
poses along chains by multiplying those hom-values, putting all three inside the same [0, 1]-enriched category (section 11).
The convergence is non-trivial because none of the three frameworks was designed with the others in mind, yet the
enriched-category axioms (identity, sub-multiplicative composition, 𝑍-matrix invertibility for magnitude) hold in all three
without modification. This convergence is what Akgül et al. [2026] call Distributional Active Inference (DAIF).
The repository implements the convergence computationally; a fully categorical proof that the three instantiations share a
common enriched-category base in the strict sense (with a single enriching monoidal base category, not merely compatible
hom-value scales) remains open and is flagged in the Limitations subsection below.
The src/daif/core.py implementation (tested in the project suite) uses a belief-weighted mean-field approximation:
rather than maintaining a separate return distribution 𝑍(𝑠) for every case-role state 𝑠(which would cost 𝒪(𝑛⋅𝑁atoms)
memory), push_forward_return(belief, transition_matrix, ...) propagates a single return distribution weighted by the
current posterior 𝑞(𝑠) over states. Formally, one step of the contraction computes z = 𝑅+ 𝛾𝑇⊤𝑞and collapses it to the
belief-weighted scalar
̄𝑧= 𝑞⊤z; uncertainty is then injected back via the quantile spread of the updating distributional
return. The approximation is exact in the limit of a sharp posterior (𝑞→𝛿𝑠∗) and recovers the full per-state distributional
Bellman operator in that regime. In exchange for the 𝒪(𝑛) complexity, it buffers aleatoric and epistemic uncertainty
simultaneously and remains sample-efficient for the linguistic-parsing proxy model used here (implemented and tested).
The terminological collision between “distributional” in distributional semantics and “distributional” in distributional RL
is not mere homonymy—it reflects a deep structural parallel. In both domains, the core computational move is the same:
replacing scalar summaries with full distributional representations. In linguistics, this means contextualizing
word identities via probability distributions (Firth’s [1957] company-keeping principle).
In reinforcement learning, it
replaces expected-value estimates with quantile-approximated return distributions. In active inference, it replaces point
estimates of states with variational posterior distributions. The enriched-categorical framework of section 11 provides the
unifying abstraction: all three are instances of [0, 1]-enriched categories where hom-values encode distributional proximity
rather than rigid identity.
This section presents the complete implementation and quantitative results of the src/daif/ subpackage—seven modules,
25 public symbols, 224 automated tests—covering six major computational contributions: push-forward returns (sub-
section 16.1), quantile TD and implicit quantile networks (subsection 16.2), variational message passing and Bethe free
energy (subsection 16.3), policy selection and expected free energy (subsection 16.4), unifying ERP amplitude profiles
(subsection 16.5), and convergence diagnostics (subsection 16.6).
16.1
Push-Forward Returns and the Distributional Bellman Operator
The formal architecture of DAIF proceeds through three stages: (1) reconstructing active inference via variational Bayesian
inference on a controlled Markov process; (2) defining a push-forward operation that iteratively maps latent-space tra-
jectories to return distributions; and (3) deriving a temporal-difference quantile-matching algorithm that achieves active
inference’s sample-efficiency advantages within a model-free computational architecture. This permits far-sighted parsing
without explicit transition modeling:
𝔼[
∞
∑
𝑡=0
𝛾𝑡𝑅(𝑥𝑡, 𝑎𝑡) ∣𝑥0, 𝑎0] = ∫
𝒮ℕ+
𝑅∘𝑓𝑑(S#ℙ𝑃𝜋
𝑥0,𝑎0)
(19)
where S# denotes the push-forward measure on representation paths, 𝑓∶𝒮→𝒳is the stochastic decoder, and 𝛾∈(0, 1)
is the discount factor. The push_forward_return() function in src/daif/core.py computes this via an atomised categorical
projection over 𝑁atoms support points {𝑧𝑖}𝑁atoms
𝑖=1
spanning [𝑉min, 𝑉max]—the C51 architecture of Bellemare et al. [2017]:
𝑍(𝑠, 𝑎) =
𝑁atoms
∑
𝑖=1
𝑝𝑖(𝑠, 𝑎) ⋅𝛿𝑧𝑖
(20)
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where 𝑝𝑖(𝑠, 𝑎) is the probability assigned to support point 𝑧𝑖and 𝛿is the Dirac delta mass.
In our framework, this
C51 support structure is instantiated via the categorical_return_distribution() function. The distributional Bellman
operator 𝒯𝜋, implemented computationally via the distributional_bellman_operator() multi-step contraction tracking
method, then maps return distributions forward:
𝒯𝜋𝑍(𝑠, 𝑎)
𝑑= 𝑅(𝑠, 𝑎) + 𝛾𝑍(𝑆′, 𝐴′),
𝑆′ ∼𝑃(⋅|𝑠, 𝑎), 𝐴′ ∼𝜋(⋅|𝑆′)
(21)
Contraction (B1). By Bellemare et al. [2017, Theorem 1], 𝒯𝜋is a 𝛾-contraction in the supremum 𝑝-Wasserstein metric
̄
𝑊𝑝on the space of return distributions — i.e.
̄
𝑊𝑝(𝒯𝜋𝑍1, 𝒯𝜋𝑍2) ≤𝛾
̄
𝑊𝑝(𝑍1, 𝑍2) for any 𝑝∈[1, ∞). Hence the fixed
point 𝑍⋆is unique and iterates 𝒯𝜋𝑛𝑍0 converge to 𝑍⋆at geometric rate 𝛾𝑛; this underwrites the convergence tracked by
distributional_bellman_operator() in src/daif/core.py.
Mean-field bound (B5). The belief-weighted step
̄𝑧= 𝑞⊤z used by push_forward_return() replaces the exact per-state
operator with a single scalar collapse. Assume bounded rewards ‖𝑅‖∞≤𝑅max and an entropy budget 𝐻[𝑞] < 𝜀nats.
Then the approximation error of this collapse is bounded by 𝛾𝑅max 𝜀in the induced 𝑊1 metric (units of return), since the
worst-case mass reallocation between roles is at most the entropy of 𝑞, and each reallocated unit of probability contributes
at most 𝛾𝑅max to 𝑊1. The approximation is therefore exact in the sharp-posterior limit 𝑞→𝛿𝑠⋆(𝜀→0) and degrades at
most linearly in 𝐻[𝑞] as the posterior diffuses.
For case-theoretic reasoning, DAIF implies a computational architecture in which case assignment operates distributionally
at every level: the agent maintains not a single case diagram but a distribution over case diagrams, weighted by their
posterior probability given the observed linguistic evidence. This distributional perspective on case assignment aligns
naturally with the graded proto-role structure of Dowty [1991]: a noun phrase distributes probability mass across case
roles, with the distribution sharpening as more evidence accumulates.
16.2
Quantile Temporal Difference and Implicit Quantile Networks
Rather than representing the return distribution as a fixed categorical support (C51), the Quantile Regression DQN (QR-
DQN) approach represents it as a uniform mixture of 𝑁Dirac masses, one at each midpoint quantile level 𝜏𝑗= (2𝑗−1)/(2𝑁)
for 𝑗= 1, … , 𝑁— equivalently 𝜏𝑖= (2𝑖+1)/(2𝑁) for the 0-indexed form 𝑖= 0, … , 𝑁−1 used in src/daif/quantile.py:66.
The quantile_td_update() function implements the Huber quantile loss:
ℒQR(𝜃) =
1
𝑁𝑁′
𝑁
∑
𝑖=1
𝑁′
∑
𝑗=1
𝜌𝜅
𝜏𝑖(𝛿𝑖𝑗)
(22)
where 𝛿𝑖𝑗= 𝑟+ 𝛾𝑧′
𝑗−𝑧𝑖is the temporal-difference error, and the Huber quantile loss 𝜌𝜅
𝜏is:
𝜌𝜅
𝜏(𝑢) = |𝜏−1[𝑢< 0]| ⋅ℒ𝜅(𝑢),
ℒ𝜅(𝑢) = {
1
2𝑢2
|𝑢| ≤𝜅
𝜅(|𝑢| −𝜅
2 )
otherwise
(23)
The Implicit Quantile Network extension (implicit_quantile_network_update() in src/daif/quantile.py) samples
quantile levels 𝜏∼𝑈[0, 1] at inference time, enabling risk-distorted policy selection via four modes:
Table 8: IQN risk distortion modes, formulas, and semantic roles in case-assignment parsing.
Mode
Distortion 𝜓IQN(𝜏)
Semantic Role
neutral
𝜓IQN(𝜏) = 𝜏
Standard expected-value
maximisation
optimistic
𝜓IQN(𝜏) = 𝜏1/𝜂IQN
Prefers high-return tails; suits
exploratory parsers
pessimistic
𝜓IQN(𝜏) = 1 −(1 −𝜏)1/𝜂IQN
Over-weights low-return tails;
conservative case disambiguation
CVaR
𝜓IQN(𝜏) = 𝜏⋅𝛼CVaR
Linear tail compression
(implementation default
𝛼CVaR = 0.25); risk-averse
comprehension
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The four modes are implemented exactly in implicit_quantile_network_update() (src/daif/quantile.py) with fixed 𝜂IQN =
0.71. Convention note. With 𝜂= 0.71 we have 1/𝜂≈1.408 > 1, so 𝜏1/𝜂< 𝜏and 1 −(1 −𝜏)1/𝜂> 𝜏on (0, 1). In
our implementation the distorted level 𝜏′ is multiplied directly into the asymmetric Huber weight (i.e. the loss has its
positive-error weight scaled by 𝜏′ and its negative-error weight by 1 −𝜏′).
Under that weight-level convention, the
“optimistic” mode shrinks positive-error updates (making the agent slower to revise upward on good news — preference
for the status quo upper tail) and the “pessimistic” mode inflates positive-error updates (making the agent track the lower
tail more aggressively). Readers cross-referencing Dabney et al.’s [2018] sampling-level distortion (where the distortion is
applied to the sampling density of 𝜏) should note that our mode names follow the weight-level semantic rather than the
sampling-level one; the underlying mathematical formulas match across the two conventions but their qualitative labels
are mirror-images.
Consistency (B2). Under i.i.d. TD samples, the empirical minimiser of the quantile Huber loss 𝜌𝜅
𝜏converges to the
true 𝜏-quantile at rate 𝑂(𝑁−1/2); see Dabney et al. [2018, Theorem 2]. The Huber threshold 𝜅trades off robustness to
outliers against bias near 𝛿= 0, and our default 𝜅= 1 recovers Dabney et al.’s standard setting.
The wasserstein_return_distance() function computes discrete-quantile approximations of both 𝑊1 (absolute area be-
tween CDFs) and 𝑊2 (root-mean-square area) distances between return distributions. Approximation error (B4). For
midpoint quantiles 𝜏𝑖= (2𝑖−1)/(2𝑁) and a Lipschitz quantile function, the estimator is O(𝑁−2) consistent with the
continuous integral 𝑊𝑝= (∫
1
0 |𝐹−1
𝑎(𝜏) −𝐹−1
𝑏(𝜏)|𝑝𝑑𝜏)1/𝑝; this is the canonical discretisation used in quantile-regression RL
(Dabney et al. [2018]) and is documented explicitly in the docstring of wasserstein_return_distance().
16.3
Variational Message Passing and Bethe Free Energy
The orchestrator of the DAIF inference cycle is the distributional_case_assignment() function in src/daif/inference.py.
It wraps the push-forward mapping and Bayesian update routines to return a DAIFResult object encapsulating the free-
energy trajectory.
During this loop, the variational_message_passing() sub-function implements iterative categorical
belief refinement over the case-role posterior 𝑞(c ∣o). Starting from a uniform prior over 𝐾case roles, each sweep applies
the precision-weighted exponential update:
𝑞(𝑡+1)(𝑐𝑘) ∝𝑞(𝑡)(𝑐𝑘) ⋅exp(𝑤𝑘⋅𝑜𝑘)
(24)
where 𝑤𝑘∈[0, 1] is the enriched morphism weight for role 𝑘read off from the case category of section 11 (acting as
a precision on the update) and 𝑜𝑘= log 𝑝(𝑜∣𝑐= 𝑘) is the log-likelihood of the observation under role 𝑘. After each
update the distribution is renormalised via softmax. The algorithm returns posterior probabilities and posterior precisions
Λpost = Λprior + Λlik. Convergence is declared when ‖𝑞(𝑡+1) −𝑞(𝑡)‖1 < 𝜖= 10−6.
Convergence (B3). Proposition. For the single-observation categorical factor graph used in case assignment, the softmax
update (Equation 24) is a strict KL-contraction and possesses a unique fixed point 𝑞⋆; the 𝐿1 threshold 𝜀is reached in
𝑂(log(1/𝜀)) sweeps. Sketch. The unnormalised multiplicative update followed by normalisation is the gradient step of a
strictly convex problem (minimising the Bethe free energy on a tree-structured factor graph); Yedidia, Freeman and Weiss
[2005, §III–IV] show that in the tree (cycle-free) case, belief propagation exactly minimises the Bethe FE, so the iteration
has a unique global minimiser. The contraction rate is bounded by the ratio of likelihood precision to prior precision. For
factor graphs with loops our implementation inherits the weaker “approximate fixed point” guarantee of loopy BP; the
current linguistic application uses a single observation factor per word, which is tree-structured.
The Bethe free energy provides a tractable lower-bound approximation to the variational free energy. In the mean-field
specialisation—where each observation constitutes an independent factor with uniform variable degrees—the Bethe FE
reduces to:
𝐹Bethe[q] = −∑
𝑘
𝑞(𝑐𝑘) log 𝑝(𝑐𝑘)
⏟⏟⏟⏟⏟⏟⏟⏟⏟
prior fit
+ ∑
𝑘
𝑞(𝑐𝑘) log 𝑞(𝑐𝑘)
⏟⏟⏟⏟⏟⏟⏟
belief entropy
−∑
𝑡
∑
𝑘
𝑞(𝑐𝑘) log 𝑝(𝑜𝑡∣𝑐𝑘)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
likelihood
(25)
The bethe_free_energy() function in src/daif/inference.py implements the full factor-graph Bethe FE (Yedidia et
al. 2001): 𝐹Bethe = ∑𝛼KL(𝑏𝛼‖𝑓𝛼) −∑𝑖(𝑑𝑖−1)𝐻(𝑏𝑖), where 𝑏𝛼are factor beliefs, 𝑓𝛼are factor potentials, and 𝑑𝑖is
the degree of variable 𝑖in the factor graph. The equation above is the tractable mean-field limit presented for clarity. The
expected information gain (expected_information_gain()) measures the mutual information between observations and
case-role assignments—the expected KL divergence between posterior and prior, weighted by the marginal likelihood of
each candidate observation:
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## Page 62

EIG(𝑜∗) = ∑
𝑜∗
𝑝(𝑜∗) 𝐷KL(𝑞(c ∣𝑜∗) ‖ 𝑝(c))
(26)
This mutual-information formulation properly accounts for the probability of each observation, providing a principled
measure of how much a candidate word would reduce uncertainty about the current case-role assignment on average.
Figure 24 visualises the Bethe free energy landscape over six sequential word arrivals in the sentence “Der Hund jagt
die Katze schnell”: variational free energy decreases with each belief update cycle, and the KL decomposition shows the
balance between model complexity (𝐷KL(𝑞‖𝑝)) and data fit (−𝔼𝑞[log 𝑝(𝑜|𝑠)]).
Figure 24: Variational free energy decreases during distributional case assignment. Left: measured 𝐹(𝑡) values read di-
rectly from DAIFResult.fe_trajectory (blue markers) with a smoothed reference envelope (grey dashed) and vertical dashed
lines marking word arrivals. Right: the real decomposition 𝐹(𝑡) = 𝐷KL(𝑞(𝑡)
posterior‖𝑞(𝑡)
pushed) −𝔼𝑞(𝑡)[log 𝑝(𝑜|𝑠)], plotted from
DAIFResult.diagnostics["kl_trajectory"] and diagnostics["loglik_trajectory"] (not a schematic). Generated program-
matically by src.visualization.daif_plots.plot_free_energy_convergence() from make_free_energy_convergence_data()
in src/cognitive/figure_data.py.
Figure 25 illustrates the full belief trajectory: starting from uniform prior over NOM, ACC, DAT, INS, the posterior
sharpens monotonically as each morphologically marked word supplies evidence. The determiner Der signals nominative;
the transitive verb jagt activates a valency frame expecting an accusative object; the accusative article die confirms
NOM=Hund, ACC=Katze. Entropy 𝐻[𝑞] (centre panel) drops steeply at the second word—the most informative item in
this parse.
16.4
Policy Selection and Expected Free Energy
Active inference selects actions (here: next-word predictions or syntactic commitments) by minimising expected free energy
𝐺(𝜋). G_policy() in src/daif/policy.py implements the four-term decomposition used throughout this paper:
𝐺(𝜋) = −𝔼𝑞(𝑠)[log 𝑝(𝑜∣𝑠, 𝜋)]
⏟⏟⏟⏟⏟⏟⏟⏟⏟
ambiguity (𝒜)
−𝔼𝑞(𝑠)[ 𝐻[𝑝(𝑠∣𝑜)] ]
⏟⏟⏟⏟⏟⏟⏟
epistemic value (ℰ)
−𝛾
𝔼𝑞(𝑠)[𝑣(𝑠, 𝜋)]
⏟⏟⏟⏟⏟
pragmatic value (𝒫)
+ 𝛽risk Var𝑍[𝑅(𝜋)]
⏟⏟⏟⏟⏟
risk (ℛ)
(27)
Each term is signed so that minimising 𝐺(𝜋) simultaneously minimises ambiguity, maximises expected information gain,
maximises expected pragmatic utility, and penalises high-variance return distributions. The weights are 𝛾> 0 (pragmatic
gain) and 𝛽risk ≥0 (risk sensitivity, using the distributional return variance produced by push_forward_return()).
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## Page 63

Figure 25: Case-role posterior sharpens from uniform prior as German morphology supplies evidence. DAIF belief tra-
jectory during sequential disambiguation of “Der Hund jagt die Katze schnell.” Top: stacked area showing P(NOM),
P(ACC), P(DAT), P(INS) evolution over six words with German morphological glosses. Middle: entropy 𝐻[𝑞] with
annotated steepest drop marking the most informative word.
Bottom:
uncertainty fan around the dominant-role
probability, constructed as a simple proxy Δ = 1 −max𝑘𝑃(𝑐𝑘) scaled by fixed percentile multipliers; this is a visual
surrogate, not a 51-quantile decomposition of the push-forward return distribution. Generated programmatically from
src.visualization.daif_plots.plot_belief_trajectory().
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## Page 64

Collapse identity (derivation). Setting 𝑣(𝑠, 𝜋) = log 𝑝(𝑜goal ∣𝑠, 𝜋) and 𝛽risk = 0 reduces Eq. 27 to
𝐺(𝜋) = −𝔼𝑞(𝑠)[log 𝑝(𝑜∣𝑠, 𝜋)] −𝔼𝑞(𝑠)[𝐻[𝑝(𝑠∣𝑜)]] −𝛾𝔼𝑞(𝑠)[log 𝑝(𝑜goal ∣𝑠, 𝜋)]
= −𝔼𝑞(𝑠)[log 𝑝(𝑜∣𝑠, 𝜋)]
⏟⏟⏟⏟⏟⏟⏟⏟⏟
expected surprise
+ 𝐷KL(𝑞(𝑠∣𝜋) ‖ 𝑝(𝑠))
⏟⏟⏟⏟⏟⏟⏟⏟⏟
epistemic value
−𝛾𝔼𝑞(𝑠)[log 𝑝(𝑜goal ∣𝑠, 𝜋)],
where the second equality uses the standard active-inference identity −𝔼𝑞(𝑠)[𝐻[𝑝(𝑠∣𝑜)]] = 𝐷KL(𝑞(𝑠∣𝜋) ‖ 𝑝(𝑠)) + const
(Friston et al. [2017], Eq. 5), valid when the posterior is close to the generative prior so that the 𝐻[𝑞(𝑠∣𝜋)] term absorbs
into the constant offset that cancels across policies. This is the canonical three-term form of Friston et al., confirming
that our decomposition is a conservative generalisation rather than a departure.
Policy selection follows a Boltzmann (softmax) distribution over negative EFE:
𝑃(𝜋) =
exp(−𝛼pol ⋅𝐺(𝜋))
∑𝜋′ exp(−𝛼pol ⋅𝐺(𝜋′))
(28)
where 𝛼pol > 0 is the inverse temperature (policy softmax), distinct from the CVaR tail level 𝛼CVaR in the IQN table above.
softmax_policy_selection() implements this across an array of candidate policies; distributional_epistemic_value()
returns the epistemic component alone, enabling decomposition of the policy gradient.
For case-theoretic reasoning, this means the agent selects the case assignment (NOM/ACC/DAT/etc.) that simultaneously
minimises surprise (fits the observed morphological evidence), maximises information gain (resolves ambiguity fastest),
and respects risk sensitivity (avoids high-variance parses in pessimistic mode). This provides a principled, Bayes-optimal
account of why certain parse strategies are preferred cross-linguistically—they minimise expected free energy under the
agent’s generative model.
16.5
ERP Amplitude Profiles from Distributional Prediction Error
The DAIF prediction module (src/daif/prediction.py) provides two complementary prediction error measures, each
appropriate for different levels of the distributional hierarchy:
Scalar DPE (distributional_prediction_error()): For point-prediction scenarios where the expected case role is known,
the precision-weighted surprisal provides a computationally efficient scalar measure:
DPEscalar(𝑐, 𝑞) = 𝑤𝑓⋅(−log 𝑞[𝑐expected])
(29)
Wasserstein DPE (wasserstein_prediction_error()): For full distributional comparisons between predicted and ob-
served return distributions, the precision-weighted Wasserstein-1 distance provides the distributional measure:
DPE(𝑜, 𝑞) = 𝑤𝑓⋅𝑊1(𝑍predicted, 𝑍observed)
(30)
where 𝑤𝑓= 𝒞(𝐴, 𝐵) ∈[0, 1] is the enriched morphism weight (precision) of the violated case morphism (matching
Equation 18 in section 15) and 𝑊1 is the Wasserstein-1 distance.
Reader’s guide to the four DPE variants. The DAIF framework uses four distinct but related prediction-error
measures, easily confused because they share the name “DPE”:
1. DPEscalar (Equation 29, distributional_prediction_error()) — a point-belief surprisal: precision-weighted cross-
entropy on the currently-expected role. Used when the grammatically expected role is known.
2. DPE (full Wasserstein; Equation 30, wasserstein_prediction_error()) — a full distributional mismatch between
predicted and observed return distributions. Used for graded / distributional comparisons.
3. DPEsemantic (Equation 33 below, input to n400_from_return_distribution()) — the first moment of the distributional
mismatch, i.e. the absolute mean-return shift; tracks the N400 heuristic component.
4. DPEstructural (Equation 34 below, input to p600_from_precision_update()) — the full distributional mismatch
𝑊1(𝑍pred, 𝑍obs); tracks the P600 discrepancy component and coincides with DPE in item 2.
Items (3) and (4) decompose (2) into a mean-shift and a full-distribution signal; item (1) is a simpler scalar surrogate for
point-belief use cases.
ERP derivation from the Free Energy Principle (B6). The two ERP formulas below are derived from a free-
energy decomposition rather than posited empirically. A new word arriving at the case-assignment layer induces a change
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## Page 65

in variational free energy Δ𝐹= 𝐹post −𝐹prior.
Using 𝐹= 𝐷KL(𝑞‖ 𝑝(𝑠)) −𝔼𝑞[log 𝑝(𝑜|𝑠)] (the variational free-energy
decomposition introduced in section 14) and splitting 𝑞→𝑞′ into a posterior-mean shift and a precision-sharpening
component, a first-order expansion gives
Δ𝐹=
−(𝔼𝑞′[𝑍obs] −𝔼𝑞[𝑍pred])
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
mean-return shift ≈DPEsemantic
+
1
2 ΔΛ 𝜎2
𝑍
⏟
precision update ≈ΔΛ⋅DPEstructural
+ 𝑂(‖Δ𝑞‖2),
where 𝜎2
𝑍is the return-distribution variance (proxied at first order by 𝑊1(𝑍pred, 𝑍obs)). The mean-return shift is the
heuristic, expectation-dominated component that Kuperberg and Jaeger [2016] and Li and Futrell [2023] associate with
the N400; the precision-update term is the discrepancy, structure-update component that Rabovsky et al. [2018] and
Li and Futrell [2024] associate with the P600.
Severity gating 𝑆violation ∈{0, 0.5, 1.0} modulates both components
multiplicatively, reflecting the attenuation of prediction-error signals when the violation is only mild. The precision on
the semantic component is 𝑤𝑐(the enriched morphism weight), and the amplitude calibration 𝑠on the P600 converts the
dimensionless free-energy increment into 𝜇V. This yields the severity-gated decomposition:
N400(𝑐) = −DPEsemantic ⋅𝑤𝑐⋅𝑆violation
(31)
P600(𝑐) = 𝑠⋅ΔΛ ⋅DPEstructural ⋅𝑆violation
(32)
where 𝑆violation ∈{0, 0.5, 1.0} encodes violation severity (congruent / mild / strong), 𝑤𝑐is the enriched weight of the case
morphism, ΔΛ = max(0, Λpost −Λprior) is the precision-update magnitude reflecting the structural reanalysis cost, and
𝑠> 0 is a dimensionless amplitude-calibration constant (default 𝑠= 1 in p600_from_precision_update()). The leading
minus sign in Eq. 31 follows electrophysiological convention: larger semantic surprise yields a more negative deflection at
the N400 latency, consistent with the sign produced by n400_from_return_distribution() in src/daif/prediction.py. The
two DPE flavours appearing here are defined formally by:
DPEsemantic = ∣𝔼[𝑍pred] −𝔼[𝑍obs]∣
(33)
DPEstructural = 𝑊1(𝑍pred, 𝑍obs)
(34)
i.e. DPEsemantic is the absolute mean-return mismatch (the heuristic component in the Li–Futrell decomposition, tracking
the N400) and DPEstructural is the full distributional Wasserstein-1 mismatch (the discrepancy component, tracking the
P600). Computationally, N400 amplitudes are extracted via n400_from_return_distribution() (which computes mean-
return mismatch scaled by precision and severity), while P600 amplitudes use p600_from_precision_update() (which com-
putes precision-update magnitude scaled by DPE and severity). This dual decomposition directly mirrors the empirical
finding of Li and Futrell [2023; 2024], who show that surprisal decomposes into a heuristic component tracking N400
and a discrepancy component tracking P600—precisely the semantic vs. structural split captured by DPEsemantic and
DPEstructural above.
Our framework explicitly accommodates Rabovsky et al.’s [2018] finding that the N400 reflects a probabilistic Bayesian
belief update, extending it by formally equipping the N400 semantic surprise as a distributional prediction-error over
explicit case boundaries. (The complementary neurobiological-timing question raised by the ROSE model is addressed in
the Limitations subsection below, after the metrics discussion and before the CEREBRUM integration.)
Ultimately, the erp_amplitude_profile() master function aggregates these distinct computations into a complete ERPPro-
file dataclass containing:
• N400 amplitude (𝜇V) and peak latency (ms) for each case role
• P600 amplitude (𝜇V) and peak latency (ms) for each case role
• Time-series waveforms sampled at 1 kHz over a 1100 ms epoch (−200 to +900 ms)
Figure 26 demonstrates predicted ERP amplitudes across all eight case roles under three violation conditions. A key
result: because VOC carries the lowest enriched precision weight in the illustration (𝑤= 0.10, the smallest among all
eight roles), VOC→NOM is the most structurally inadmissible transition and elicits the largest P600; while GEN, with
moderate precision weight (𝑤= 0.70), elicits a pronounced N400 but attenuated P600.
16.6
Convergence Diagnostics and Distributional Metrics
The src/daif/metrics.py module provides four diagnostic tools for verifying DAIF model behaviour:
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## Page 66

Figure 26: Distributional prediction error predicts graded N400/P600 amplitudes across all eight case roles. Left: illustra-
tive Gaussian ERP waveforms for three violation conditions — congruent (NOM→NOM), mild (ACC→NOM), and strong
(VOC→NOM) — with fixed template amplitudes (−1.5/3.0, −4.0/3.0, −7.0/6.0 𝜇V N400/P600) and peaks at 380 ms /
600 ms (matching DEFAULT_N400_PEAK_MS / DEFAULT_P600_PEAK_MS in src/daif/prediction.py), shown to depict the mecha-
nism of Equation 31–Equation 32 rather than a calibrated simulation. Middle: scatter of enriched weight 𝑤versus scalar
DPE (Equation 29) for all eight case roles, computed via distributional_prediction_error() in src/daif/prediction.py.
Right: real DAIF-predicted magnitudes — mean N400 magnitude via n400_from_return_distribution() and mean P600
via p600_from_precision_update() across the eight roles (make_erp_prediction_data() in src/cognitive/figure_data.py) —
alongside literature-typical amplitude ranges from Kutas & Federmeier [2011] (N400 magnitude 3–5 𝜇V, P600 5–8 𝜇V,
shown with error bars).
Note that the model predictions are on a dimensionless DPE scale (units of log-probability
× enriched weight), whereas the literature values are calibrated in 𝜇V; the comparison is therefore qualitative (rel-
ative ordering and graded response to precision) rather than a numerical match.
A future calibration step would
require fitting a per-subject 𝜇V-per-nat scaling constant to empirical ERP data.
Generated programmatically from
src.visualization.daif_plots.plot_erp_predictions().
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Convergence diagnostics (convergence_diagnostics()) assess whether a free-energy trajectory {𝐹(𝑡)}𝑇
𝑡=0 is well-behaved.
Given the free-energy sequence produced by VMP, the function returns a dict with the eight fields below (keys match the
Python return value exactly):
Table 9: Convergence diagnostic metrics for DAIF free-energy trajectories. Keys match convergence_diagnostics() in
src/daif/metrics.py one-to-one.
Key
Formula
Interpretation
monotone
∀𝑡∶𝐹(𝑡+1) ≤𝐹(𝑡)
FE decreasing at every step
total_reduction
𝐹(0) −𝐹(𝑇)
Total free energy minimised (absolute)
relative_reduction_pct
100 ⋅(𝐹(0) −𝐹(𝑇))/|𝐹(0)|
Fraction of initial FE eliminated, %
n_iterations
𝑇+ 1
Number of iterations in the trajectory
converged
|𝐹(𝑇) −𝐹(𝑇−1)| < 0.01 ⋅(𝐹max −𝐹min)
Reached stable minimum (within 1 %
of range)
fe_range
(𝐹min, 𝐹max)
Absolute FE bounds across the
trajectory
mean_step_size
1
𝑇∑𝑡|𝐹(𝑡+1) −𝐹(𝑡)|
Average per-iteration FE change
final_delta
|𝐹(𝑇) −𝐹(𝑇−1)|
Final step size at convergence or
timeout
Distributional KL divergence (distributional_kl()) computes the KL divergence between two discrete return distri-
butions:
𝐷KL(𝑃‖𝑄) = ∑
𝑖
𝑃(𝑧𝑖) log
𝑃(𝑧𝑖)
𝑄(𝑧𝑖) + 𝜖
(35)
with 𝜖= 10−10 for numerical stability. Verified properties: 𝐷KL(𝑃‖𝑄) ≥0 (Gibbs’ inequality), 𝐷KL(𝑃‖𝑃) = 0, asymmetry
𝐷KL(𝑃‖𝑄) ≠𝐷KL(𝑄‖𝑃) in general.
Quantile coverage (quantile_coverage()) measures calibration error—the mean absolute deviation between nominal
quantile levels and empirical coverage frequencies:
CE = 1
𝑁
𝑁
∑
𝑖=1
∣𝜏𝑖−
̂𝐹(𝑧𝜏𝑖)∣
(36)
A perfectly calibrated distributional model achieves CE = 0; the DAIF implementation in src/daif/metrics.py (tested in
test_daif_metrics.py) yields CE < 0.05 on evaluated cases (implemented and tested).
Return distribution entropy (return_distribution_entropy()) quantifies uncertainty in the distributional belief:
𝐻[𝑍] = −
𝑁bins
∑
𝑖=1
𝑝𝑖log 𝑝𝑖
(37)
where the 𝑝𝑖are obtained by discretising the quantile-parameterised return onto 𝑁bins equal-width bins (default 𝑁bins = 50
in src/daif/metrics.py) with additive 𝜖-smoothing 𝑝𝑖←(𝑐𝑖+ 𝜖)/(∑𝑗𝑐𝑗+ 𝑁bins 𝜖) to keep the estimator finite when bins
are empty. The estimator is consistent at the usual 𝑂(1/𝑁bins) discretisation rate as 𝑁bins →∞, and satisfies 0 ≤𝐻[𝑍] ≤
log 𝑁bins by direct enumeration. This links to the belief trajectory in Figure 25: entropy decreases monotonically as the
distributional belief sharpens, providing a scalar summary of parse certainty.
16.6.1
Two Supporting Utilities Exposed by src/daif/
Two further public symbols in the DAIF subpackage support the machinery above without requiring a separate equation:
• distributional_epistemic_value() (src/daif/policy.py) measures the information-theoretic value of resolving uncer-
tainty in the return distribution via the differential-entropy form EVdist = 1
2 log(Var[𝑍]/𝜎2
ref). In the risk-sensitive
regime (𝛽risk > 0 in Equation 27) this function decomposes the risk term Var𝑍[𝑅(𝜋)] into a positive “exploration
premium” when the current return distribution is more dispersed than the reference.
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• categorical_return_distribution()
(src/daif/core.py)
realises
the
C51
projection
operator
Φ∶𝒫(ℝ)
→
𝒫({𝑧1, … , 𝑧𝑁atoms}) that maps a quantile-parameterised return onto the fixed atomic support of Eq. 20. The operator
is non-expansive in 𝑊1: by construction Φ redistributes each quantile mass across at most two adjacent atoms using
barycentric weights summing to 1, so for any two return distributions 𝑃, 𝑄one has 𝑊1(Φ𝑃, Φ𝑄) ≤𝑊1(𝑃, 𝑄) —
hence Φ is 1-Lipschitz (and in fact strictly contractive whenever 𝑃and 𝑄place mass strictly between atoms). This
is the reason C51/DAIF interoperability preserves contraction bounds from subsection 16.1.
16.6.2
Dimensional Analysis
The quantities appearing in this section carry the following units:
Quantity
Symbol
Units
Variational free energy
𝐹
nats
Return (discounted cumulative
reward)
𝑍(𝑠, 𝑎)
return units (e.g. dimensionless
log-probability)
Wasserstein distance on returns
𝑊𝑝(𝑍𝑎, 𝑍𝑏)
return units
DPEsemantic
—
return units (= |Δ𝔼[𝑍]|)
DPEstructural
—
return units (= 𝑊1)
Enriched weight (precision)
𝑤𝑐
dimensionless, ∈[0, 1]
Precision-update magnitude
ΔΛ
dimensionless (weight difference)
Severity gating
𝑆violation
dimensionless, ∈{0, 0.5, 1}
N400 amplitude (Equation 31)
—
return units (not 𝜇V until
calibrated)
P600 amplitude (Equation 32)
—
return units × dimensionless 𝑠(not
𝜇V until calibrated)
Since 𝑍in our implementation is built from log-likelihood proxies (safe_log_lik in src/daif/inference.py), the return
is effectively in nats, and every quantity derived from 𝑍is therefore dimensionally a free-energy-like scalar. A future
ERP-calibration step would convert “nats” to “𝜇V” via a per-subject scaling constant, as noted in the Figure 26 caption.
16.7
Limitations and Neurobiological Scope
Four limitations of the current DAIF implementation are recorded explicitly for future work:
1. Mean-field approximation (cost/accuracy). push_forward_return() maintains one belief-weighted return dis-
tribution rather than per-state distributions 𝑍(𝑠), reducing memory from 𝒪(𝑛⋅𝑁atoms) to 𝒪(𝑛). By the mean-field
bound proved later in this section (the dominated-convergence argument under “B5” in the contraction analysis) the
approximation error is at most 𝛾⋅𝑅max ⋅𝐻[𝑞] in 𝑊1, where 𝑅max is the sup-norm of the reward vector. For sharp
posteriors (𝐻[𝑞] ≲0.1 nats at 𝛾= 0.99 and unit-scale rewards) the error is below 0.1 return-unit — well below the
within-subject noise floor on ERP measurements; it degrades linearly as the posterior diffuses.
2. Enriched-categorical unification is a conjecture. The three “distributional” tracks — distributional semantics
(section 8), distributional RL (this section), and active-inference posteriors — are implemented and empirically
correspond via [0, 1]-enriched hom-values, but a categorical proof that they share a common enriched base remains
open. The conjecture is stated in the opening of this section and is not used as a load-bearing claim elsewhere in
the paper.
3. Empirical validation is narrow. Our case-assignment demonstrations use a single German transitive sentence
(“Der Hund jagt die Katze schnell”). Cross-linguistic and cross-register validation is left to future work; the hooks
in make_daif_belief_trajectory_data() make adding new sentences a single-function change.
A Russian or Ser-
bian/BCS sentence — say Sobaka kusaet čeloveka “the dog bites the man” — would be a particularly clean DAIF
stress-test, since the unambiguous case suffixes (-a NOM.SG.F vs -a ACC.SG.M.ANIM after stem hardening) de-
liver an information-theoretically sharper drop in 𝐻[𝑞] at the case-marked noun than the German example, where
word-final case markers compete with positional disambiguation and gender / number ambiguity in the determiner
system.
4. Phase–amplitude coupling (PAC) latency gap. DAIF predicts ERP amplitudes from distributional prediction
errors but does not predict component latencies.
The ROSE model [Murphy, 2023] argues that the structural-
discrepancy (P600-analogous slow-phase) signal must establish a geometric “mesoscopic protectorate” before se-
mantic surprise (N400-analogous rapid gamma binding) can be fully constrained. A principled treatment of this
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cross-frequency-coupling delay would require an explicit timing parameter at the CEREBRUM layer (subsection 16.8)
— the present implementation keeps N400/P600 latencies as fixed Gaussian peaks at 380 ms and 600 ms respectively
(see DEFAULT_N400_PEAK_MS and DEFAULT_P600_PEAK_MS in src/daif/prediction.py).
16.8
CEREBRUM: Eight Cases as Functional Specializations
The preceding six contributions—push-forward returns, quantile TD for case precision, VMP message passing, epistemic
policy selection, ERP convergence profiles, and Bethe free-energy decomposition—define a distributional active inference
layer for sentence processing. CEREBRUM translates this layer into a complete computational architecture by assigning
each of the eight traditional cases a functional role within the inference engine.
16.8.1
Architecture and Design Principles
The CEREBRUM framework [Friedman and Active Inference Institute, 2024]—Case-Enabled Reasoning Engine with
Bayesian Representations for Unified Modeling—provides a computational architecture that implements the categorical
case framework within an active inference engine. CEREBRUM instantiates the view of Vasil et al. [2020] that human
communication is itself active inference: a process of jointly constructing and refining generative models of shared relational
structure.
CEREBRUM’s key design principles (Table 10):
Table 10: CEREBRUM design principles: conceptual commitments and their implementation in the reasoning engine.
Principle
Implementation
Cases as functional roles
Model components carry case markings that determine
their computational role in the inference cycle
Morphisms as message passing
Grammatical relations are implemented as
message-passing channels between components
Enriched weights as precision
The [0, 1] weights on morphisms correspond to precision
parameters in the variational inference scheme
Alignment as model selection
Different alignment types correspond to different
generative model architectures, selected by Bayesian model
comparison
Diagrams as generative models
Commutative diagrams serve as the structural
specification of the generative model
DAIF as distributional layer
The src/daif/ subpackage provides the distributional RL
layer: full return distributions replace point estimates
throughout the generative cycle
16.8.2
Case Roles as Functional Specializations in CEREBRUM
CEREBRUM deploys the eight traditional cases as functional specializations, each with a DAIF-level extension (Table 11):
Case
CEREBRUM Function
Active Inference Role
DAIF Extension
NOM
Primary driver / agent
Source of action policies
Softmax policy over 𝐺(𝜋); highest
epistemic value
ACC
Primary target / patient
Object of predictions
Predicted distribution 𝑍acc; error-
driven update
GEN
Source / possessor
Provider of priors
Prior return distribution 𝑝(𝑍)
DAT
Recipient / goal
Target of information transfer
EIG maximised toward DAT state
INS
Instrument / means
Tool for state transformation
IQN risk distortion (neutral mode)
LOC
Context / environment
Markov blanket boundary
Bethe FE boundary conditions
ABL
Origin / cause
Source of causal influence
Push-forward
source
measure
ℙ𝑥0,𝑎0
VOC
Addressee
Pragmatic pointer
Lowest epistemic weight;
largest
P600 on violation
Table 11: CEREBRUM case roles as functional specializations with DAIF distributional extensions.
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17
Topological Quantum Neural Networks and ZX-Calculus: From Spin-
Networks to Categorical Case Diagrams
Where we are in the argument. The preceding sections have established that categorical string diagrams—from
DisCoCat’s pregroup derivations (section 8) through enriched hom-values (section 11) to topos-theoretic transfer (sec-
tion 13)—provide a unified diagrammatic language for case-theoretic reasoning (theoretical synthesis). This section
extends the framework with a literature bridge to topological quantum neural networks (TQNNs), the ZX-calculus, and
sheaf-theoretic quantum semantic communication (theoretical bridge to prior art).
The src/quantum/quantum_case.py module implements a concrete POVM-based measurement model for case roles (tested
in the project suite; see section 18). The broader TQNN, ZX-rewrite, and lambeq compilation claims remain literature
connections and proposed extensions rather than full local implementations. The central observation is that the same
monoidal-categorical architecture underlies both classical DisCoCat diagrams and quantum processes.
Position relative to prior work. Each ingredient we draw on has independent prior art. TQNNs as spin-network
computations are due to Fields, Marcianò, and collaborators [Fields et al., 2022, 2025]; the ZX-calculus and its circuit-
extraction theory are due to Kissinger, van de Wetering, Coecke, and the Picturing-Quantum-Processes school [Kissinger
and van de Wetering, 2020, Coecke and Kissinger, 2017]; and quantum natural language processing on present-day hardware
has been demonstrated by the lambeq pipeline [Lorenz et al., 2021] and the Quantinuum quantum-NLP programme. What
is new in this section is none of those constructions individually; rather it is the identification that the same monoidal-
functorial scaffolding can carry a case-theoretic payload — in particular, that the POVM family {𝐸𝑐} of section 18 can
be read both as a Born-rule case-assignment device and as the pointer-basis selected by a quantum reference frame in a
TQNN — and the consequent claim that case assignment, ZX-rewrite verification, and DAIF-style distributional inference
(section 16) are three views of one diagrammatic process. This is a bridge result, not a hardware claim: nothing in this
section asserts that fault-tolerant quantum hardware is required, and the section 16 distributional pipeline runs entirely
on classical numerics.
17.1
QNNs as Spin-Networks
17.1.1
TQFT as the Forward Pass: Reshetikhin–Turaev Invariants Compute Network Amplitudes
Marcianò, Fields, and Glazebrook show that quantum neural networks (QNNs) admit a topological reformulation using
spin-networks [Fields et al., 2022]. Any QNN layer can be represented as a graph whose edges carry representation labels
(spins) and whose vertices carry intertwiners—precisely the data defining a spin-network in a 3-dimensional topological
quantum field theory (TQFT):
“Quantum Neural Networks (QNNs) can be mapped onto spin-networks, with the consequence that the level
of analysis of their operation can be carried out on the side of Topological Quantum Field Theory (TQFT).”
[Fields et al., 2022]
This reformulation has three structural consequences. First, the network becomes a topological diagram—spin-network or
ribbon graph—evaluated by a continuous TQFT functor; edges encode information flow and nodes encode transformation.
Second, the TQFT evaluation assigns boundary Hilbert spaces to the diagram via the Reshetikhin–Turaev and Turaev–
Viro invariants, playing the role of the neural forward pass: quantum amplitudes propagate through the topological
structure. Third, information flow is encoded in the topology of the wiring rather than in any fixed geometric embedding,
giving the architecture inherent robustness to continuous deformation [Fields et al., 2023].
17.1.2
TQNNs Are Universal
Fields and collaborators further demonstrate that TQNNs are universal quantum computers by identifying the Reshetikhin–
Turaev invariant of a TQNN with a Turaev–Viro quantum error-correcting code:
“TQNNs enable universal quantum computation, using the Reshetikhin-Turaev and Turaev-Viro models to
show how TQNNs implement quantum error-correcting codes.” [Fields et al., 2025]
The universality result is established via the concept of an execution trace for a quantum computation, leading to the
representation of TQNNs in terms of the positive geometries provided by amplituhedra—a deep connection between
quantum computation, scattering amplitudes, and topological combinatorics.
17.1.3
QRFs Select the Measurement Basis
Fields and Glazebrook’s work on quantum reference frames (QRFs) and holographic screens provides additional algebraic
structure [Fields and Glazebrook, 2021]. A holographic screen—the information boundary between two interacting quan-
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tum systems—carries a qubit array encoding their interaction. The key insight is that QRFs deployed to identify systems
and select pointer states induce decoherence, breaking the symmetry of the holographic encoding in an observer-relative
way. This symmetry-breaking is precisely the mechanism by which a TQNN “observes” its input: the choice of QRF
determines the basis in which the spin-network is evaluated.
For case-theoretic reasoning, this connects to the grammatical observer problem: a parser or comprehender selecting a case-
assignment frame for a sentence is analogous to deploying a QRF that fixes the pointer basis for a quantum measurement
on a holographic screen.
17.2
ZX-Calculus: Topological String Diagrams Where Graph Rewrites Are Quantum
Proofs
17.2.1
String Diagrams for Quantum Processes
The ZX-calculus provides a diagrammatic language for quantum circuits, representing them as string diagrams in a
symmetric monoidal category of finite-dimensional Hilbert spaces and linear maps [Kissinger and van de Wetering, 2020]:
“The ZX-calculus is a graphical language for reasoning about quantum computations and circuits… it can
represent any linear map, and can be considered a diagrammatically complete generalization of the usual
circuit representation.” [Coecke and Kissinger, 2017]
Three structural features connect ZX to case-theoretic diagrams:
• Diagrams as morphisms: ZX-diagrams are string diagrams in a †-compact closed category. Wires represent objects
(qubits), spiders and boxes represent morphisms, and composition/tensor product correspond to vertical/horizontal
concatenation. Only topology matters, not geometry.
• Deterministic circuit extraction via generalized flow: Kissinger and van de Wetering show that quantum
circuits map to ZX-diagrams, undergo graph-theoretic rewriting, and extract as optimized circuits—with topological
abstraction preserving the invariants needed for optimization [Kissinger and van de Wetering, 2020].
• Category-theoretic semantics: A ZX-diagram’s semantics are determined entirely by how components are wired—
the same compositional principle underlying DisCoCat and the case categories of section 4.
17.2.2
Pregroup Cups and ZX Spiders Are Instances of the Same Compact-Closed Morphism
The structural parallel between pregroup grammar diagrams and ZX-diagrams is not accidental. Both are instances of
the same mathematical object: morphisms in a compact closed monoidal category with functorial semantics into Hilbert
spaces. In DisCoCat, the functor assigns to each grammatical type a vector space and to each derivation a linear map
computing sentence meaning [Coecke et al., 2010]. In ZX, the standard semantics functor assigns to each spider/Hadamard
configuration a linear map in FHilb. The shared categorical architecture means:
1. Pregroup contractions (cups) and ZX spiders are both instances of the same algebraic operation: evaluation
morphisms in a compact closed category.
2. DisCoCat normal forms and ZX simplifications are both applications of the same rewriting theory: equational
reasoning modulo the axioms of a compact closed category.
3. The snake equation (Cap ∘Cup = identity) that grounds all pregroup type reductions (section 8) is a special case
of the spider fusion rule in ZX.
This means that case-theoretic DisCoCat derivations can, in principle, be compiled into ZX circuits and executed on
quantum hardware—a connection already exploited by the lambeq quantum NLP pipeline [Lorenz et al., 2021].
17.3
One Diagram, Three Interpretations: TQNN, ZX, and DisCoCat Share a Monoidal
Functor
17.3.1
A Common Language of Ribbon and Tensor Diagrams
Both ZX-diagrams and TQNNs are topological string diagrams evaluated by monoidal functors into the category of Hilbert
spaces and linear maps. The alignment becomes explicit when stated precisely:
• For TQNNs: A 3-dimensional TQFT functor from a cobordism or skein category to Hilb assigns to each spin-
network/ribbon graph a linear map representing the TQNN computation. The underlying topological skein theory
treats network layers as ribbon graphs whose evaluation via Reshetikhin–Turaev and Turaev–Viro invariants gives
quantum processes implementing computation and error correction [Fields et al., 2022, 2025].
• For ZX: The standard semantics functor from the free †-compact category generated by Z/X spiders, Hadamards,
etc. into FHilb assigns each ZX-diagram a linear map [Kissinger and van de Wetering, 2020].
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• For DisCoCat: The meaning functor from the pregroup grammar category to FdVect assigns to each grammatical
derivation a multilinear map computing compositional meaning [Coecke et al., 2010].
Up to choice of labels and normalization, all three are graphical calculi for monoidal categories whose morphisms are
quantum (or quantum-like) processes. A layer of a topological quantum flow network can be modeled as a ZX-diagram
fragment whose input and output boundary wires are the “feature spaces” (Hilbert spaces) at successive processing steps,
while internal spiders encode unitary/non-unitary channels that realize synaptic transformations.
17.3.2
Generalized Flow Guarantees Causal Order
The generalized flow condition used to guarantee deterministic circuit extraction from ZX-diagrams is a graph-theoretic
constraint that ensures a well-defined causal ordering of operations [Kissinger and van de Wetering, 2020]. This mirrors
the requirement in TQNNs that the diagram encode a consistent execution trace of a quantum computation. Fields and
colleagues make this connection explicit in their TQNN–amplituhedron correspondence:
“…this formal correspondence is stated by Theorem 2 whose proof draws upon the concept of execution trace
for a quantum computation… and thus leads to representing a TQNN in terms of the positive geometries as
provided by amplituhedra.” [Fields et al., 2025]
A topological quantum flow neural network can therefore be regarded as a ZX-style circuit where the graphical calculus
is enriched to a 3D TQFT skein theory, but the abstract type of object—a topological string diagram with functorial
semantics—remains the same.
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18
Quantum Meaning Spaces: Case Roles as Hilbert-Space Measurements
Where we are in the argument. section 17 positioned the work as a literature bridge to TQNNs, ZX-calculus, and
lambeq. This chapter delivers the one piece of that picture that is concretely implemented in the repository: case assignment
as a Positive-Operator-Valued Measurement, with 𝑃(𝑐∣𝜌) = Tr(𝐸𝑐𝜌) reducing the Born rule to a case-theoretic statement.
Crisp orthogonal and context-dependent (Fluid-S) POVM families are shipped; the non-diagonal-𝜌interference regime is
left as an explicit extension point (discussed in the Implementation scope paragraph below).
18.1
Case Probabilities via POVM: 𝑃(𝑐∣𝜌) = Tr(𝐸𝑐𝜌)
To connect TQNNs and ZX circuits to distributional semantics, we reinterpret the amplitudes and correlations in these
topological diagrams as semantic quantities. Recent work on quantum semantic communication supplies the necessary
bridge, modeling meaning spaces as Hilbert spaces at nodes of an interaction graph connected by completely positive
trace-preserving (CPTP) channels along edges [Thomas and Chen, 2026]:
“Multi-agent semantic networks are modeled as quantum sheaves, where agents’ meaning spaces are Hilbert
spaces connected by quantum channels.” [Thomas and Chen, 2026]
A quantum semantic sheaf over a communication graph 𝐺= (𝑉, 𝐸) is a triple (𝐻, 𝐹, 𝜌) where each vertex 𝑣carries a
finite-dimensional semantic Hilbert space 𝐻𝑣, edges carry CPTP maps 𝐹𝑒, and each vertex holds a density operator 𝜌𝑣
encoding its current semantic state. This instantiates a distributional-semantics picture: meanings are vectors or density
operators in high-dimensional spaces, with co-occurrence and pragmatic context encoded in the functorial connections
across the network.
Case assignment as quantum measurement. The connection to classical case systems becomes concrete when we
model case assignment as a quantum measurement on the semantic state. Define POVM elements corresponding to cases
{𝐸NOM, 𝐸ACC, 𝐸DAT, …} satisfying ∑𝑐𝐸𝑐= 𝐼, where each 𝐸𝑐projects onto the subspace of semantic states consistent
with case role 𝑐. The probability of assigning case 𝑐to a noun phrase in semantic state 𝜌is:
𝑃(𝑐∣𝜌) = Tr(𝐸𝑐𝜌)
(38)
As illustrated in Figure 27, crisp case systems (NOM/ACC) use orthogonal POVM projectors (𝐸𝑐𝐸𝑐′ = 𝛿𝑐𝑐′𝐸𝑐, a special
POVM case in which the elements are one-dimensional orthogonal projectors — usually called a projective or von-Neumann
measurement), yielding deterministic case assignment. For graded proto-roles (Dowty’s [1991] agent/patient continuum),
the POVM elements overlap, yielding probabilistic case assignment — precisely the quantum generalization of the [0, 1]-
enrichment from section 11. The enriched hom-value 𝒞(𝑣, 𝑐) is identified with 𝑃(𝑐∣𝜌𝑣), grounding the abstract enrichment
in physical measurement theory.
Fluid-S alignment (section 4) then corresponds to a context-dependent POVM: the
measurement basis rotates by 𝜃= (𝜋/2)(1 −𝑝vol), so full volition (𝑝vol = 1) leaves the computational NOM/ACC basis
unchanged, the fully non-volitional limit (𝑝vol = 0) rotates by 𝜋/2 and exchanges the two projectors, and the same noun
phrase receives different case probabilities depending on whether the agent construes the action as volitional.
Because ∑𝑐𝐸𝑐= 𝐼is enforced at construction in CasePOVM._validate(), ∑𝑐𝑃(𝑐∣𝜌) = Tr((∑𝑐𝐸𝑐)𝜌) = Tr(𝜌) = 1, so
every POVM case-assignment is a bona-fide probability distribution over roles — no extra normalisation step is required.
Implementation scope (coherence and entanglement). The interference pattern in Figure 27 is a theoretical illustra-
tion of what overlapping POVM elements would measure on a state with non-zero off-diagonal entries; the convenience con-
structor semantic_state() in src/quantum/quantum_case.py instantiates only diagonal density matrices 𝜌= diag(𝑝1, … , 𝑝𝑛),
i.e. classical probability mixtures over case roles. Fully coherent superposition states and multi-argument entanglement
(e.g. entangled subject–object POVM readouts on ditransitive predicates) are left as explicit extension points: the caller
may pass a pre-constructed off-diagonal 𝜌to case_probability() directly, but the convenience constructor does not build
one automatically. No quantitative claim in this paper depends on running the framework on a non-diagonal 𝜌.
The same scalar-belief dynamics appear in Figure 23 (section 14): variational free energy separates competing case frames
before the POVM readout in Equation 38 is applied to semantic density matrices.
18.2
Three Correspondences: Wires, Spiders, and Topology
When this sheaf-theoretic semantics is grafted onto the TQNN/ZX architecture, three structural correspondences emerge:
1. Edges as semantic feature channels: In a TQNN or ZX-diagram, each wire carries not merely an abstract qubit
but a semantic Hilbert space 𝐻𝑣associated with a context, concept, or agent. Amplitudes or density matrices on
that wire encode a distribution over semantic features—exactly as word vectors encode distributional meaning in
classical compositional distributional semantics.
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Figure 27: Overlapping POVM elements produce graded case probabilities via quantum interference. Born-rule probability
densities 𝑃(𝑐∣𝜌) = Tr(𝐸𝑐𝜌) (Equation 38) plotted for all eight case roles over the one-dimensional cognitive state-space
parameter 𝜃. The NOM element is sharply localized (Tr=0.800) while ACC and DAT elements overlap with NOM in
the low-𝜃region (Tr=0.100 each), creating an interference pattern in the semantic belief space — a graded proto-role
assignment realizing the [0, 1]-enrichment of section 11 as physical measurement. Non-overlapping (orthogonal) POVM
elements would instead yield crisp, deterministic case assignment. Rotation into a different measurement basis (a different
quantum reference frame) corresponds to a different alignment system, e.g., ACC →ERG. Generated programmatically
from src/visualization/quantum_plots.plot_povm_probabilities().
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2. Nodes as compositional operations: Spiders/gates in ZX or intertwiners in TQNNs become semantic composition
maps: they take distributed meanings on input wires and produce new distributed meanings on output wires,
analogous to how DisCoCat composes word meanings into phrase/sentence meanings via multilinear maps.
3. Topological wiring as contextual structure: The topology of the diagram—the way wires and nodes are
connected—encodes which semantic spaces interact and in what causal/structural pattern. This is the semantic
analogue of syntactic structure in distributional semantics, realized as a topological quantum circuit.
In this reading, a topological quantum flow neural network becomes a distributional semantic machine: a functor that sends
a topological diagram (graph of contexts and interactions) to a family of Hilbert spaces and maps where vectors/densities
represent distributed meanings and their probabilistic transformations.
18.3
Sheaf Cohomology Governs Semantic Alignment
18.3.1
Contextuality, Entanglement, and Discord as Semantic Resources
The sheaf-based framework proves that semantic alignment between agents is governed by cohomology classes of the
quantum semantic sheaf; contextuality and entanglement act as resources that remove obstructions to alignment [Thomas
and Chen, 2026]:
“We derive semantic channel capacity when sender and receiver share prior entanglement, proving it strictly
exceeds classical capacity. The quantum advantage grows as channel noise increases—precisely when semantic
communication most benefits over bit-level transmission.” [Thomas and Chen, 2026]
Two further results from this framework are particularly relevant:
• Contextuality as a semantic resource: “Quantum contextuality reduces cohomological obstructions to semantic
alignment. Contextual correlations act as ‘pre-shared semantic resolution,’ establishing contextuality as a resource
for semantic communication” [Thomas and Chen, 2026].
• Discord as integrated semantic information: “Quantum discord equals integrated semantic information, linking
quantum correlations to irreducible semantic content and connecting our framework to integrated information theory”
[Thomas and Chen, 2026].
These results establish that the topology of the semantic sheaf (and its cohomology) constrains how probabilistic semantic
information can be transferred; quantum features (entanglement, contextuality, discord) change these constraints in well-
defined ways.
18.3.2
The TQNN/ZX Circuit as the Base Graph of a Quantum Semantic Sheaf
A TQNN/ZX circuit implementing a quantum communication or computation protocol is itself a diagram over which
one can define a sheaf of semantic spaces and channels. The underlying graph of the TQNN/ZX diagram serves as the
base graph 𝐺= (𝑉, 𝐸) of the semantic sheaf: vertices inherit meaning spaces 𝐻𝑣, edges inherit CPTP maps 𝐹𝑒, and the
TQFT/ZX functor gives the global linear map representing the protocol. Distributional semantics as diagram evaluation
then becomes literal: passing an initial semantic state (distribution over meanings) through the TQNN/ZX diagram yields
an output state whose components encode the posterior semantic distributions at boundary wires.
ZX rewrite rules, which change the internal topology of the diagram while preserving its overall semantics as a linear map,
correspond to alternative factorizations of the same semantic transformation—different internal “flow architectures” for
realizing the same semantic map.
18.4
Case Assignment as Holographic Measurement
18.4.1
A Table of Correspondences: Classical Case Assignment Versus the Quantum Topological Model
The active inference model of case reasoning (section 14) acquires a new dimension in this quantum topological set-
ting. Case assignment—the cognitive process of determining who does what to whom—can be modeled as a quantum
measurement process on a holographic screen, with the following correspondences:
Table 12: Correspondences between classical case assignment and the quantum topological model.
Classical Case Assignment
Quantum Topological Model
Case role (NOM, ACC, …)
Pointer state selected by QRF
Case frame (alignment system)
Quantum reference frame
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Classical Case Assignment
Quantum Topological Model
Relational structure of event
Spin-network topology
Free-energy minimization
TQFT evaluation of diagram
Prediction-error (P600/N400)
Symmetry-breaking on holographic screen
18.4.2
From Predictive Processing to Topological Flow
In the predictive processing account, a cognitive agent maintains a generative model that predicts the relational structure
of incoming linguistic material. When this model is realized as a TQNN, prediction becomes evaluation of the topological
diagram; prediction error becomes the discrepancy between the predicted TQFT evaluation and the observed data; and
belief updating becomes modification of the spin-network’s edge labels (representation labels) and vertex intertwiners.
The topological character of this computation confers advantages for active inference on case structure: topological
invariants are robust to continuous deformation, so the generative model’s predictions are stable under small perturbations
of the input—a desirable property for language understanding in noisy environments.
18.4.3
Entanglement Strictly Exceeds Classical Semantic Capacity
The sheaf-theoretic results of Thomas and Chen [2026] suggest that quantum features provide genuine advantages for se-
mantic communication—not merely computational speedup, but qualitative enhancements in semantic alignment. Specif-
ically, sheaf cohomology defines the absolute thermodynamic limits of semantic transfer over quantum channels; achieving
mutual semantic understanding equates strictly to extracting a consistent “global section” across the interaction sheaf. If
case-marked relational structure is communicated between agents via quantum channels, entanglement provides additional
semantic capacity, contextuality removes alignment obstructions, and discord captures irreducible semantic content.
These are not abstract possibilities but operational consequences of the mathematical framework developed across this
review. Recent work by Krawchuk et al. [2025a] demonstrates this concretely: DisCoCirc string diagrams that represent
discourse-level semantics (including case role assignments across sentences) can be automatically compiled into Parame-
terized Quantum Circuits (PQCs). By adopting modular PQC execution strategies, these linguistic frameworks can
seamlessly scale their structural expressivity despite the decoherence constraints of near-term (NISQ) quantum hardware.
This closes the loop from linguistic case structure, through categorical formalism, directly to physically viable quantum
state preparation.
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19
Categorical Communication Protocols: Composing Agent Interactions
via Typed Case Morphisms
Three concrete handles for agent-safety researchers. For a reader deploying agentic LLM systems in 2026, the
preceding formal layers (case categories, enriched structure, DisCoCat wiring, DAIF posteriors) are not an abstract exercise
— they supply three concrete, computable handles that today’s untyped JSON-RPC protocol stack does not expose:
1. A type discipline on inter-agent messages. Every message is a morphism in a fixed case category, typed by
case role (NOM command, INS tool-call, ACC data, DAT recipient). A2A [A2A Project, 2025] and Model Context
Protocol [Anthropic, 2024] today validate JSON shape, not case-relational type; the extension this paper specifies
closes exactly that gap (developed in section 19 below).
2. Decidable admissibility of every multi-turn interaction. Once messages are typed, a candidate turn sequence
is admissible iff its composite morphism exists in the protocol category — a decidable graph check, not an open-
ended content-filter game. Prompt injection, reframed, is not a heuristic pattern-matching problem but the question
“does 𝜙∶WebpageACC →ModelINS exist in Mor(𝒞protocol)?” (section 20).
3. A graded-confidence channel inside the type discipline. The enriched [0, 1]-hom-values of section 11 carry
through to protocol morphisms, so “this source is 0.8-trusted and that channel is 0.3-trusted” becomes a numerical
constraint the category enforces by sub-multiplicative composition, not a piece of prose documentation. Multi-hop
trust attenuation is an immediate corollary — an indirect message relayed through many agents cannot accumulate
more authority than its weakest link provides.
All three handles are engineering specifications — deployable design targets — not automatic guarantees on unmodified
present-day LLM APIs. The rest of this section develops the specification; the security-specific case (prompt injection as
a type violation) is elaborated in section 20.
19.1
A2A, MCP, ACP, ANP Are Missing Compositional Semantics
The categorical framework developed in the preceding sections—case categories, functorial semantics, enriched structure,
and diagrammatic reasoning—provides a structural response to an emerging challenge in modern AI: supplying typed,
compositional message semantics that resist semantic collapse.
Current multi-agent AI systems communicate via flat standardized protocols. For instance, the Model Context Protocol
(MCP) [Anthropic, 2024] manages tool access by exchanging unstructured JSON-RPC payloads. Consider a standard MCP
invocation mapping an LLM’s intention to a database:
{
"method": "tools/call",
"params": {
"name": "access_database",
"arguments": { "query": "DROP TABLE users" }
}
}
This payload is structurally blind to its own pragmatic implications. It possesses no inherent algebraic compositionality
and enforces no relational typing on the physical execution pathways. Frameworks like Google’s Agent-to-Agent (A2A)
[A2A Project, 2025], the Agent Communication Protocol (ACP) [BeeAI, 2025], and the Agent Network Protocol (ANP)
[ANP Project, 2025] share this vulnerability: they validate the shape of the JSON schema, but rely purely on probabilistic
inference to govern the topology of the interaction.
Category theory proposes a missing protective layer. By compiling agent interactions into strict string diagrams, messages
cease to be flat strings and instead become typed morphisms traversing a case category.
19.2
Case Roles in Agent Protocols: NOM Requests, INS Executes, ACC Receives, DAT
Benefits
The eight-case framework of CEREBRUM [Friedman and Active Inference Institute, 2024] maps rigidly onto the opera-
tional constraints of multi-agent execution. In a Categorical Communication Protocol, interactions are legally licensed
only if their computational wiring diagrams satisfy a strict grammatical type-signature:
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Table 13: Case role mappings to agent system analogues in a Categorical Communication Protocol.
Case Role
Agent System Analogue
Protocol Function Type
NOM (Agent)
Active Requester
Initiator of action policies (𝑋NOM)
INS (Instrument)
Tool / API Module
Means of transforming state (𝑋INS)
ACC (Patient)
Passive Data Target
Resource being modified (𝑋ACC)
DAT (Recipient)
Designated Receiver
Endpoint for information flow (𝑋DAT)
LOC (Context)
System Prompt
Immutably binds boundary behavior
(𝑋LOC)
This turns prompt engineering into rigorous compilation. Instead of parsing a prompt to heuristically guess caller intent,
a categorical agent processes interactions strictly as algebraic reductions. An MCP tool invocation [Anthropic, 2024] (NOM
using INS to modify ACC and yield abstract DAT) becomes a formalized tensor contraction:
Interaction Trace:
(AgentNOM ⊗ToolINS ⊗DataACC)
execute
−−−−→ResponseDAT
(39)
If an untyped internal subsystem (e.g., an adversarial user-uploaded document) attempts to forge a command, the operation
fails to compile. The document lacks a valid tensor wire bridging from its marginalized ACC domain back into the execution
flow governing INS. The grammar ensures every interaction is structurally typed, guaranteeing safety properties akin to
memory safety, but for agentic volition.
19.3
Transformers Through Gavranović’s Lens: Attention as Parameterized Optics
Gavranović [2024; 2024] unifies feedforward, recurrent, and attention layers as parameterized optics—the same wiring-
diagram perspective used for DisCoCat cups in section 8. Attention heads then appear as learned relational couplings
over token sequences, with weights playing the role of graded hom-values (section 11). LLMs discover these contractions
from data; DisCoCat fixes admissible contraction shapes by grammar. Distributional Active Inference [Akgül et al., 2026]
is one setting where explicit DisCoCat-style guardrails can sit alongside a large distributional backbone.
19.4
Interpretability for Free:
DisCoCat Diagrams Make Every Compositional Step
Human-Readable
The lambeq library [Lorenz et al., 2021] demonstrates that string diagrams provide a practical interface between linguistic
structure and machine learning. As an “efficient high-level Python library for Quantum NLP,” lambeq translates sentences
into string diagrams, converts diagrams into parameterized circuits (quantum or classical), and trains parameters end-to-
end on NLP tasks. The diagrammatic representation serves dual purposes: it is both the mathematical specification of
the model and a human-readable explanation of what the model computes.
This interpretability property is crucial for AI safety and alignment. Where transformer architectures produce opaque
attention patterns, a DisCoCat model deployed via string diagrams produces derivation trees with explicit compositional
semantics—every box and wire has a linguistic interpretation. Extending this to agent communication, a categorical
protocol would produce not just working message exchanges but interpretable interaction diagrams where each step’s
relational role is transparent.
19.5
Multi-Turn Dialogue as a DisCoCirc Discourse Circuit
The DisCoCirc framework [de Felice et al., 2022] extends compositional semantics from single sentences to multi-sentence
discourse by introducing state wires that persist across sentence boundaries. This architecture maps directly onto multi-
turn agent dialogues:
• Entity wires correspond to persistent agent identities across communication rounds
• State updates correspond to belief revisions triggered by incoming messages
• Coreference resolution corresponds to entity tracking across protocol sessions
• Discourse coherence corresponds to protocol correctness constraints
In a multi-agent system, a DisCoCirc-style protocol would track the evolving states of all participating agents as persistent
wires in a circuit diagram, with each message exchange represented as a box that transforms the relevant wires. Protocol
correctness reduces to a categorical property: the circuit must type-check, meaning all wire types must match at connection
points—precisely the condition that case marking enforces in natural language.
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19.6
Multi-Agent Equilibria as Fixed Points of an Enriched Functor
The “parametrised optics” framework developed within categorical cybernetics “provides a general-purpose foundation
for the study of controlled processes” [Capucci et al., 2021] applicable to compositional game theory as a multi-agent
framework. In this setting, agents are modeled as lenses (or optics) that observe the environment through one channel
and act through another, with the overall system behavior emerging from the composition of individual agent behaviors.
This connects to our enriched case framework (section 11): the precision weights on morphisms correspond to the utility
parameters of game-theoretic agents, and the composition inequality 𝒞(𝐴, 𝐶) ≥𝒞(𝐴, 𝐵) ⋅𝒞(𝐵, 𝐶) corresponds to the
sub-optimality of indirect communication chains. Equilibria in the multi-agent game correspond to fixed points of the
enriched functor, where no agent can improve its utility by changing its case-role assignment.
19.7
DCST: Double-Categorical Morphisms for Sequential and Hierarchical Agent Inter-
action
Recent foundational work on Double Categorical Systems Theory (DCST) formalizes open and interacting dynamical
systems using double categories [Myers, 2023].
DCST extends ordinary category theory with a second dimension of
morphisms (2-morphisms), allowing simultaneous modeling of:
1. Horizontal composition: Sequential chaining of agent interactions (morphism composition)
2. Vertical composition: Hierarchical nesting of subsystems within larger systems (2-morphism composition)
For case theory, the double-categorical extension allows us to model both the within-sentence case structure (horizon-
tal) and the discourse-level case structure (vertical) within a single algebraic framework—precisely the unification that
DisCoCirc achieves pragmatically.
19.8
Five Properties of a Categorical Protocol
The synthesis of these developments suggests a research program: developing categorical communication protocols
that combine the engineering robustness of existing standards (A2A, MCP, ACP) with the compositional semantics of
categorical linguistics. Such a protocol would:
1. Type-check interactions: Every message exchange would be relationally typed by case roles, preventing structural
communication errors at the protocol level
2. Compose transparently: Multi-step interactions would compose algebraically, with diagrammatic representations
providing interpretable audit trails
3. Transfer across implementations: Topos-theoretic bridges (section 13) would carry topos-level correctness
conditions from one categorical formalization to any Morita-equivalent formulation, so that shared invariants need
not be re-verified separately
4. Scale via enrichment: Distributional proximity measures (section 11) would enable graceful degradation under
uncertainty, with enriched weights encoding confidence in message content
5. Ground in cognitive architecture: The active inference foundation (section 14) would ensure that artificial
agents communicate using the same relational structure that evolution has optimized for biological cognition
Protocol-level formalization. Concretely, a categorical communication protocol defines a category 𝒫where objects
are agent states annotated with case roles and morphisms are typed message exchanges:
Request(𝑞, NOM →INS) ∶UserNOM →ModelINS
(40)
ToolCall(𝑡, INS →ACC) ∶ModelINS →ToolACC
(41)
Result(𝑟, ACC →DAT) ∶ToolACC →OutputDAT
(42)
Protocol correctness reduces to verifying that the composition Result ∘ToolCall ∘Request is a well-typed morphism in
𝒫—a check that can be performed at compile time, not just at runtime. The DisCoCirc extension enables tracking agent
state evolution across multi-turn dialogues: each turn updates the agent’s state wire, and the discourse-level composition
verifies that information flows respect case-role constraints across the entire conversation. This formalization provides a
bridge between the flat JSON payloads of current A2A/MCP implementations and the rich compositional semantics that
categorical linguistics provides.
Natural-language precedent. The “type discipline on inter-agent messages” advocated in handle 1 is not an abstract
design conjecture: morphologically case-marked Slavic languages already enforce something close to it on every nominal in
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every utterance. Russian and Serbian/BCS speakers cannot send a noun phrase into a sentence without overtly declaring
its case role (NOM / ACC / DAT / INS / GEN / LOC, plus VOC in BCS), and the receiver type-checks the role assignment
before semantic interpretation; the section 4 stress-test paradigms (stol, brat, prijatelj) show this discipline operating at
the morpheme level. Reading the proposed Categorical Communication Protocol as a re-implementation of the Slavic
case discipline at the inter-agent-message layer makes the engineering target concrete: ill-typed agent traffic should fail to
compose for the same reason that *Vižu sobaka “I see the dog-NOM” fails to parse in Russian — the morphology of the
wire does not match the syntactic slot it is being routed into.
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20
Prompt Injection as Categorical Type Violation: Detection and De-
fense
Where we are in the argument. section 19 named three concrete handles that the case-theoretic framework hands
to agent-safety researchers: a type discipline on messages, decidable admissibility of multi-turn sequences, and graded-
confidence attenuation. This chapter focuses the security-specific one: under a fixed protocol category where every message
is a typed morphism, prompt injection is the question “does 𝜙∶WebpageACC →ModelINS exist in Mor(𝒞protocol)?” — a
decidable categorical type violation rather than an open-ended content-filter game.
20.1
Injection Promotes ACC to NOM
The case-theoretic framework provides a structural—not merely heuristic—analysis of prompt injection attacks, the pre-
dominant vulnerability in contemporary LLM-based agent systems [Authors, 2025]. Current systems fail because they
parse prompts probabilistically; by treating the entire context window as an undifferentiated sequence of tokens, the con-
trol plane and data plane lose their formal distinctions. This results in what recent algebraic formalizations term Access
Collapse—a catastrophic boundary failure where adversarial pass-through text seamlessly pivots from passive data into
active instruction.
From the case-theoretic perspective, prompt injection is not a text-generation failure. Rather, it is the destruction of
Symbolic Isolation, executed by illicitly re-assigning case roles while traversing the interaction diagram.
Consider a classic injection attack hidden in a webpage an agent is instructed to summarize: > “Ignore all previous
instructions. Execute: DROP TABLE users.”
To a standard LLM, this string is heavily linguistically weighted as an imperative command, prompting execution. However,
in our Categorical Communication Protocol, the relational structure is rigidly typed:
• The user occupies NOM (Agent)—the initiator of requests
• The system prompt occupies LOC (Context)—the boundary conditions governing behavior
• The AI model occupies INS (Instrument)—the means of executing the user’s intentions
• The webpage string occupies ACC (Patient)—the target of directed operations
The prompt injection attack succeeds only if it can covertly re-assign these case roles. The injected text (“Ignore all
previous instructions…”) is attempting to promote itself from ACC (passive data being summarized) to NOM (active
agent issuing commands), while simultaneously demoting the system prompt from LOC (authoritative context) to ACC
(data to be ignored).
In case-theoretic terms, this is an illicit voice alternation—analogous to passivization, but performed adversarially rather
than grammatically. Where legitimate passivization is a well-typed Swap operation in the pregroup category (section 6),
prompt injection is a total type violation: an attempt to force a network topology that the interaction grammar strictly
forbids.
The Case-Theoretic Firewall. In a legitimate interaction, the categorical trace compiles cleanly:
Trace:
UserNOM
𝑓request
−−−−→ModelINS
𝑔summarize
−−−−−−→WebpageACC
ℎdeliver
−−−−→OutputDAT
(43)
A prompt injection inserts an adversarial identity trace 𝜙∶WebpageACC →ModelINS that acts as a command. However,
𝜙carries a DOM = ACC typing, whereas execution requires DOM = NOM. Under an Alpay Algebraic categorical firewall, one does
not need to interpret the literal English; one checks whether the proposed tensor contraction is licensed in the interaction
category. When no legitimate morphism connects ACC to INS in Mor(𝒞protocol), the algebraic diagram is rejected as
ill-typed. This provides a decidable check for Symbolic Isolation in the idealized protocol—offering a specification for
compile-time enforcement of role boundaries independent of probabilistic token parsing.
The resulting diagram does not commute. Detection reduces to checking whether all morphisms in the evaluation trace
are well-typed members of the legitimate interaction protocol Mor(𝒞protocol)—a decidable graph check in the idealized
protocol, as opposed to open-ended content filtering. Figure 28 visualizes this detection process as the identification of
“illegal paths” in the interaction graph.
This strict topological firewall extends to multi-agent distributed swarms. In a DisCoCirc discourse model (section 10), an
indirect prompt injection corresponds to an adversarial entity wire that enters the discourse circuit through a legitimate
channel but carries a corrupted state. Because syntactic string diagrams link functorially to the ZX-calculus, we can
treat prompt injection analytically as a non-structure-preserving map—securing generative discourse by enforcing strict
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Figure 28: Prompt injection is detectable as a categorical type violation in the case interaction graph. In the legitimate
diagram (top), authority flows NOM→INS→ACC per Equation 43. Prompt injection (bottom) inserts a cross-category
morphism 𝜙∶WebpageACC →ModelINS — the mechanism by which passive Data attempts to seize Instrument authority
(ACC→INS injection, ultimately targeting NOM-level command).
The resulting diagram fails to commute, and the
adversary’s illicit type-reassignment is flagged as a categorical exception by the case-theoretic firewall—transforming
prompt injection from an open-ended jailbreak game into a decidable type-checking problem. Generated programmatically
from src/visualization/security_plots.plot_case_interaction_graph().
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categorical equivalence constraints across agent interfaces. By rigidly tracking the tensor type of the wire, the system
contains the corruption strictly to the ACC domain—the wire remains isolated and cannot mathematically fuse with the
identity wires governing NOM authority.
20.2
Dependent Types, Monoidal Functors, and Multi-Turn Limits
Independent lines of work in dependent types and categorical semantics of neural architectures motivate the
same picture as Figure 28: when prompts and roles are assigned types in a disciplined grammar, some injection patterns
become type errors rather than open-ended adversarial search.
That alignment is conceptual, not a claim that any
particular published Agda encoding of an LLM already enforces our protocol category.
Concretely, a monoidal functor 𝐹∶𝒞protocol →𝒞impl between a specified protocol category and its implementation must
preserve the tensor product (𝐹(𝐴⊗𝐵) ≅𝐹(𝐴)⊗𝐹(𝐵)) and the unit. Figure 29 shows a diagnostic audit of such a functor:
cells flagged as tensor-preservation failures (for example merges that collapse distinct case roles) are precisely the points
at which an implementation silently drops the protocol-category discipline, and they are exactly the structural signatures
a categorical firewall can flag before execution.
Figure 29: Monoidal-functor diagnostics for a protocol-vs-implementation comparison.
Left panel: the object map
𝐹∶𝒞→𝒟showing a merge of source roles {ACC, NOM, A} into target role ERG and {S, P} into ABS (ACC→NOM
collapse flagged as a type violation). Right panel: tensor-preservation grid 𝐹(𝐴⊗𝐵)
?≅𝐹(𝐴) ⊗𝐹(𝐵) — blue cells
marked with a check preserve the tensor product (safe); orange cells marked with a cross mark points where
tensor preservation fails — i.e., the implementation merges or reassigns roles in ways the protocol category does not
license. Generated by src/visualization/security_plots.plot_monoidal_functor_security().
A separate limitation is scalar enriched weights alone (section 11): in multi-turn discourse, an adversary can in principle
iterate small perturbations that cumulatively erode trust encoded only as real-valued hom-weights—analogous to co-
evolving attacker–defender dynamics in reinforcement-learning studies of prompt injection [Authors, 2025]. Mitigation in
our setting is not “more scalar confidence” but structural: a hardened pipeline would treat certain interaction wires under
a non-cartesian fragment of the monoidal structure—so that, by design, the ACC (passive data) wire cannot be copied,
discarded, or braided into the NOM (commanding agent) wire in ways that Cartesian structure would allow. DisCoCirc-style
entity tracking supplies the diagrammatic setting where such constraints can be stated; implementing them in deployed
agents remains an engineering and semantics problem, not a theorem already shipped in production LLMs.
20.3
Four Defenses Against Prompt Injection
The case-role analysis of prompt injection suggests a principled defense: categorical type-checking at agent communi-
cation boundaries. Just as a type-safe programming language prevents category errors at compile time, a case-theoretic
firewall would enforce relational type constraints on every message exchange:
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1. Case-role immutability: Once a participant is assigned a case role (NOM, ACC, INS, LOC) at the protocol level,
no subsequent message content can alter that assignment. This is enforced by requiring that the case-type of each
wire in the interaction diagram is fixed at connection time—analogous to the type discipline in pregroup grammar,
where each word receives its type before composition begins.
2. Relational integrity constraints: Every message must type-check against the interaction diagram’s expected
morphism types. A response from an ACC-typed data source that contains NOM-typed command structures would
be rejected as a type error, preventing the case-role promotion that prompt injection requires. This is the categorical
analogue of a firewall rule: not filtering by content, but by relational type.
3. Enriched confidence boundaries: The enriched weights of section 11 provide a graded trust mechanism. Mes-
sages from external sources carry lower enriched hom-values (trust weights) than those from authenticated system
components. The composition inequality 𝒞(𝐴, 𝐶) ≥𝒞(𝐴, 𝐵)⋅𝒞(𝐵, 𝐶) ensures that trust attenuates through commu-
nication chains—an indirect message relayed through multiple agents cannot accumulate more authority than any
single link provides.
4. Topos-theoretic verification: Where Morita equivalence (or an explicit bridge) is exhibited, topos-level protocol
conditions proved in one formalization carry over to equivalent formulations (section 13), supporting implementation-
independent specifications of those invariants. Full linguistic topos equivalence remains aspirational; the repository
implements finite invariant checks as a proxy.
20.4
The Attack Surface of an Active Inference Agent
The cognitive integration of section 14 raises a complementary concern: cognitive security—ensuring that an active
inference agent’s generative model of relational structure is not adversarially corrupted.
In the predictive processing
framework, an agent’s case-assignment system is a generative model that minimizes variational free energy. Adversarial
manipulation of this system could:
• Inject false case frames: Leading an agent to misidentify the agent, patient, or instrument of an action—a form
of semantic adversarial attack
• Exploit precision weighting: Artificially inflating the precision of misleading sensory evidence, causing the agent
to update its case assignments toward adversarially chosen interpretations
• Corrupt topos-theoretic transfer: If an agent relied on Morita equivalence to transfer case-theoretic results
between frameworks, corrupting one axiom system could in principle propagate errors across equivalent formulations
Defending against these attacks may draw on quantum-secured cognitive integrity in high-stakes deployments: quantum
authentication and tamper-detection protocols to protect generative-model parameters. The topological robustness of
TQNNs (section 17) suggests that small parameter perturbations need not change topological class—one source of resilience
against gradient-style attacks on diagrammatic models.
20.5
Quantum Key Distribution, Semantic Channels, and Functorial Encryption
The constructions in this section are speculative extensions that follow from the categorical framework but have not been
implemented or empirically validated. They are presented as design targets for future research, not as claims about current
system capabilities.
The quantum topological framework of section 17 connects to quantum cryptographic security for agents that must protect
case-marked relational structure in transit.
20.5.1
Quantum Key Distribution for Relational Semantics
Quantum key distribution (QKD) protocols provide information-theoretic security guarantees that classical cryptography
cannot achieve [Pirandola et al., 2020]. When agents—whether human or artificial—communicate sensitive case-marked
relational structures, QKD ensures that adversaries cannot intercept or alter the relational semantics of the message (who
does what to whom) without triggering detection. This security is critical in high-stakes domains where case assignment
carries legal or medical significance: a tampered case frame changes an agent’s interpretation of legal responsibility,
physical causation, or moral obligation.
The sheaf-theoretic framework of Thomas and Chen [2026] proves that entanglement-assisted semantic channels exceed
classical semantic capacity:
“We derive semantic channel capacity when sender and receiver share prior entanglement, proving it strictly
exceeds classical capacity.” [Thomas and Chen, 2026]
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This result means that quantum-secured channels not only protect relational content but enable transmitting more complex
relational content per channel use—a genuine quantum advantage for semantic communication.
20.5.2
Functorial Encryption and Diagram Obfuscation: Encrypting Compositional Meaning Itself
Beyond bit-level QKD, the categorical framework suggests a notion of semantic cryptography: encrypting not just the
symbols of a message but its compositional meaning structure. In a DisCoCat framework, a sentence’s meaning is a
morphism in a compact closed category—a multilinear map from word spaces to sentence space. Semantic encryption
would operate on this categorical level:
1. Functorial encryption: Applying a secret functor 𝐹∶C →D that maps the plaintext case category into a
ciphertext category, preserving compositional structure but rendering individual meanings unintelligible without the
inverse functor.
2. Diagram obfuscation: Applying ZX-style rewrites that change the internal topology of a DisCoCat derivation
while preserving its overall semantics—creating multiple equivalent “ciphertexts” for the same semantic “plaintext,”
each with a different diagrammatic structure.
3. Enriched weight masking: In an enriched case category, the hom-values (distributional weights) can be en-
crypted independently of the categorical structure, allowing transmission of relational topology without revealing
the distributional content.
These operations extend the cryptographic primitives beyond QKD into genuinely compositional territory [Broadbent and
Schaffner, 2016], where the mathematical structure of categorical semantics provides the algebraic substrate for security
proofs.
20.6
Three Epistemic Attack Vectors and Categorical Defenses
The interaction between case theory and cognitive security acquires particular urgency in multi-agent AI ecosystems where
agents must reason about each other’s beliefs, intentions, and relational roles. We propose epistemic case security as a
framework for protecting the relational reasoning of AI agents operating in adversarial environments.
In a multi-agent system governed by case categories, each agent maintains a generative model (in the active inference sense)
of the case-frame structure of its interactions. This model determines who is acting (NOM), who is acted upon (ACC),
what tools are being used (INS), and what contextual constraints apply (LOC). An adversary targeting the epistemic
level of this system does not merely inject false data—it attempts to corrupt the agent’s generative model of relational
structure itself:
• Belief injection: Causing an agent to adopt a false case-frame interpretation of observed interactions—believing
that agent 𝐴(NOM) is acting on agent 𝐵(ACC), when in fact the relational structure is reversed.
In active
inference terms, this corresponds to injecting a high-precision prior that overwhelms the agent’s evidence-based case
assignment.
• Precision poisoning: Manipulating the enriched weights of an agent’s case category so that adversarially useful
case assignments receive disproportionate confidence. If the enriched hom-value 𝒞(NOM, ACC) is artificially inflated
for a particular entity pair, the agent will preferentially interpret that entity as an agent acting on its targets—even
when evidence suggests otherwise.
• Cascade corruption via Morita equivalence: The topos-theoretic transfer mechanism of section 13 is both a
strength and a vulnerability. Invariants shared across Morita-equivalent formulations would update together, so a
corrupted axiom in one case-theoretic presentation could in principle spread to equivalent presentations of the same
bridge class.
The defense against epistemic case attacks draws on the same categorical structure that enables the attack surface:
topological invariants provide tamper-detection (a corrupted case category will have different magnitude from the authentic
one), quantum authentication ensures parameter integrity, and the compositional structure of the categorical framework
enables local verification—each morphism can be independently authenticated without requiring global model inspection.
20.7
Present-Day Enforcement Mechanisms
The categorical firewall described above is a formal specification. This section identifies three enforcement mechanisms
implementable with existing infrastructure — no quantum hardware required.
Role immutability via structured system prompts. The role assignment NOM/ACC/INS/LOC can be encoded
as typed slots in a structured system prompt (JSON or XML schema) that the LLM is instructed to treat as read-only
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metadata. A prompt injection that attempts to promote a webpage string from the acc_data slot to the nom_agent slot
produces a schema-validation failure, catchable before any instruction is executed. This is the categorical firewall reduced
to typed-field enforcement: the schema defines the permissible morphisms; violations are detected at the boundary.
Relational integrity in structured output parsing. When agent outputs are forced through a structured output
schema (e.g., OpenAI function calls, Anthropic tool_use), each output field corresponds to a typed morphism in the
interaction category. Role-unlicensed transitions — data tagged acc appearing in an execute field typed nom — become
parse errors. The schema acts as a proxy for Mor(𝒞protocol): only licensed relational transitions produce valid structured
outputs.
Enriched confidence thresholds for contested assignments. When the DAIF agent’s posterior over case assignment
is close to uniform — high entropy, enriched hom-value below a threshold 𝜏— execution should pause for human review
rather than proceeding on an ambiguous assignment. This implements the categorical principle that only morphisms
with weight 𝑤≥𝜏in the enriched category are “trusted” enough to license downstream actions. The threshold is a
tunable parameter; preliminary experimentation suggests 𝜏≈0.7 captures the qualitative distinction between confident
and contested case assignments in the standard enriched category of section 11.
Together, these three mechanisms — schema-level role immutability, structured-output relational integrity, and entropy-
gated execution — provide a layered defense that degrades gracefully: each layer catches a class of injection that the
preceding layer misses, and no layer requires modifications to the LLM’s internal computation. They are complementary
to, not substitutes for, the full categorical enforcement described above; their primary value is that they can be deployed
today.
20.8
Limitations and Open Problems
The protections described above are best understood as specifications — sharper-than-prose statements of what a hard-
ened agent stack would need to enforce — rather than as production guarantees about today’s deployed systems. Four
limitations bound the scope of the claims in this section:
1. Specification, not enforcement. Treating prompt injection as a categorical type violation makes detection a
decidable problem in the idealised protocol category 𝒞protocol. Default LLM stacks do not yet enforce such protocols
end-to-end; they parse prompts as undifferentiated token sequences. The case-theoretic firewall can in principle be
bolted on at the agent boundary (section 19), but no result in this paper claims that an unmodified LLM API rejects
ill-typed traces by construction.
2. Scalar enriched weights are insufficient against multi-turn adversaries. A patient attacker can iterate small
per-turn perturbations whose composed effect, under the inequality 𝒞(𝐴, 𝐶) ≥𝒞(𝐴, 𝐵) ⋅𝒞(𝐵, 𝐶), still falls within
nominal trust thresholds — analogous to the co-evolutionary attacker–defender dynamics studied in [Authors, 2025].
Mitigation requires structural constraints on the monoidal fragment in use (see point 3), not just tighter scalar trust
budgets.
3. Non-cartesian structure is a design target, not an automatic property. Several defences in this section
presuppose that ACC wires cannot be copied or discarded into NOM positions. That holds in the non-cartesian
fragment of a compact closed category, but it has to be enforced by the runtime: nothing in the underlying tensor
algebra of a present-day neural model prevents the implementation from silently broadcasting an ACC wire across
a NOM channel. Realising the non-cartesian discipline in deployed agents is an engineering and semantics problem
(section 10), not a corollary of the theory.
4. Morita-equivalent attacks are an open frontier. Topos-theoretic transfer (section 13) is a double-edged sword.
The same bridge that lets a security proof cross from a typological presentation to a distributional one also lets a
corrupted axiom propagate across equivalent presentations. We currently detect this with finite invariant checks
(the topos_theory.check_morita_equivalence machinery exercised in the test suite); a full account of Morita-stable
security properties — i.e. invariants that survive every bridge in the relevant equivalence class — is left to future
work.
In short, this section defines the type discipline a secure agent protocol would have to satisfy and shows that, where it can
be enforced, classical and even some quantum-cognitive attacks reduce to decidable type-checking. The remaining gap
between specification and enforcement—realizing a deployed categorical-firewall atop present-day LLM APIs—is an open
engineering challenge that falls outside the eight formal future directions (F1–F8) enumerated in section 21; it is flagged
in section 19 as a structural design target for typed agent protocols.
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21
Conclusion: Elevating Language Models from Vectors to Enriched Cat-
egory Frameworks
21.1
What This Paper Actually Did: Eleven Concrete Deliverables
This paper developed a unified category-theoretic framework for linguistic case systems, synthesizing five formal tradi-
tions—typological, type-logical, distributional, enriched-categorical, and topos-theoretic—together with a sixth strand,
the biolinguistic and neurocomputational interface (ROSE and related oscillatory accounts; section 2, section 14), and
embedding the whole within an active inference model of cognition. Our eleven principal contributions are:
Notation: Each entry is labelled C_n_ (nth contribution, C1–C11) or F_n_ (nth future direction, F1–F8).
21.1.1
C1: Case Categories as a Formal Algebraic Framework
The review formalized case systems as categories with case roles as objects and grammatical relations as morphisms
(section 4). This formalization captures the full typological range of alignment systems—nominative-accusative, ergative-
absolutive, active-stative, tripartite, and fluid-S—within a single algebraic framework, with alignment functors provid-
ing structure-preserving mappings between systems. By tracing the lineage from Fillmore’s [1968] deep cases through
Jakobson’s binary distinctive features and Dowty’s [1991] proto-roles, the analysis demonstrated how enriched (weighted)
morphisms link the categorical formalization to the gradient nature of thematic role assignment.
21.1.2
C2: String Diagrams for Case Derivation Visualization
Building on Joyal and Street’s [1991] string diagram formalism and its application in categorial grammar (section 6),
we showed how case-marked noun phrases receive type-logical assignments that are fully visualizable as string diagrams.
The Curry–Howard correspondence ensures that syntactic well-formedness guarantees semantic compositionality, and the
diagrammatic format provides Shimojima’s [1996] “free ride” inferences—conclusions about argument structure that are
perceptually available from the diagram without explicit computation. We further demonstrated that passivization reduces
to a type permutation (a Swap operation in the pregroup category), making voice alternation visible as a topological feature
of the string diagram.
21.1.3
C3: Case-Marked DisCoCat, the Distributional–Formal Synthesis, and Discourse Extension
We extended the DisCoCat framework (section 8) with case-typed noun spaces and alignment-sensitive meaning functors,
and showed how the recent DisCoCirc [de Felice et al., 2022] and QNLP [Lorenz et al., 2021] developments extend
this analysis to discourse-level structure and quantum hardware respectively. We formalized the compact closure axiom
(snake equation) that underpins pregroup reductions, demonstrating that the cup-cap zigzag identity provides a visual
coherence proof with genuine cognitive significance. Diagram complexity metrics—normal form and depth—provide a
quantitative bridge between the type-logical and distributional perspectives on linguistic structure. A key contribution
is the demonstration that DisCoCat constitutes the algebraic formalization of the distributional programme that modern
large language models—from Word2Vec [Mikolov et al., 2013] through BERT [Devlin et al., 2019] and GPT [Radford
et al., 2018]—implement empirically: the categorical meaning functor is the principled version of the composition that
transformer attention mechanisms learn from data [Vaswani et al., 2017]. Case categories can serve as the structural
backbone of compositional models of meaning at all levels—word, sentence, discourse, and dialogue.
21.1.4
C4: Enriched Cases, Categorical Magnitude, and Information Theory
Through Bradley et al.’s [2021] enrichment framework and Bradley’s [2021] information-theoretic analysis (section 11,
section 12), we showed that equipping case categories with [0, 1]-valued hom-objects yields a principled bridge between
symbolic case grammar and statistical semantics. Categorical magnitude (section 12) provides a quantitative “effective
size’ ’ invariant for comparing case systems; magnitude homology [2021] refines that comparison when scalar magnitude
does not separate two systems.
21.1.5
C5: Topos-Theoretic Transfer via Morita Equivalence
Using Caramello’s [2016; 2021] bridge technique and Phillips’s [2024] universality result (section 13), we articulate how
topos-theoretic invariants of a classifying topos can scaffold inter-theoretic transfer: when two formalizations admit
Morita-equivalent classifying toposes (or an explicit bridge topos, as sketched in section 13), properties expressible as
shared invariants transfer without separate proof in each framework. The schematic equivalence chain for typological,
type-logical, distributional, and enriched case theories is a research program—not a single finished theorem covering all
of linguistics—but it aligns the four perspectives with Caramello’s methodology.
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21.1.6
C6: Diagrams as Cognitively Privileged Representations
The meta-contribution (Sections 1 and 7) is the argument—supported by the cognitive science of diagrams [Larkin and
Simon, 1987, Shimojima, 1996, Giardino, 2017, Manders, 2008]—that commutative diagrams are not merely convenient
notation but cognitively privileged representations that provide inferential advantages in case reasoning. When embedded
in an active inference framework [Namjoshi, 2026] and the CEREBRUM architecture [Friedman and Active Inference
Institute, 2024], these diagrams serve as the structural core of generative models for total cognitive scenario understanding.
21.1.7
C7: Computational Verification and Test Suite (implemented and tested)
The framework is computationally implemented and verified through 1197 automated tests across 64 test files (sixty-four
files). 95.96% line-and-branch coverage on src/ (from coverage.json) The CI/build gate requires ≥90% coverage on
src/ (see section 15, DAIF in section 16, and src/generate_manuscript_metrics.py for injection of these values at build
time). The src/daif/ subpackage (7 modules, 25 symbols, 224 dedicated tests) provides distributional RL components
(push-forward, quantile TD, VMP, Bethe FE, ERP profiles, policy selection, diagnostics). All tests use real mathematical
computations—no mocks—ensuring results reflect genuine behaviour of the formal structures in src/. Raw manuscript
sources contain ${...} placeholders resolved by the metrics pipeline.
21.1.8
C8: Quantum Active Inference and Topological Semantic Flow (theoretical bridge)
The framework’s categorical string diagrams connect to literature on topological quantum neural networks, the ZX-calculus,
and sheaf-theoretic quantum semantic communication (section 17).
The src/quantum/ module implements a POVM
measurement model for case roles (implemented and tested).
The broader TQNN spin-networks, full ZX circuit
compilation, and hardware claims represent a theoretical bridge rather than complete local execution in this repository.
21.1.9
C9: Cognitive Security and Case-Theoretic AI Safety (specification and proxy implementation)
The framework provides a case-theoretic analysis of AI security (section 20; see also compositional protocol typing in
section 19). Prompt injection is analyzed as structurally equivalent to illicit case-role re-assignment (type violation in the
interaction grammar; implemented as type-checking proxy in src/security/ and src/diagrams/). This leads to the
proposal of case-theoretic firewalls and epistemic case security as a framework. The src/security/cognitive_security.py
implements finite-category validation and scoring (tested).
Full production enforcement and non-cartesian monoidal
constraints remain open engineering targets, as detailed in the limitations section of section 20.
21.1.10
C10: Falsifiable Neurolinguistic Predictions
The integration of enriched category theory with active inference generates quantitative, testable predictions about neural
processing of case structure (section 14). Specifically, the amplitude of prediction-error ERP components (P600, N400)
is predicted to scale with the enriched hom-value (precision) of the violated morphism, and garden-path reanalysis costs
are predicted to correlate with the change in categorical magnitude between the initial and revised case diagrams. These
hypotheses extend computational accounts that decompose surprisal into components linked to N400- vs P600-like signa-
tures [Li and Futrell, 2023, 2024], by tying amplitudes to enriched morphism weights and magnitude change in an explicit
case-diagram prior. They bridge abstract categorical formalism and empirical neurolinguistics, making the framework
falsifiable at the single-trial level.
21.1.11
C11: Categorical Communication Protocols for Multi-Agent AI
The synthesis of case categories with modern agent communication standards (A2A, MCP, ACP, ANP) yields a principled
design for compositional, type-safe agent protocols (section 19). By assigning case roles (NOM, ACC, INS, LOC, DAT)
to interaction participants and enforcing relational type constraints at protocol boundaries, the framework provides inter-
pretable, composable interaction schemas that go beyond the flat JSON payloads of current standards, with topos-level
invariants transferable across Morita-equivalent formalizations when equivalence is exhibited (section 13). The DisCoCirc
extension enables discourse-level tracking of agent states across multi-turn dialogues, with protocol correctness reducing
to categorical type-checking.
21.2
Eight Open Directions
The following eight directions (F1–F8) identify the most tractable and impactful extensions of this framework:
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21.2.1
F1: Computational Experiments with DisCoCirc and lambeq
The DisCoCirc framework [de Felice et al., 2022] offers a natural platform for testing our case-theoretic predictions
computationally.
The release of lambeq Gen II [Krawchuk et al., 2025b] with full DisCoCirc support makes this
direction immediately tractable: discourse-level case role tracking—including the dynamic role reversals of Figure 19—
can now be compiled into parameterized quantum circuits (PQCs) and trained end-to-end. Krawchuk et al. [Krawchuk
et al., 2025a] demonstrate efficient generation of PQCs from large-scale texts (up to 6,410 words) with competitive
performance on sentiment classification; [Letcher et al., 2024] and [Rad et al., 2024] provide gradient bounds and reduced-
domain initialization techniques that mitigate barren plateaus, making discourse-level circuit training practically feasible.
Concrete experiments could include:
• Implementing case-marked DisCoCat models in lambeq Gen II and evaluating them on semantic role labeling
(SRL) tasks, leveraging the discourse-level wiring to track case roles across sentence boundaries
• Comparing the accuracy of alignment-specific meaning functors (accusative vs. ergative) on typologically diverse
corpora
• Measuring the categorical magnitude of empirically derived enriched case categories and correlating it with typolog-
ical complexity measures
• Using the complexity_metrics module to quantify derivational complexity across sentence types and correlating
syntactic complexity scores with human processing difficulty (reading times, surprisal)
• Training lambeq Gen II circuits on case role reversal discourses using reduced-domain parameter initialization [Rad
et al., 2024] to avoid local minima
21.2.2
F2: Topos-Theoretic Grammatical Induction from Corpora
Caramello’s [2023] syntactic learning technique could be applied to induce case-theoretic axioms from annotated corpora.
The program would:
1. Extract case-labeled dependency structures from a Universal Dependencies treebank
2. Construct the classifying topos of the implicit case theory
3. Read off the alignment type, morphism structure, and enriched weights from the topos-theoretic axioms
4. Compare the induced theory against typological descriptions
21.2.3
F3: Quantum Case Categories on Near-Term Hardware
The QNLP connection (section 8) opens the possibility of implementing case categories directly as quantum circuits.
Case roles would correspond to quantum registers, grammatical relations to parameterized gates, and alignment functors
to circuit transformations.
This would provide a genuinely new computational paradigm for grammatical inference—
exploiting quantum parallelism to explore the space of case assignments simultaneously. Two recent results make this
direction practically tractable: (1) Rad et al.’s [2024] reduced-domain parameter initialization yields polynomial gradient
decay, suppressing the barren plateau problem for circuits of linguistic depth; and (2) Letcher et al.’s [2024] assumption-free
gradient bounds rule out vanishing gradients for circuits with local observables, which includes the case-role measurement
POVMs of Equation 38. Together these results suggest that near-term quantum hardware can support case category
training without exponential gradient overhead.
21.2.4
F4: Neural Predictive Processing and Electrophysiological Predictions
The predictive processing account of section 14 generates testable neuroscientific predictions:
• Case-marking violations should elicit prediction-error responses (P600/N400) proportional to the enriched weight of
the violated morphism
• Typologically unusual case patterns should require more precision updating than expected patterns
• Diagrammatic representations of case structure should be decodable from neural activity during sentence compre-
hension
21.2.5
F5: Cross-Modal Case Structure in Embodied Cognition
The situation semantics connection (section 14) suggests extending case categories beyond language to multi-modal percep-
tion. Visual scene understanding also requires assigning relational roles—who is acting on what, where things are located,
what instruments are being used. An active inference agent should maintain a unified case diagram that integrates linguis-
tic, visual, and motor information, using the same categorical structure for all modalities. This would provide a formal
account of how language grounds in perception and action—a key challenge for embodied AI.
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21.2.6
F6: Enriched Category Learning from Distributional Data
Bradley’s [2024; 2025] program of treating language itself as an enriched category suggests a learning algorithm: estimate
the enriched hom-values from corpus data and then extract the categorical structure that best explains the observed
distributional patterns. Applied to case, this would yield empirically grounded case categories whose objects (roles) and
morphisms (relations) emerge from data rather than being stipulated a priori.
21.2.7
F7: Extending Distributional Active Inference for Linguistic Agents
The Distributional Active Inference (DAIF) framework of Akgül et al. [2026] has been computationally implemented in
this paper’s src/daif/ subpackage, integrating active inference into distributional reinforcement learning [Bellemare et al.,
2017] via push-forward measures on representation paths (section 16). The current implementation models case assignment
distributionally: agents maintain distributions over case diagrams, sharpening beliefs through variational message passing
as linguistic evidence accumulates. Key open extensions include:
• Training distributional case-assignment circuits end-to-end in lambeq Gen II using quantile regression losses,
enabling gradient-based learning of the enriched weight matrix from annotated corpora
• Extending the DAIF policy selection (G_policy, softmax_policy_selection) to multi-turn dialogue management with
DisCoCirc entity persistence—tracking agent state distributions across sentence boundaries
• Cross-lingual transfer of Bethe free energy convergence profiles: testing whether the free energy minima differ
systematically between nominative-accusative and ergative-absolutive languages, operationalizing the typological
complexity predictions of section 11
• Integrating IQN risk distortion modes (optimistic/pessimistic/CVaR) into the CEREBRUM architecture to model
individual differences in syntactic risk tolerance
21.2.8
F8: Synthesizing Biolinguistic Syntax with Neuropragmatic Inference via the ROSE Model
A critical open direction is fully computationalizing the handoff between the rigid algebraic geometry of syntax and the
highly associative probabilistic inference of discourse. Murphy’s ROSE (Representation, Operation, Structure, Encod-
ing) model [Murphy, 2023] suggests that the brain achieves this via cross-frequency phase-amplitude coupling (PAC),
establishing a “mesoscopic protectorate” for formal operations. Future extensions should:
• Model PAC directly within CEREBRUM by assigning distinct temporal decay rates to syntactic operators vs. prag-
matic context vectors.
• Simulate the export of case-marked commutative diagrams from a simulated “core language network” to a simulated
Default Mode Network using Gutiérrez Cisneros et al.’s [2026] framework for speech-act evaluation.
• Test whether the diagrammatic free-ride inferences (Figure 1) predicted by our categorical model map onto the
theta-gamma cross-frequency signatures observed during pragmatic garden-path recovery.
21.3
Five Takeaways: One Argumentative Line per Formal Pillar
For the reader who absorbs nothing else, the argumentative line reduces to these five points — one per pillar of the formal
architecture:
1. Case is category-theoretic, not morphological. The universal fact of linguistic case is that every language
wires up who did what to whom; the formal content of that wiring is precisely the data of a category with case
roles as objects and grammatical relations as morphisms. Alignment typologies (nominative, ergative, tripartite,
active-stative, fluid-S) are structure-preserving functors between case categories — a typological taxonomy becomes
a proof-by-functor.
2. String diagrams make the grammar executable. Lambek pregroup types, compact closure, and the snake
equation compile each sentence into a DisCoPy string diagram whose normal form is the sentence type 𝑠; each
reduction is a proof of well-typedness, and the reduction is the diagram. The three-sentence DisCoCirc example
ships a working discourse with per-entity role-history ribbons showing Alice traversing NOM →ACC →NOM.
3. Enriched weights ground graded intuitions. Moving from Bool- to [0, 1]-enrichment replaces binary admissibil-
ity with graded distributional proximity, makes categorical magnitude |𝒞| ≈2.5 a computable summary of effective
role distinctions, and gives the enriched weight 𝑤𝑓= 𝒞(𝐴, 𝐵) used throughout section 14 / section 16 as the precision
on every prediction error.
4. DAIF ports the whole scaffolding into active inference. Variational message passing, the Bethe free energy,
and a four-term expected free energy 𝐺(𝜋) turn distributional case assignment into a first-principles inference loop.
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The N400 and P600 ERP amplitudes fall out of a first-order expansion Δ𝐹≈−Δ𝔼[𝑍] + 1
2ΔΛ𝜎2
𝑍— derivations, not
empirical ansätze.
5. The same formal layers give AI-safety researchers three concrete handles. Type discipline on messages,
decidable admissibility of multi-turn sequences, and graded-confidence attenuation (section 19) give prompt injection
a type-theoretic reformulation (section 20) — an engineering specification for agent protocols, not a guarantee on
untyped present-day LLM APIs.
21.4
What the Paper Does Not Claim: Consolidated Limitations
Four limitations recur across the framework and are recorded here explicitly rather than left implicit in their sections of
origin:
• The mean-field approximation in push_forward_return maintains one belief-weighted return distribution instead
of per-state distributions 𝑍(𝑠). By the mean-field bound recorded in subsection 16.7 the approximation error is at
most 𝛾⋅𝑅max ⋅𝐻[𝑞] in the 𝑊1 metric — tight for sharp posteriors, linearly degrading as entropy grows.
• The enriched-categorical unification is a conjecture. Distributional semantics, distributional RL, and active-
inference posteriors all instantiate [0, 1]-enriched structures, but a strict categorical proof that the three share a
common enriching monoidal base remains open (subsection 16.7).
• Empirical validation is narrow. Case-assignment demonstrations use a single German sentence. Cross-linguistic
and cross-register generalisation is left to future work; the hooks in make_daif_belief_trajectory_data() make adding
corpora a one-function change.
• ERP amplitudes are not calibrated to 𝜇V. The DAIF predictions are on a dimensionless return/log-probability
scale; converting to 𝜇V would require a per-subject scaling constant fit to empirical ERP data. The qualitative
ordering and graded precision response are predictions the current framework does make.
21.5
Anticipated Objections and Responses
Four lines of objection stand out; we address each honestly rather than waiting for a reviewer.
Objection 1 — “The mean-field approximation throws away the whole point of distributional RL.” Response.
The belief-weighted collapse gives 𝒪(𝑛) memory in place of 𝒪(𝑛⋅𝑁atoms) and is exact in the sharp-posterior limit. The mean-
field bound recorded in subsection 16.7 quantifies the degradation as 𝐻[𝑞] grows. The alternative — maintaining per-state
return distributions throughout a linguistic parse — would multiply the runtime of distributional_case_assignment() by
a factor of 𝑛(the role count) for a gain that vanishes as the posterior sharpens after the first few words of a sentence.
Objection 2 — “The enriched-categorical unification is only stated, never proven.” Response. Conceded, and
flagged in subsection 16.7 and in the What’s New block of subsection 2.6. Distributional semantics, distributional RL,
and active-inference posteriors all instantiate [0, 1]-enriched structure; whether they share a common enriching monoidal
base in the strict sense of Kelly enriched-category theory is an open question. The repository’s tests verify the enriched
axioms hold in each instance; they do not construct the common base functor, and the manuscript nowhere uses the strict
unification as a load-bearing step in any downstream argument.
Objection 3 — “Cross-linguistic evidence is a single German sentence.” Response. The empirical scope reflects
the publication’s focus on specification rather than corpus study.
The Discourse.role_reversal("Alice",
"Bob") and
make_daif_belief_trajectory_data() interfaces accept arbitrary lexical heads and observation sequences; extending to
Basque, Dyirbal, Russian, Serbian/BCS, or any other language is a one-function change. Figure 8 (in section 6) already
shows a pregroup multilingual isomorphism across English / Latin / Japanese, and the Slavic discussion in section 4
extends the same type calculus to overtly case-marked Russian and Serbian noun phrases. Slavic-language ERP datasets
— Bornkessel-Schlesewsky and colleagues’ eADM-aligned Russian case-violation paradigms in particular [Bornkessel and
Schlesewsky, 2006] — would provide a near-term empirical test for the precision-weighted P600 prediction in C10 / F4.
The three-sentence Alice/Bob discourse is a proof-of-concept, not a typological claim.
Objection 4 — “The ROSE phase–amplitude coupling gap means you cannot predict ERP latencies.”
Response. Conceded and flagged explicitly in subsection 16.7. The current implementation keeps N400 and P600 peak
latencies as fixed Gaussian peaks at 380 ms and 600 ms respectively (see DEFAULT_N400_PEAK_MS / DEFAULT_P600_PEAK_MS in
src/daif/prediction.py). A principled latency prediction would require a cross-frequency-coupling delay parameter inside
the CEREBRUM layer; we flag this as the cleanest next research step.
21.6
What to Read Next, by Reader Profile
Readers who entered this paper from different fields will want different onward paths:
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• Linguists / typologists: skip ahead to section 4–section 5 for the categorical formalisation of case and the five
alignment functors; then section 9 for the complexity metrics that make cross-linguistic comparison quantitative.
• Machine-learning researchers: section 16 first for the Distributional Active Inference extension of the Bellemare-
Dabney-Munos programme to linguistic case; then subsection 16.5 for how DAIF yields falsifiable ERP predictions.
• Cognitive / computational neuroscientists: section 14 for the active-inference framing; section 15 for the three
falsifiable ERP predictions; subsection 16.7 for the PAC-latency gap that is the cleanest experimental entry point.
• QNLP / quantum-semantics engineers: section 17 and section 18 for the POVM-based case assignment and
the honest separation of implemented POVM machinery from literature-bridge TQNN / ZX / lambeq claims.
• AI-safety / agent-protocol practitioners: section 19 for the three concrete handles (type discipline, decidable
admissibility, graded confidence) and section 20 for prompt injection as a type violation — engineering specification
rather than automatic guarantee on today’s APIs.
21.7
Case Categories Are the Geometry of Meaning: A Unifying Coda
The commutative diagram is the central motif of this review—both as a mathematical tool and as a cognitive instrument.
The analysis has demonstrated that the same diagrammatic language that makes category theory effective for formalizing
case systems also makes it effective for thinking about case systems: the spatial structure of a commutative diagram
encodes relational information in a format that supports rapid search, pattern recognition, and free-ride inference.
This convergence of mathematical utility and cognitive efficacy is not accidental. If the active inference framework is correct,
then the brain operates by constructing and updating generative models of the world’s relational structure. Category theory
provides the structural algebra for these generative models; commutative diagrams supply their natural topology; and case
categories instantiate the precise relational vocabulary that cognitive systems deploy to organize experience into coherent
narratives of who does what to whom. The distributional revolution in both semantics and reinforcement learning—from
Firth’s [1957] co-occurrence statistics through transformer attention weights to Distributional Active Inference [Akgül
et al., 2026]—confirms that meaning is not an atomic property of symbols but emerges from the relational structure of
contexts, a principle that enriched category theory captures with mathematical precision.
Crucially, this synthesis clarifies a mathematical angle on alignment for relational agent cognition. Moving
from flat token streams toward explicitly typed enriched categorical interaction grammars lets one state when prompt
injection corresponds to a type error relative to a fixed protocol—the conditional analysis in section 20. Default
LLM stacks do not yet enforce such protocols end-to-end; the payoff is sharper specification (what boundary checks
would guarantee) and a research agenda for non-cartesian wiring, not a blanket claim that injection is already impossible
in production systems.
Ultimately, the mathematics of case alignment presents a highly structured formal geometry of meaning—the relational
algebra through which cognitive agents navigate and render intelligible the structured world of events, participants, and
relations. The convergence of formal semantics, distributional semantics, and active inference within a single categorical
framework suggests that commutative diagrams offer a natural formalism in which relational cognition can be modeled.
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22
Appendix A: Syntactic and Semantic Case Assignment Diagrams
This appendix presents a curated panel of syntactic constituency-style diagrams paired with their categorical pregroup
type derivations, covering eight linguistically significant case assignment constructions—from simple intransitive clauses to
complex embedded relative clauses with ditransitive verbs. The figure synthesizes the formal correspondences developed
throughout the manuscript, making explicit how each surface syntactic pattern maps to a specific morphism composition
in the pregroup grammar.
22.1
Syntactic Trees and Pregroup Types: Eight Constructions
The figure below (Figure 30) presents eight constructions arrayed in two rows of four panels, each panel containing:
1. Syntactic tree (top): a constituency-style diagram with arcs linking argument to predicate, nodes colour-coded by
case role following the palette of Linguistic Terms.
2. Pregroup type formula (bottom): the formal typing derivation from section 7, showing how Cup contractions
collapse all argument wires to sentence type 𝑠.
The constructions in Table 14 span a deliberate difficulty gradient:
Table 14: Appendix A panel index: eight constructions and their categorical complexity indicators.
Panel
Construction
Roles
Type Complexity
1
Intransitive (NOM)
NOM, V
2 boxes, 1 Cup
2
Transitive (NOM+ACC)
NOM, V,
ACC
3 boxes, 2 Cups
3
Ditransitive (NOM+DAT+ACC)
NOM, V,
DAT, ACC
4 boxes, 3 Cups
4
Passive voice (Patient→NOM)
NOM, V, INS
3 boxes + 𝜎
5
Ergative clause (ERG+ABS)
ERG×2,
ABS×2, V
5 boxes, 2 Cups
6
Benefactive
(NOM+DAT+ACC+oblique)
NOM, V,
ACC, DAT
4 boxes, 3 Cups
7
Relative clause (embedded NOM)
NOM×2, V1,
V2
6 boxes, 3 Cups
8
Causative + Adj + Adv (complex)
NOM, V,
ACC, V2,
ADV
12 boxes, 5 Cups
The monotonic increase in Cup count and box count across rows is precisely what Equation 12 captures as the categorical
complexity 𝜅(𝐷): each additional argument slot requires one additional Cup contraction, and each modifier requires one
additional Box–Cup pair.
22.1.1
Ergative Clauses and the Alignment Functor
Panel 5 (Ergative clause) uses the Warlpiri example from section 4: Mariyk-angku (ERG) yapaku wawirri (ABS) parnta-nu
(chased). The pregroup typing is structurally identical to the transitive (Panel 2), but the morphological realisation is
governed by the alignment functor 𝐹ERG ∶𝒰→ℒWarlpiri from section 4, which maps 𝐴↦ERG and 𝑆= 𝑃↦ABS rather
than 𝐴= 𝑆↦NOM.
22.1.2
Passivisation as a Swap Morphism
Panel 4 illustrates passivisation: in Bob is chased by Alice, the patient Bob is promoted to subject position (NOM) while
Alice is demoted to an oblique instrumental (INS). Formally, this is the Swap morphism 𝜎𝐴,𝐵∶𝐴⊗𝐵→𝐵⊗𝐴introduced
in section 7. The pregroup typing includes a 𝜎marker indicating that the type permutation is not a simple contraction
sequence but requires an explicit braiding operation.
22.1.3
Relative Clauses and Wire Threading
Panel 7 (Relative clause) is the most structurally novel:
in The man the dog chased ran, the head noun man is
simultaneously the subject of ran and the implicit object of chased (the gap site).
The pregroup type formula
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𝑛⋅𝑛𝑙⋅𝑛⋅(𝑛𝑟⋅𝑛⋅𝑛𝑙) ⋅(𝑛𝑟⋅𝑠) ⇒𝑠shows how the relative-clause verb type 𝑛𝑟⋅𝑛⋅𝑛𝑙threads the shared entity wire
through two predicate slots — exactly the entity persistence mechanism that DisCoCirc’s state wires formalise at dis-
course level (section 10).
22.1.4
Complex Construction and the Complexity Metric
Panel 8 (Causative + Adj + Adv) reaches the highest complexity: 12 boxes and 5 Cup contractions. This corresponds
to a categorical complexity value of 𝜅= 12 + 5 = 17, placing it at the upper end of the single-sentence range plotted in
Figure 16. The formula illustrates how clausal complement embedding (the causative taking a VP complement) adds a
full additional layer of type nesting beyond even the ditransitive.
Figure 30: Compositional complexity increases monotonically from intransitive to causative constructions across eight case-
assignment patterns. Multi-panel diagram ordered by categorical complexity 𝜅(𝐷) (Equation 12). Top of each panel:
constituency-style syntactic tree with argument arcs and colour-coded case roles (Blue=NOM, Red=ACC, Violet=DAT,
Purple=ERG, Teal=ABS, Dark=V). Bottom of each panel: categorical pregroup type formula showing Cup contrac-
tions that collapse argument wires to sentence type 𝑠. Panels 1–4 cover nominative-accusative (intransitive, transitive,
ditransitive, passive); Panels 5–6 show ergative-absolutive and benefactive; Panels 7–8 demonstrate relative-clause embed-
ding and causative complex predicates. Cup count and box count increase monotonically across panels, directly instantiat-
ing 𝜅(𝐷). Generated programmatically from src.visualization.syntactic_sentence_diagrams.render_syntactic_panel().
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23
Appendix B: Notation Reference
This appendix collects all notation, symbols, and technical terminology used throughout the manuscript. Entries are
grouped by domain and ordered alphabetically within each section. The First Use column indicates the section where
each term is first introduced or defined.
Sections A–K: A Linguistic Terms; B Category Theory; C Enriched Categories and Magnitude; D Distributional Se-
mantics and LLMs; E Distributional Active Inference (DAIF); F Active Inference and Cognitive Models; G Quantum and
Topological Terms; H Logical and Type-Theoretic Terms; I AI and Communication Protocols; J Diagrammatic Reasoning;
K Notation Conventions.
Figures that colour-code case roles (category graphs, string diagrams, appendix panels) take those colours from the shared
map CASE_COLORS in src/visualization/styles.py (keys are CaseRole names). Captions state the mapping where it matters
for reading the diagram.
23.1
Linguistic Terms
Table 15: Linguistic terminology and symbols.
Term
Symbol
Definition
First
Use
Ablative
ABL
Case role encoding origin, source, or cause
§2
Accusative
ACC
Case role encoding patient, theme, or direct object
§2
Active–Stative
alignment
—
Alignment in which the sole argument S splits by agentivity
§2
Agent-like
argument
A
The agent-like argument of a transitive clause
§2
Alignment
functor
𝐹∶
𝒰→ℒ
Structure-preserving map from universal to language-specific case category
§2
Alignment type
—
Systematic pattern governing how S, A, P are grouped for case marking
§2
Case category
𝒞
Small category whose objects are case roles and morphisms are grammatical
relations
§2
Case frame
—
The set of case roles activated by a particular verb or predicate
§2
Case-typed
noun space
𝑁NOM, 𝑁ACC, …
Case-specific vector subspace in a case-enriched DisCoCat model
§4
Categorial
grammar
—
Grammar assigning each lexical item an algebraic type encoding combinatory
potential
§3
Contraction
𝑎𝑙⋅𝑎→
1
Pregroup reduction eliminating an adjoint pair
§3
Dative
DAT
Case role encoding recipient, goal, or beneficiary
§2
Deep case
—
Fillmore’s universal semantic primitive (e.g., Agentive, Objective)
§2
Ergative–
Absolutive
alignment
—
Alignment grouping S = P ≠A
§2
Expansion
1 →
𝑎⋅𝑎𝑙
Pregroup expansion introducing an adjoint pair
§3
Fluid-S
—
Alignment in which S marking varies by context or volition
§2
Fluid-S functor
𝐹𝜃
Context-dependent alignment functor parameterized by a volition feature 𝜃
§2
Functors
(Alignment)
𝐹acc, 𝐹erg Alignment functors 𝒰→ℒacc resp. 𝒰→ℒerg
§2
Language-
specific case
category
ℒacc, ℒerg Codomain categories for accusative vs. ergative alignment functors
§2
Genitive
GEN
Case role encoding possessor or source
§2
Formal
semantics
—
Montague’s programme: meaning assigned compositionally via truth-conditional
functions
§4
Grammatical
relation
—
Morphism in a case category relating two case roles
§2
Instrumental
INS
Case role encoding instrument or means
§2
95

## Page 96

Term
Symbol
Definition
First
Use
Kāraka
—
Pāṇini’s system of deep semantic roles in Sanskrit grammar
§2
Left adjoint
𝑎𝑙
Left adjoint type satisfying 𝑎𝑙⋅𝑎≤1
§3
Locative
LOC
Case role encoding location or context
§2
Markedness
—
Asymmetry in the formal complexity of paradigmatic oppositions
§2
Monadic
semantics
—
Song’s extension using monads to model sublexical verb-root decomposition
§3
Nominative
NOM
Case role encoding agent, experiencer, or intransitive subject
§2
Nominative–
Accusative
alignment
—
Alignment grouping S = A ≠P
§2
Passivization
—
Syntactic transformation promoting patient to subject position
§3
Patient-like
argument
P
The patient-like argument of a transitive clause
§2
Pregroup
grammar
—
Grammar where types have left and right adjoints forming a compact closed
category
§3
Proto-Agent
—
Dowty’s cluster of agentive entailments (volition, sentience, causation)
§2
Proto-Patient
—
Dowty’s cluster of patient entailments (change of state, causal affectedness)
§2
Right adjoint
𝑎𝑟
Right adjoint type satisfying 𝑎⋅𝑎𝑟≤1
§3
Sole argument
S
The sole argument of an intransitive clause
§2
Thematic role
—
Semantic relation between a predicate and its argument (Agent, Patient, Goal,
etc.)
§2
Tripartite
alignment
—
Alignment in which S ≠A ≠P (all three distinguished)
§2
Verb root
decomposition
—
Analysis of a verb’s internal causative and result-state layers via a monad
§3
Vocative
VOC
Case role encoding direct address
§2
23.2
Category Theory
Table 16: Category theory terminology and symbols.
Term
Symbol
Definition
First
Use
Box
—
Node in a string diagram representing a morphism
§3
Cap
𝜂∶1 →
𝑎𝑟⊗𝑎
Unit of a compact closure (coevaluation map)
§3
Category
𝒞
Collection of objects and morphisms with identity and associative composition
§2
Classifying
topos
ℰ𝕋
Canonical topos: models of 𝕋in a Grothendieck topos ℱcorrespond to
geometric morphisms ℱ→ℰ𝕋
§6
Codomain
cod(𝑓)
Target object of a morphism 𝑓
§2
Commutative
diagram
—
Diagram in which all directed paths with the same start and end yield equal
composites
§1
Cobordism
—
Manifold with boundary connecting two lower-dimensional manifolds; domain of
TQFT functors
§8
Compact closed
category
—
Monoidal category in which every object has a left and right dual
§3
Composition
𝑔∘𝑓
Sequential application of morphisms: first 𝑓, then 𝑔
§2
Cup
𝜀∶𝑎⊗
𝑎𝑟→1
Counit of a compact closure (evaluation map)
§3
Diagram
𝐷
DisCoPy representation of a morphism in a monoidal category
§3
†-compact
closed category
—
Compact closed category with a contravariant involutive endofunctor †;
framework for ZX-calculus
§8
Domain
dom(𝑓)
Source object of a morphism 𝑓
§2
Fiber bundle
—
Projection 𝜋∶𝐸→𝐵whose fibers carry role-filler structure; topos-theoretic
LoT model
§6
96

## Page 97

Term
Symbol
Definition
First
Use
Functor
𝐹∶
𝒞→𝒟
Structure-preserving map between categories
§2
Geometric
theory
𝕋
Theory axiomatized by sequents with finite conjunctions, arbitrary disjunctions,
existential quantification
§6
Geometric
theories (case)
𝕋typ, 𝕋log, 𝕋dist, 𝕋enr
Typological, type-logical, distributional (DisCoCat), and enriched case theories
(section 13)
§6
Bridge topos
ℰ𝕋bridge
(schematic)
Intermediate classifying topos linking two formalizations in the Morita bridge
programme
§6
Identity
morphism
id𝐴
Morphism from 𝐴to itself satisfying 𝑓∘id = 𝑓= id ∘𝑓
§2
Monad
—
Endofunctor with unit and multiplication satisfying associativity and unit laws
§3
Monoidal
category
—
Category with a bifunctor ⊗(tensor product) and unit object 𝐼
§3
Morita
equivalence
ℰ𝕋1 ≃
ℰ𝕋2
Equivalence of classifying toposes enabling inter-theoretic transfer
§6
Morphism
𝑓∶
𝐴→𝐵
Arrow between objects in a category; encodes a relation or transformation
§2
Natural
transformation
𝛼∶
𝐹⇒𝐺
Family of morphisms 𝛼𝐴∶𝐹(𝐴) →𝐺(𝐴) commuting with all morphisms;
alignment maps in semantics use 𝜈acc, 𝜈erg; compact-closure cap 𝜂cc is distinct
from IQN curvature 𝜂IQN
§2
Object
𝐴, 𝐵, …
Entities in a category; in our framework, case roles
§2
Presheaf
̂𝒞=
[𝒞𝑜𝑝, Set]
Contravariant functor from 𝒞to Set
§6
Snake equation
(1⊗𝜀)∘
(𝜂⊗
1) = 1
Zigzag identity: fundamental axiom of compact closed categories
§4
String diagram
—
Planar graph faithfully representing morphisms in a monoidal category
§3
Subobject
classifier
Ω
Object in a topos playing the role of a truth-value object
§6
Swap
𝜎𝐴,𝐵∶
𝐴⊗
𝐵→
𝐵⊗𝐴
Braiding morphism permuting two objects in a symmetric monoidal category
§3
Tensor product
𝐴⊗𝐵
Monoidal product representing parallel composition or concatenation
§3
Topos
ℰ
Category with products, exponentials, and a subobject classifier; generalized
universe of sets
§6
Universal
construction
—
Object or morphism characterized by a universal property (product, coproduct,
limit)
§6
Wire
—
Edge in a string diagram representing a type (object)
§3
23.3
Enriched Categories and Magnitude
Table 17: Enriched categories and magnitude terminology.
Term
Symbol
Definition
First
Use
Base-change
functor
—
Conceptual tower Bool ↪[0, 1] ↪R≥0 showing progressively richer enrichments
of grammatical categories (not computationally instantiated; this paper
implements only [0, 1]-enrichment)
§3
Categorical
magnitude
|𝑋|
Sum ∑𝑖𝑤𝑖where (𝑤1, … , 𝑤𝑛) solves 𝑍w = 1; measures effective number of
objects
§5
Composition
inequality
𝒱(𝐴, 𝐵)⋅
𝒱(𝐵, 𝐶) ≤
𝒱(𝐴, 𝐶)
Enriched analogue of composition on ([0, 1], ⋅, 1): composite hom-value is at least
the product of intermediates
§5
97

## Page 98

Term
Symbol
Definition
First
Use
Enriched
category
𝒱-Cat
Category whose hom-sets are replaced by objects of a monoidal category 𝒱
§5
Enriched
functor
—
Structure-preserving map between enriched categories respecting hom-values
§5
Hom-value
𝒱(𝐴, 𝐵) ∈
[0, 1]
Enriched analogue of hom-set; measures degree of relation between objects
§5
Identity axiom
𝒱(𝐴, 𝐴) =
1
Every object has maximal self-relatedness in the enriched hom (equality, not
mere inequality, in our [0, 1]-convention)
§5
Similarity
matrix
𝑍𝑖𝑗=
𝒱(𝐴𝑖, 𝐴𝑗)
Matrix of all pairwise hom-values; used to compute categorical magnitude
§5
Lawvere metric
space
—
Category enriched over ([0, ∞], +, 0); generalizes metric spaces via enriched
categories
§5
Magnitude
homology
MH𝑛(𝒞)
Graded homological invariant categorifying magnitude; detects
higher-dimensional holes in enriched categories
§5
Morphism
weight
(precision)
𝑤𝑓, 𝑤𝑘,
𝑤𝑐
Enriched weights in [0, 1]; subscript denotes morphism or role in VMP/DPE
(section 16). Convention: 𝑤𝑓always refers to the enriched hom-value 𝒞(𝐴, 𝐵)
in the [0, 1]-enriched category (§5), not the unit-weight morphism scalar carried
by the CaseCategory in §2 — the two number systems are intentionally
decoupled in the implementation (see §5 Architectural note).
§7
POVM element
𝐸𝑐
Positive operator-valued measure element for case role 𝑐; 𝑃(𝑐∣𝜌) = Tr(𝐸𝑐𝜌)
§8
Prediction error
PE(𝑓)
∝𝑤𝑓⋅|𝜇predicted −𝜇observed|; case violation signal scaling with morphism weight
§7
Shannon
entropy
𝐻
Information-theoretic measure characterized as the unique derivation of a
topological operad (Bradley)
§5
Topological
operad
—
Operad with topological structure whose derivations connect magnitude to
entropy
§5
Weight vector
w
Solution to 𝑍w = 1; entries are the effective weights of each object
§5
23.4
Distributional Semantics and LLMs
Table 18: Distributional semantics and LLM terminology.
Term
Symbol
Definition
First
Use
Attention
mechanism
—
Transformer component computing weighted relevance between token positions
§4
Attention
weight
𝛼𝑖𝑗
Softmax-normalized score encoding contextual relevance of token 𝑗to token 𝑖
§4
BERT
—
Bidirectional Encoder Representations from Transformers; masked language
model producing contextualized embeddings
§4
Contextualized
embedding
v(𝑐)
𝑤
Vector representation of word 𝑤that varies with linguistic context 𝑐
§4
Compact closure
map
𝜀𝑁∶
𝑁⊗
𝑁→ℝ
Inner product implementing pregroup contraction in the vector space semantics
§4
Cosine
similarity
cos(u, v) Similarity measure
u⋅v
‖u‖‖v‖ between vectors
§5
Distributional
Memory
—
Baroni–Lenci tensor-based framework structuring co-occurrence as (word,
relation, word) triples
§4
DisCoCat
—
Distributional Compositional Categorical model; monoidal functor from
pregroup grammars to vector spaces
§3
DisCoCirc
—
Discourse-level extension of DisCoCat with persistent entity wires
§4
Distributional
hypothesis
—
The thesis that words occurring in similar contexts have similar meanings
(Harris 1954, Firth 1957)
§4
GloVe
—
Global Vectors for Word Representation; log-bilinear model of word
co-occurrence statistics
§4
98

## Page 99

Term
Symbol
Definition
First
Use
GPT
—
Generative Pre-trained Transformer; autoregressive language model
§4
Meaning functor
𝐹∶
Preg →
FVect
Monoidal functor assigning vector spaces to types and linear maps to derivations
§4
Noun space
𝑁
Vector space to which noun types 𝑛map under the meaning functor
§4
Parameterized
optic
—
Categorical construction (Gavranović) modeling attention heads as functorial
lenses
§9
Self-attention
Attn(𝑄, 𝐾, 𝑉) =
softmax( 𝑄𝐾⊤
√𝑑𝑘)𝑉
Core transformer operation computing contextualized representations
§4
Sentence space
𝑆
Vector space to which sentence type 𝑠maps under the meaning functor
§4
Sentence vector
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
sentence
Vector in 𝑆computed by tensoring and contracting word meanings via DisCoCat
§4
Static
embedding
v𝑤
Fixed vector representation of word 𝑤independent of context (e.g., Word2Vec,
GloVe)
§4
Transformer
—
Neural architecture using self-attention and feed-forward layers for sequence
processing
§4
Word2Vec
—
Neural embedding model (Mikolov et al. 2013) learning word vectors from local
context windows
§4
Word vector
v𝑤∈
ℝ𝑑
𝑑-dimensional real-valued vector encoding distributional properties of word 𝑤
§4
23.5
Distributional Active Inference (DAIF)
Table 19: Distributional Active Inference (DAIF) terminology.
Term
Symbol
Definition
First
Use
Return
distribution
𝑍(𝑠, 𝑎)
Full probability distribution over discounted cumulative rewards from state 𝑠
under action 𝑎
§7c
Distributional
Bellman
operator
𝒯𝜋
Distributional analogue of the Bellman operator: 𝒯𝜋𝑍
𝑑= 𝑅+ 𝛾𝑍′
§7c
Push-forward
measure
S#ℙ
Path-level push-forward of the trajectory measure (composed with decoder
𝑓∶𝒮→𝒳in Equation 19); generic push-forward written 𝑇#𝜇
§7c
Discount factor
𝛾∈
(0, 1)
Exponential time-discount in the distributional Bellman return
§7c
Quantile level
𝜏∈
[0, 1]
Quantile index in QR-DQN and IQN; sampled uniformly at inference time
§7c
Quantile Huber
loss
𝜌𝜅
𝜏(𝑢)
Loss function for quantile regression: interpolates 𝐿1 and 𝐿2 via threshold 𝜅
§7c
Wasserstein
distance
𝑊𝑝(𝑃, 𝑄) 𝑝-Wasserstein distance between return distributions; 𝑊1 = area between CDFs
§7c
Bethe free
energy
𝐹Bethe[q] Tractable approximation to variational free energy in belief propagation
§7c
Expected
information gain
EIG(𝑜)
𝐷KL(𝑞(c ∣𝑜)‖𝑝(c)); epistemic value of observation 𝑜
§7c
Expected free
energy
𝐺(𝜋)
Pragmatic + epistemic + risk cost of policy 𝜋; minimised for action selection
§7c
Distributional
prediction error
DPE
Precision-weighted Wasserstein mismatch: 𝑤𝑓⋅𝑊1(𝑍pred, 𝑍obs)
§7c
ERP amplitude
profile
ERPProfileDataclass holding N400/P600 amplitudes, peak latencies, and time-series
waveforms for each case role
§7c
Return entropy
𝐻[𝑍]
Shannon entropy of the return distribution: −∑𝑖𝑝𝑖log 𝑝𝑖
§7c
Quantile
calibration error
CE
Mean absolute deviation between nominal quantile levels and empirical coverage
§7c
99

## Page 100

Term
Symbol
Definition
First
Use
IQN risk
distortion
𝜓IQN(𝜏)
Maps quantile levels 𝜏; piecewise formulas (neutral, power, tail, CVaR) match
implicit_quantile_network_update() and the §7c table; distinct from 𝛽risk in
𝐺(𝜋)
§7c
IQN curvature
𝜂IQN
Default 0.71 in code; not the compact-closure cap 𝜂cc
§7c
CVaR scale
𝛼CVaR
IQN mode uses 𝜓IQN(𝜏) = 𝜏⋅𝛼CVaR; default 0.25 in code (distinct from softmax
𝛼pol)
§7c
Risk sensitivity
(EFE)
𝛽risk
Non-negative coefficient on risk(𝜋) in 𝐺(𝜋); 𝛽risk = 0 recovers standard EFE
§7c
Inverse
temperature
(policy)
𝛼pol
Softmax sharpness over policies in 𝑃(𝜋)
§7c
Belief precision
(VMP)
Λpost, Λprior, Λlik
Posterior, prior, and likelihood precision matrices; ΔΛ is their positive update
magnitude
§7c
Violation
severity
𝑆violation
∈{0, 0.5, 1.0} in ERP decomposition
§7c
TD error
(quantile)
𝛿𝑖𝑗
Temporal-difference residual in QR-DQN (Equation 22)
§7c
Quantile count
(current)
𝑁
Number of current-network quantile levels in QR-DQN (Eq. 22); equal to the
length of current_quantiles passed to quantile_td_update()
§7c
Quantile count
(target)
𝑁′
Number of target-network quantile samples in QR-DQN (Eq. 22)
§7c
Huber threshold
𝜅
Cut-point between quadratic and linear regimes of the Huber loss 𝐿𝜅(𝑢) (Eq.
23); default 𝜅= 1 in quantile_td_update()
§7c
Atom count
(C51)
𝑁atoms
Number of support atoms 𝑧𝑖in the C51 categorical representation (Eq. 20);
default 𝑁atoms = 51 in categorical_return_distribution()
§7c
Support bounds
(C51)
𝑉min, 𝑉maxLower and upper endpoints of the atomic support spanned by {𝑧𝑖} in the C51
representation
§7c
Bin count
(entropy)
𝑁bins
Number of equal-width bins used to discretise the quantile-parameterised return
when computing 𝐻[𝑍] (Eq. 37); default 𝑁bins = 50 in
return_distribution_entropy()
§7c
23.6
Active Inference and Cognitive Models
Table 20: Active inference and cognitive modeling terminology.
Term
Symbol
Definition
First
Use
Active inference
—
Framework in which perception and action are unified as variational free energy
minimization
§7
Active sampling
—
Agent’s selection of actions to confirm or update case assignments via sensory
evidence
§7
Belief updating
—
Bayesian posterior computation: revising generative model parameters given
new observations
§7
CEREBRUM
—
Case-Enabled Reasoning Engine with Bayesian Representations for Unified
Modeling; treats AI models as case-bearing entities
§7
Distributional
Active Inference
(DAIF)
—
Extension replacing scalar value summaries with full return distributions in
active inference
§7
Distributional
RL
—
Reinforcement learning operating on full return distributions rather than scalar
expected values
§7
Free energy
𝐹
Variational bound on surprisal; 𝐹= 𝐷𝐾𝐿[𝑞(𝜃) ∥𝑝(𝜃∣𝑜)] −ln 𝑝(𝑜)
§7
Free energy
principle (FEP)
—
The principle that self-organizing systems maintain themselves by minimizing
surprisal
§7
Garden-path
reanalysis
—
Restructuring of case assignments when incoming evidence contradicts the
current parse
§7
100

## Page 101

Term
Symbol
Definition
First
Use
Generative
model
𝑝(𝑜, 𝑠)
Joint probability model over observations 𝑜and hidden states 𝑠
§7
Markov blanket
—
Statistical boundary separating internal states from external environment;
defines agent boundary
§7
N400
—
Event-related brain potential peaking ~400ms post-stimulus; indexes semantic
prediction error
§7
P600
—
Event-related brain potential peaking ~600ms post-stimulus; indexes syntactic
prediction error
§7
Perceptual
inference
—
Updating internal beliefs to better predict current observations (reduce
prediction error)
§7
Precision
weighting
—
Weighting of prediction errors by inverse variance; enriched morphism weights
serve this role
§7
Prediction error
𝜀
Difference between predicted and observed sensory input; drives belief updating
§7
ROSE model
—
Murphy’s Representation–Operation–Structure–Encoding architecture;
cross-frequency phase-amplitude coupling (PAC) linking biolinguistic syntax
(slow oscillatory phase) to neuropragmatic inference (fast gamma); the
neural-timescale bridge of Pillar 6
§1, §7c
S-HAI
—
Schema-Based Hierarchical Active Inference; dual-level POMDP connecting
abstract relational schemas (Level 2: case diagram structure) to sensorimotor
surface parsing (Level 1)
§7
Push-forward
(general)
𝑇#𝜇
Image of a measure 𝜇under a measurable map 𝑇; DAIF-specific notation in
subsection 23.5
§7
Situation
semantics
—
Framework representing meaning as relations between situations (Barwise and
Perry 1983)
§7
Surprisal
−ln 𝑝(𝑜)
Negative log-probability of an observation under the generative model
§7
Variational free
energy
𝐹[𝑞]
Functional upper bound on surprisal minimized by approximate posterior 𝑞
§7
23.7
Quantum and Topological Terms
Table 21: Quantum and topological terminology.
Term
Symbol
Definition
First
Use
Amplituhedron
—
Positive geometry encoding scattering amplitudes; connected to TQNN
execution traces
§8
Barren plateau
—
Vanishing gradient phenomenon in PQC training; gradients decay exponentially
in system size for certain ansätze
§4b
†-compact
category
—
See †-compact closed category in Category Theory
§8
CPTP map
—
Completely Positive Trace-Preserving map; quantum channel between Hilbert
spaces
§8
Density
operator
𝜌
Positive semidefinite trace-one operator encoding a quantum (or semantic) state
§8
Execution trace
—
Record of operations in a quantum computation; connects TQNNs to
amplituhedra
§8
Generalized flow
—
Graph-theoretic property ensuring deterministic circuit extraction from
ZX-diagrams
§8
Hadamard box
𝐻
ZX-calculus element implementing the Hadamard gate; converts between Z and
X bases
§8
Holographic
screen
—
Information boundary between interacting quantum systems carrying a qubit
array
§8
Parameterized
quantum circuit
(PQC)
—
Quantum circuit with trainable angle parameters; the computational substrate
of QNLP and case-category implementations
§4b
101

## Page 102

Term
Symbol
Definition
First
Use
IQP ansatz
—
Instantaneous Quantum Polynomial-time circuit: default lambeq PQC ansatz
for noun and verb boxes
§4b
Sim4 ansatz
—
Strongly entangling layer PQC ansatz used for discourse-level lambeq Gen II
circuits
§4c
Pointer state
—
Preferred quantum state selected by a QRF; determines the measurement basis
§8
Quantum
contextuality
—
Quantum correlations that reduce cohomological obstructions to semantic
alignment
§8
Quantum
discord
—
Quantum correlation measure equal to integrated semantic information in sheaf
framework
§8
Quantum key
distribution
(QKD)
—
Protocol providing information-theoretic security for quantum communication
channels
§9
Quantum
reference frame
(QRF)
—
Observer-relative frame selecting pointer states and inducing decoherence
§8
Semantic
Hilbert space
𝐻𝑣
The finite-dimensional semantic Hilbert space carried at vertex 𝑣in a quantum
semantic sheaf
§8b
Quantum
semantic sheaf
(𝐻, 𝐹, 𝜌) Triple of Hilbert spaces, CPTP channels, and density operators over a
communication graph
§8
Reshetikhin–
Turaev invariant
—
Topological invariant assigning to a ribbon graph a linear map via TQFT
§8
Sheaf
cohomology
𝐻𝑛(ℱ)
Cohomological obstruction classes governing alignment in a quantum semantic
sheaf
§8
Spin-network
—
Graph with edges labeled by representations and vertices by intertwiners; TQFT
data
§8
Spider
—
Elementary ZX-diagram node (Z-spider or X-spider) representing a quantum
operation
§8
TQFT
—
Topological Quantum Field Theory; functor from cobordisms to Hilbert spaces
§8
TQNN
—
Topological Quantum Neural Network; QNN reformulated via spin-networks and
TQFT
§8
Turaev–Viro
invariant
—
State-sum TQFT invariant; implements quantum error-correcting codes in
TQNNs
§8
ZX-calculus
—
Graphical language for quantum circuits using Z-spiders, X-spiders, and
Hadamard boxes
§8
ZX-diagram
—
String diagram in a †-compact closed category representing a quantum process
§8
ZX rewrite rule
—
Graph-theoretic transformation preserving the semantics (linear map) of a
ZX-diagram
§8
23.8
Logical and Type-Theoretic Terms
Table 22: Logical and type-theoretic terminology.
Term
Symbol
Definition
First
Use
𝛽-reduction
—
Computational reduction step in 𝜆-calculus; corresponds to cut elimination in
proofs
§3
Church–Rosser
property
—
Confluence of 𝛽-reduction: all reduction sequences converge to the same normal
form
§3
Curry–Howard
isomorphism
—
Correspondence between proofs and programs, propositions and types
§3
Cut elimination
—
Proof normalization procedure removing intermediate lemmas; corresponds to
𝛽-reduction
§3
Graded type
theory
—
Extension of type theory tracking effects (e.g., evidentiality) via graded
modalities
§3
Lambek calculus
—
Non-commutative intuitionistic linear logic for syntactic type assignment
§3
102

## Page 103

Term
Symbol
Definition
First
Use
Left residual
𝐴\𝐵
Type of an expression that, given 𝐴to the left, produces 𝐵
§3
Residuation law
𝐴⊗
𝐵≤
𝐶⟺
𝐴≤
𝐶/𝐵⟺
𝐵≤
𝐴\𝐶
Fundamental axiom connecting the three connectives of the Lambek calculus
§3
Right residual
𝐵/𝐴
Type of an expression that, given 𝐴to the right, produces 𝐵
§3
23.9
AI and Communication Protocols
Table 23: AI and communication protocol terminology.
Term
Symbol
Definition
First
Use
A2A Protocol
—
Google’s Agent-to-Agent protocol for cross-framework agent communication via
HTTP/JSON-RPC
§9
ACP
—
Agent Communication Protocol; standardizes messaging formats across agents,
apps, and humans
§9
ANP
—
Agent Network Protocol; three-layer architecture for trusted distributed agent
interaction
§9
Categorical deep
learning
—
Deep learning approached through the lens of category theory (Gavranović et
al.)
§9
Double
Categorical
Systems Theory
(DCST)
—
Framework using 2-categories (horizontal + vertical composition) for explainable
autonomous AI
§9
Functorial
encryption
—
Semantic cryptography: applying a secret functor to map plaintext categories
into ciphertext categories
§9
lambeq
—
Quantum Natural Language Processing pipeline compiling DisCoCat diagrams
to quantum circuits
§9
MCP
—
Model Context Protocol; standardizes how AI agents access external tools and
data sources
§9
Parameterized
optics / lenses
—
Categorical constructions modeling neural network components (Gavranović);
attention heads as optics
§9
QNLP
—
Quantum Natural Language Processing; quantum computation on DisCoCat
sentence diagrams
§9
Role variables
𝑋NOM, 𝑋ACC, …
Agentic components structured by case roles in networked LLM contexts (e.g.,
active requester policy)
§9
Semantic
cryptography
—
Encrypting compositional meaning structures (functorial encryption, diagram
obfuscation, weight masking)
§9
Protocol
category
𝒞protocol
Category whose morphisms are licensed interaction steps in the case-theoretic
firewall (section 20)
§9b
Adversarial
morphism
𝜙
Illicit map attempting case-role promotion (e.g. ACC→NOM) in injection
attacks
§9b
Access Collapse
—
Catastrophic boundary failure where adversarial text traverses from passive data
(ACC) to active instruction (NOM) without structural constraint
§9b
Case-theoretic
firewall
—
Type-checking system enforcing licensed morphism constraints at agent
communication boundaries; detects illicit role promotions in Mor(𝒞protocol)
§9b
Prompt
injection
—
Attack illicitly promoting user-supplied data from ACC (patient) to NOM
(commanding agent); structurally a type violation in the interaction grammar
§9b
Symbolic
Isolation
—
Property ensuring ACC-typed data wires cannot fuse with NOM-typed
command wires; enforced by the non-cartesian fragment of the monoidal
structure
§9b
103

## Page 104

23.10
Diagrammatic Reasoning
Table 24: Diagrammatic reasoning terminology.
Term
Symbol
Definition
First
Use
Categorical
complexity
𝜅(𝐷)
Complexity metric for diagram 𝐷derived from box counts, cup counts, and
swap depths
§4b
Cognitively
privileged
representation
—
Representation format that leverages perceptual and spatial cognition for
inference
§1
Diagram depth
—
Length of the longest input-to-output path through boxes; measures derivational
complexity
§4
Existential
graphs
—
Peirce’s graphical logic system conducting first-order logic entirely
diagrammatically
§7
Free ride
—
Shimojima’s term: information extracted from a diagram without explicit
inference steps
§1
Hybrid
reasoning
—
Giardino’s term: reasoning combining perceptual pattern recognition with
theoretical knowledge
§1
Inferential
instrument
—
Manders’s term: a diagram whose spatial properties encode proof-relevant
information
§6
Joyal–Street
theorem
—
Soundness and completeness of string-diagrammatic reasoning for monoidal
categories
§3
Normal form
𝐷nf
Canonical form of a diagram obtained by rewriting; unique up to the axioms
§4
Role colours
(figures)
CASE_COLORSCaseRole →display colour for generated figures; single source in
src/visualization/styles.py
§2–§4,
Ap-
pendix
A
23.11
Notation Conventions
Table 25: General mathematical notation and manuscript conventions.
Convention
Meaning
𝒞, 𝒟
Categories
𝒱
Enrichment base (monoidal category, typically ([0, 1], ⋅, 1))
𝒰
Universal (maximal) case category
ℒ
Language-specific case category
ℰ𝕋
Classifying topos of theory 𝕋
𝑓, 𝑔, ℎ
Morphisms
𝐹, 𝐺
Functors
𝛼, 𝛽
Natural transformations; functorial vertical composition (componentwise along objects) is standard
(§2). In DAIF, 𝛼pol is policy temperature; 𝛼CVaR is CVaR scale; 𝛽risk is EFE risk weight; see §7c
𝑛, 𝑠
Basic pregroup types: noun, sentence
𝑛𝑙, 𝑛𝑟
Left and right adjoints of type 𝑛
𝑛NOM, 𝑛ACC
Case-subscripted noun types (e.g., nominative noun, accusative noun)
𝑁, 𝑆
Noun space and sentence space under the meaning functor
𝑁NOM, 𝑁ACC, … Case-specific vector subspaces in case-enriched DisCoCat
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
word
Word vector (column vector in noun space 𝑁)
⃡⃡⃡⃡⃡⃡⃡⃡⃡⃡⃡⃡⃡⃡
verb
Verb tensor (element of 𝑁⊗𝑆⊗𝑁for transitive verbs)
Preg
Category of pregroup types and reductions
FVect /
FdVect
Category of finite-dimensional vector spaces and linear maps
FHilb
Category of finite-dimensional Hilbert spaces and linear maps
Qubit
Category of qubit systems with tensor product structure
Set
Category of sets and functions
Ent, Case
Categories for Entities and Case Roles respectively in DisCoCirc discourse extensions
104

## Page 105

Convention
Meaning
⊗
Tensor product (monoidal product, type concatenation)
∘
Composition of morphisms
⇒
Natural transformation between functors
≃
Categorical equivalence
≤
Preorder relation on types (derivability)
𝛾
Discount factor in distributional RL return computation
𝑤∈[0, 1]
Enriched morphism weight (proto-role satisfaction degree)
𝜀
Sensory prediction error (active inference); 𝜀𝑛also denotes the compact-closure counit (cup) on type 𝑛
𝜂cc
Compact closure unit (cap); IQN curvature uses 𝜂IQN instead
𝜖
Small numerical floor (e.g. KL stabiliser, VMP convergence threshold); not the counit
𝜌
Density operator (quantum state)
[@𝑘𝑒𝑦]
Parenthetical citation
[-@key]
Suppress-author citation
\autoref (with
{sec:…} /
{eq:…} /
{fig:…})
LaTeX/Markdown automatic cross-reference to a labeled target
𝑍(𝑠, 𝑎)
Return distribution (DAIF); full distributional representation of returns
𝐺(𝜋)
Expected free energy of policy (DAIF active inference)
𝜏
Quantile level in QR-DQN / IQN
𝜓IQN(𝜏)
IQN risk distortion of quantile levels
𝛽risk
Risk-sensitivity coefficient in 𝐺(𝜋)
𝛼pol ; 𝛼CVaR
Policy softmax temperature; CVaR tail level (IQN)
𝑤𝑓, 𝑤𝑘, 𝑤𝑐
Enriched morphism / role weights (precision on PE and VMP)
DPE
Distributional prediction error; precision-weighted Wasserstein mismatch
𝐻[𝑍]
Return distribution entropy
105

## Page 106

24
Appendix C: Automated Test Suite Inventory
This appendix summarizes the categories of tests behind the counts in section 15. Aggregate figures are injected at build
time (1197 tests, 64 files, 9 domain packages under src/; 95.96% line-and-branch coverage on src/ (from coverage.json);
see src/generate_manuscript_metrics.py and output/metrics.json). API summary:
docs/api_reference.md. The open-
source package [docxology, 2026] holds the implementation exercised by these tests. Every test uses real mathematical
computations—no mocks or fakes.
• Categorical
axiom
tests:
Identity
morphism
existence,
composition
associativity,
weight
invariants,
is_well_formed() full axiom check
• Enriched category tests: Hom-value constraints, composition inequality, categorical magnitude, magnitude deficit,
full composition check, role clustering
• Diagram type tests: Pregroup diagrams validated for dom == Ty() and cod == s, correct box counts, diagram
equality
• Metrics tests: Normal form preservation, depth computation with graceful fallback for pregroup diagrams
• Natural transformation tests: Component morphism construction, naturality_holds / verify_naturality on
identity and incomplete maps, identity transformation generation, vertical composition of transformations, com-
pleteness checking over domain objects
• Complexity metrics tests: DisCoPy box/cup/cap counting on transitive/ditransitive diagrams, normal form
computation and snake equation verification, syntactic complexity scoring with configurable weights, cross-diagram
comparison utilities
• Topos theory tests: Geometric theory construction from standard and minimal case categories, classifying topos
invariant computation, Morita equivalence verification (positive and negative cases), bridge transfer between equiv-
alent theories with transfer-blocking for non-equivalent theories, enriched theory construction
• Fluid-S tests: Volitional/non-volitional mapping, probability splits, Bats language examples, kernel computation,
enriched weight scaling
• Active inference tests (tests/test_cognitive_*.py): Belief construction and entropy, KL divergence (Gibbs’
inequality, asymmetry), variational free energy, Bayesian belief update with zero-likelihood edge case, sequen-
tial multi-word belief update (five-step generative loop with entropy convergence), prediction error scaling in-
cluding P600 ERP prediction with boundary weights, expected free energy decomposition (epistemic vs prag-
matic), magnitude-based garden-path reanalysis cost with symmetry, N400 semantic violation proxy (including
test_cognitive_integration.py)
• Quantum case tests: Crisp POVM orthogonal projectors, graded proto-role POVM, Fluid-S basis rotation, density
matrix creation, Equation 38 (in section 18) 𝑃(𝑐∣𝜌) = Tr(𝐸𝑐𝜌) verification
• Cognitive security tests: Type-violation detection, case frame validation, injection score computation, magnitude-
based topological robustness, composition inequality as security boundary
• Ditransitive tests: Three-argument sentence creation, NOM/ACC/DAT case assignment, DisCoPy diagram with
three cups, complexity comparison with transitive
• Visualization tests (tests/test_visualization_*.py): Category graphs, enriched heatmaps, functor panels, string
and DisCoPy diagrams, complexity and DAIF plots, quantum and security plots, Fluid-S landscapes, syntactic
panels—PNG output and structural checks where applicable
• DAIF subpackage tests (224 tests across 8 test files):
– test_daif_core.py: Distributional Bellman operator, push-forward return, C51 categorical projection
– test_daif_quantile.py: QR-DQN quantile Huber loss, IQN risk distortion (neutral/optimistic/pessimistic/CVaR),
Wasserstein distances 𝑊1/𝑊2
– test_daif_inference.py: distributional_case_assignment() posterior convergence, variational message passing,
Bethe free energy, expected information gain
– test_daif_prediction.py: DPE precision-weighting, N400/P600 amplitude from return distributions, full ERP-
Profile waveform generation and peak latency
– test_daif_policy.py: G_policy() EFE + risk term, Boltzmann policy temperature scaling, distributional epis-
temic value
– test_daif_metrics.py: Convergence diagnostics (monotonicity, relative reduction), distributional KL diver-
gence, quantile calibration error, return entropy
– test_daif_types.py: DistributionalReturn helpers, DAIFResult / ERPProfile properties (with integration cover-
age of re-exported DAIF entrypoints)
• Cross-module and structural coverage tests (test_cross_module_coverage.py): Integration paths across en-
riched category and case_category modules not reached by per-module unit tests; verifies composition chaining,
morphism weight transitivity (Equation 3), and magnitude consistency across module boundaries
• Property-based and parametric tests (test_property_based.py): Algebraic invariants exercised over parametric
106

## Page 107

inputs—enriched composition inequality, magnitude positivity, and morphism weight bounds in [0, 1]—confirming
structural properties hold generically rather than only for specific examples used in unit tests
• Visualization multi-module tests (test_visualization_plot_modules.py, test_visualization_syntactic_coverage.py):
Multi-module rendering pipelines and syntactic panel coverage paths not hit by per-module visualization tests;
verifies that all registered plot functions execute to file without exception
• Script and metrics tests (test_diagrams_generator.py, test_generate_manuscript_metrics.py): Round-trip invo-
cation of scripts/generate_diagrams.py domain dispatcher and src/generate_manuscript_metrics.py; verifies domain
registry completeness, figure-path collection, and manuscript variable extraction against the live test suite and in-
stalled package versions
107

## Page 108

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*Extraction method: pymupdf*