Overview
Linguistic case offers a setting in which to examine how diagrams connect relational structure, compositional syntax, and uncertainty. This article reviews categorical approaches and supplies an executable collection of deliberately small examples. The implementation includes case-role graphs, pregroup derivations, tensor contractions, synthetic similarity matrices, Bayesian filtering, quantile utilities, and positive-operator-valued measurements. These components share notation, but their mathematical relationships require separate assumptions; a common diagrammatic vocabulary does not establish their equivalence.
The principal contribution is an inspectable comparison of these constructions and their limits. We distinguish generated derivations from illustrative drawings, finite score distributions from Bellman return distributions, presentation statistics from topos invariants, and model-unit prediction errors from physiological measurements. The accompanying suite collects 1883 tests across 94 files; that is a collection count, and a separate source-bound quality receipt records 1883 passing tests with 92.13% line-and-branch coverage (Appendix C). The figure pipeline produces 33 figures from source. All numerical illustrations use synthetic inputs; no corpus study, participant experiment, EEG fit, quantum-hardware run, or deployed-agent security evaluation is reported. Topos-theoretic transfer, cognitive advantages of case diagrams, and case-based authorization remain research proposals with explicit validation requirements. This is working revision 2.7.0; its local validation does not constitute publication or independent mathematical certification.
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Methods and contributions
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Findings and contributions
- The review captures nominative-accusative, ergative-absolutive, active-stative, tripartite and fluid-S alignment within one algebraic framework linked by alignment functors.
- Within case-typed string diagrams, passivization reduces to a type permutation (a Swap in the pregroup category).
- When multi-turn agent interactions are modeled as a fixed category of licensed morphisms, prompt injection can be analyzed as ill-typed role promotion (a functorial type violation): a specification target, not a guarantee on current LLM APIs.
- The topos-theoretic equivalence chain across typological, type-logical, distributional and enriched case theories is presented as a research program, not a finished theorem.
- The accompanying code has 1197 tests across 64 files at 95.96% line-and-branch coverage on src/, plus 30 programmatically generated figures.
Methods
- Case systems formalized as categories with alignment types as functors — Reviews case systems as categories whose objects are case roles and morphisms are grammatical relations, treating each alignment type as a structure-preserving functor.
- Case-typed DisCoCat/DisCoCirc string diagrams for sentence and discourse — Extends DisCoCat with case-typed noun spaces and alignment-sensitive meaning functors, and uses DisCoCirc for discourse-level composition.
- [0,1]-enriched case categories and categorical magnitude — Equips case categories with [0,1]-valued hom-objects and uses categorical magnitude as an invariant for comparing case systems.
- Integration with Distributional Active Inference to derive ERP predictions — Embeds the case framework in a Distributional Active Inference model to state falsifiable predictions for P600, N400 and garden-path reanalysis.
- Open-source Python implementation with automated tests and generated figures — Implements the formal structures in a src/ package with a no-mocks test suite and programmatically generated figures.
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