# Full Text: Active Skillference: A Validated Prerequisite Graph, Computational Claim Registry, and SkillTree Delivery Contract

> Extracted from `Active_Skillference_v1.0.0_DOI-10.5281-zenodo.21865644.pdf`

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Active Skillference
A Validated Prerequisite Graph, Computational Claim Registry, and SkillTree Delivery Contract
Daniel Ari Friedman
Active Inference Institute
daniel@activeinference.institute
ORCID: 0000-0001-6232-9096
Edition 1.0.0 – 2026 DOI: 10.5281/zenodo.21865644

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Publishing Information
Active Skillference
A Validated Prerequisite Graph, Computational Claim Registry, and SkillTree Delivery Contract
Daniel Ari Friedman
Active Inference Institute
daniel@activeinference.institute
ORCID: 0000-0001-6232-9096
Edition 1.0.0 – 2026
Text license: MIT
Source-code license: MIT
DOI: 10.5281/zenodo.21865644
Source repository: https://github.com/ActiveInferenceInstitute/Active_Skillference
Named Skill Atlas
All exported skill names are tokenized into a deterministic front-matter atlas; visible labels are readable representatives rather than a complete printed index.
Suggested citation: Daniel Ari Friedman (2026). Active Skillference: A Validated Prerequisite Graph, Computational Claim Registry, and
SkillTree Delivery Contract (Edition 1.0.0). Active Inference Institute. https://github.com/ActiveInferenceInstitute/Active_Skillference.
https://doi.org/10.5281/zenodo.21865644.
This open textbook is generated from version-controlled Markdown, tested Python modules, programmatic figures, and rendered Mermaid
diagrams. Corrections and improvements may be submitted via the source repository linked above.
Accessibility note: the compact PDF is optimized for dense print. Reader-profile builds, HTML output, and source Markdown can be
generated from the same manuscript materials.

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Contents
1 Abstract: Active Skillference as Provenance-Bound Curriculum Infrastructure 4
2 Introduction: Why Active Inference Needs an Auditable Learning Path 5
2.1 Problem: prerequisite opacity, numerical drift, and evidence-status confusion . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Historical lineage and model boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Situation awareness: theory, model, software, and education are different objects . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1 Education and delivery boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Approach: provenance-bound prerequisites delivered through SkillTree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Contributions: graph validation, provenance binding, scholarship audit, and delivery . . . . . . . . . . . . . . . . . . . . . . 6
3 Methods Overview: F rom Mathematical Kernels to a V alidated Learning Artifact 8
3.1 Evidence classes and construction boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 Methods: Computational Kernels, Model Scope, and Quantitative Provenance 9
4.1 Computational kernels as the sole authority for quantitative examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.1.1 Scope of the implemented kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.1.2 Evidence boundary and analytic checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.2 Statistical boundary: deterministic validation without learner-outcome inference . . . . . . . . . . . . . . . . . . . . . . . . 9
4.3 Provenance binding from kernel registry to learner-facing prose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5 Methods: Evidence Roles, Curriculum Graph, and Stratum Design 11
5.1 Source-role matrix for citation coverage and source health . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.2 Curriculum graph model, DAG validation, and quality gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.2.1 Pedagogical quality constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.3 Stratum order as an inspectable historical and pedagogical design choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
6 Methods: SkillT ree Export, Dashboard Ownership, and Publication Surfaces 13
6.1 SkillTree export as a dependency-safe project, subject, skill, and quiz map . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
6.2 Learner dashboard as a local lesson surface and read-only SkillTree handoff . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
6.3 Publication surfaces for cover, page-two atlas, figures, and artifact audits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7 Results Overview: Structural, Provenance, Scholarship, and Delivery V alidation 14
8 Results: Curriculum Graph and Assessment Audits 15
8.1 Graph validity, prerequisite depth, and assessment-quality gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
9 Results: Scholarship Coverage and Evidence-Status Audits 20
9.1 Citation coverage, locator health, and evidence-class auditing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
10 Results: Kernel-Backed T eaching Examples and Model Boundaries 26
10.1 Kernel-generated mathematical examples and model boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
11 Results: SkillT ree Export and Learner-Surface Handoff 27
11.1 SkillTree export, learner-surface handoff, and generated front matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
12 Discussion and Conclusion: What the Artifact Establishes—and What It Does Not 33
12.1 Principal findings: a claim-calibrated curriculum artifact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
12.2 Interpretation: infrastructure rather than intelligence or personalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
12.3 Implications: future adaptive learning systems and open-science interoperability . . . . . . . . . . . . . . . . . . . . . . . . . 33
12.4 Limitations and planned empirical evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
12.5 Conclusion: a bounded, reproducible contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
13 Evaluation Setup: Deterministic Artifact T ests and Planned Learner Study 34
13.1 Deterministic model suite and fixed inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
13.2 Curriculum export and runtime ownership boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
13.3 Reproduction, delivery, and release checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
13.4 Planned learner-outcome study (design only) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
13.4.1 Browser-facing validation path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
14 Reproducibility and Release Evidence: Regenerating the Artifact Chain 37
14.1 Deterministic kernel outputs, stable graph ordering, and rerunnable counts . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
14.2 Verification gates for code, prose, artifacts, and rendered surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
14.2.1 Coverage floor caveat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
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14.3 Release-evidence ledger and publication blockers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
14.3.1 Release inventory comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
14.4 Provenance chain from kernel function to learner-facing value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
14.5 Disposable generated artifacts and output inventory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
15 Scope and Related W ork: Model, Platform, and Outcome Boundaries 40
16 Scope: Curriculum Platform, Runtime Ownership, and Non-Claims 41
16.1 Curriculum platform scope: not an inference engine or hosted LMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
17 Related W ork: Historical F oundations, Active Inference Evidence Status, and Information Sources 42
17.1 Active Inference and FEP lineage: historical context and curriculum scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
17.1.1 Formal lineage and model commitments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
17.2 Ancient, medieval, and early-modern foundations of inference, perception, and sequenced learning . . . . . . . . . . . . . . . 42
17.2.1 Demonstration, vision, and ordered instruction before modern probability . . . . . . . . . . . . . . . . . . . . . . . . 42
17.2.2 Early-modern method, probability, and experimental perception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
17.2.3 Eighteenth-century association, education, and structured cognition . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
17.3 Evidence-status critiques and empirical boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
17.4 Information, variational, filtering, and application sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
17.4.1 Information and filtering sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
18 Related W ork: SkillT ree Delivery , Security Posture, Reporting, and Open Science 44
18.1 SkillTree delivery as prerequisite learning and gamified practice context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
18.2 SkillTree provenance, deployment security, and access-control boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
18.3 Reproducible reporting, accessible artifacts, and source-role auditing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
18.4 Open-science education ecosystem, interoperability, and community handoff . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
19 Limitations and Planned Evaluation: Delivery Dependencies, Kernel Scope, and Learner Outcomes 46
19.1 Delivery dependencies, fixed-kernel scope, and learner-outcome boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
19.1.1 Delivery substrate and kernel scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
20 Supplement: Model-Bounded V ariational and Expected F ree Energy Examples 47
20.1 SkillTree bridge from derivations to quiz-gated subjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
20.1.1 Bridge figure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
20.2 Noisy-sensor posterior and the tight variational-free-energy bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
20.3 Expected free energy: epistemic value and bounded policy comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
20.3.1 Preference convention and parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
20.3.2 EFE parameter table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
20.4 Sensor precision as a sample-versus-commit policy boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
20.5 Exact inference on a chain with sum-product belief propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
20.5.1 Sum-product parameter table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
20.6 Conjugate parameter learning from Dirichlet beliefs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
20.7 Bayesian model reduction: whether data require added structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
20.8 Ambiguity as a state-dependent component of expected free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
20.9 Multi-step planning and model-dependent look-ahead value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
20.10Continuous predictive coding as model-specific prediction-error descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
21 Code and Data A vailability 54
22 References: Bibliography and Source-Role Audit 55
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1 Abstract: Active Skillference as Provenance-Bound Curriculum Infrastructure
Active Inference and the Free Energy Principle (FEP) provide model-based accounts of belief updating, learning, and action under
uncertainty. We present Active Skillference, a provenance-bound curriculum-generation and SkillTree-export system for teaching those
formal ideas. The paper evaluates structural validity, quantitative provenance, citation-role coverage, and artifact reproducibility; it does
not evaluate learner outcomes, establish a new theory of Active Inference, or present an intelligent tutoring system.
The curriculum is expressed as code: a typed, validated directed acyclic graph of 630 skills across 111 subjects spanning all 8 strata
(mathematics -> probability -> information theory -> variational methods -> the FEP -> active inference -> computation -> applications),
connected by 1199 prerequisite edges with a maximum dependency depth of 75 (of which the substantive concept chain accounts for 33;
the remaining depth is per-stratum review and mastery sequencing rather than conceptual prerequisite, as the methodology details).
Its defining feature is content-provenance binding: every quantitative value shown to a learner is produced by a tested computational
kernel and inserted through a typed claim token, never hand-typed, and the build refuses to export if a claim is unbacked or if a bare
result number appears in learner prose, manuscript prose, or correct numeric quiz answers. The contribution is therefore a systems and
curriculum-infrastructure artifact: it makes a formal subject inspectable and deliverable, but does not claim that the resulting path is
optimal for every learner.
The validated graph exports directly into SkillTree’s data model (Project -> Subjects -> Skills with learning-path dependencies and
quiz-gated completion), includes a scripted REST seeding path for a configured instance, and is mirrored by a local dashboard that
exposes generated artifacts, figures, claim ledgers, scholarship audits, and graph diagnostics without taking ownership of learner progress
or scoring from SkillTree. The result is a curriculum with a validator-backed artifact chain: re-running the kernels regenerates the claim
ledger, figures, manuscript variables, SkillTree export, and learner-facing numbers, so the platform’s teaching claims remain bounded by
what the code, citations, validators, and documented limitations actually support.
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2 Introduction: Why Active Inference Needs an Auditable Learning Path
2.1 Problem: prerequisite opacity, numerical drift, and evidence-status confusion
2.1.1 Historical lineage and model boundary
The Free Energy Principle is often situated in a historically bounded perception-as-inference context that includes Helmholtz’s account
of unconscious inference, Bayesian-brain accounts of neural uncertainty, and tutorials that make prediction error and variational bounds
teachable [von Helmholtz, 1867, Knill and Pouget , 2004, Bogacz, 2017, Sprevak and Smith , 2023]. Its deeper background is selective rather
than a single direct descent: ancient accounts of demonstration and ordered instruction, medieval optical work, early-modern method
and probability, and eighteenth-century accounts of association and education all expose related questions about evidence, perception,
uncertainty, and sequence [ Aristotle, 1901, Ibn al Haytham , 1989, Quintilian, 95, Bacon, 1620, Descartes, 1628, Huygens, 1657, Newton,
1704, Rousseau, 1762]. Bernoulli, de Moivre, Bayes, and Laplace then anchor more specific histories of probability and inverse-probability
reasoning, while Berkeley, Hume, Condillac, Hartley, Locke, Comenius, and Kant provide distinct traditions of perception, association,
habit, experience, and structured cognition [ Bernoulli, 1713, de Moivre, 1718, Bayes and Price , 1763, Laplace, 1774, Berkeley, 1709, Hume,
1739, Condillac, 1754, Hartley, 1749, Locke, 1690, Comenius, 1657, Kant, 1781]. These sources are historical context and design analogies,
not evidence that modern variational free energy, Bayesian brain models, or Active Inference already existed in their period; the fuller
selective history appears in sec. 17.
The modern FEP frames self-organizing systems in terms of variational free energy, with review work clarifying both the principle’s scope
and the need to avoid oversimplified readings [ Friston et al. , 2006, Friston, 2010, Friston et al. , 2023, Zhang and Xu , 2024]. In the discrete
formulations used here, Active Inference supplies model-based rules for belief updating and policy evaluation: variational free energy is
used for current inference, while expected free energy evaluates policies under stated preferences and observation models [ Friston et al. ,
2017a, Da Costa et al. , 2020, Parr et al. , 2022, Pezzulo et al. , 2024]. The word agent therefore names a modeled decision-making system;
it does not by itself establish consciousness, intelligence, autonomy, or empirical adequacy.
2.2 Situation awareness: theory, model, software, and education are different objects
2.2.1 Education and delivery boundary
The field is easiest to misread when four layers are collapsed into one label. At the theory layer, the FEP is a broad explanatory and
mathematical framing whose scope, generality, and distinctive empirical predictions remain matters of active debate [ Friston et al. , 2023,
Aguilera et al. , 2022, Gershman, 2019, Hodson et al. , 2024]. At the model layer, a discrete Active-Inference process theory specifies beliefs,
likelihoods, preferences, policies, and an inference or planning scheme; changing those assumptions changes the model’s behavior, even
when authors use the same vocabulary [ Friston et al. , 2017a, Da Costa et al. , 2020, Parr et al. , 2022, Millidge et al. , 2021b]. A software
implementation then adds a third layer: libraries such as pymdp and this project’s kernels are executable realizations with different goals,
tests, and coverage, not interchangeable evidence for the theory or for biological adequacy [ Heins et al. , 2022, van Oostrum et al. , 2025].
The education layer adds a fourth boundary. Active-learning and prepared-environment proposals make the framework relevant to
instruction, while knowledge-space and curriculum-aware cognitive-diagnosis work describe stronger learner-model commitments that
require response data and an explicit account of learner state [ Di Paolo et al. , 2024, Doignon and Falmagne , 1985, Falmagne and Doignon ,
2011, Fu and Fang, 2025]. Learning analytics can support such future evaluation, but its use also creates privacy, data-protection, validity,
and human-oversight obligations [ Liu and Khalil , 2023, Sclater and Bailey , 2015]. SkillTree is a delivery substrate with its own runtime
semantics; it does not convert a curriculum export into an intelligent tutor or turn a model demonstration into learner-outcome evidence
[SkillTree, 2026, National Security Agency , 2020].
A modeled Active-Inference agent is therefore a formal decision-making object with specified states, observations, preferences, and policies;
it is not by itself an intelligent system. An intelligent system would require a separate capability and evaluation argument, while an adaptive
learner-facing tutor would additionally require learner-state data, intervention logic, and prospective outcome evidence. Active Skillference
occupies the intersection of these layers as a provenance-bound curriculum artifact: it makes a selected model family representable,
teachable, and inspectable without claiming to settle the theory, personalize instruction, or measure learning.
That compact summary hides a long prerequisite chain. To interpret expected free energy, a learner needs KL divergence; to interpret
KL divergence, entropy and expectation; and to work with those quantities, probability, logarithms, optimization, and the geometry of
distributions. A learner can encounter apex terms such as “expected free energy” or “epistemic value” before the mathematical foundations
that make them derivable. The pedagogical risk is conceptual miscalibration: a learner may know the words before being able to compute
the quantities or distinguish mathematical identities from process-theory commitments, empirical hypotheses, and philosophical scope
claims. Active Inference tutorials and expected-free-energy derivation reviews show why the formal sequence cannot be replaced by slogans,
while education-oriented and associative-learning Active Inference work frames learning as structured exploration of prepared material,
attention, and prediction-error reduction rather than passive term recognition [ Smith et al. , 2022, Millidge et al. , 2021b, Champion et al. ,
2026, Di Paolo et al. , 2024, Anokhin et al. , 2024].
A second problem is quantitative integrity. Hand-maintained numerical examples can drift from the equations they illustrate. Without an
executable source of truth, a reader cannot tell whether a displayed value was computed, rounded, copied, or invented. This is a design
risk that the platform addresses; it is not a prevalence claim about every tutorial or textbook.
A third problem is scholarship drift. A manuscript can accumulate citations that are syntactically valid but weakly connected to the
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claims they are supposed to support, while newer reviews or critiques change the evidential status of a phrase without changing its syntax.
That matters here because technical critiques of FEP generality and recent empirical-status reviews, psychology-and-psychiatry progress
reviews, and distinctive-prediction critiques belong in the evidential context [ Aguilera et al. , 2022, 2023, Hodson et al. , 2024, Badcock
and Davey, 2024, Gershman, 2019]. This project treats the bibliography as part of the build contract: every cited source is mapped to a
support lane and role, then checked before figures or learner artifacts are emitted.
2.3 Approach: provenance-bound prerequisites delivered through SkillTree
Active Skillference addresses the three failure modes with one construction: a validated directed acyclic graph whose nodes are atomic skills
and whose edges are explicit prerequisites. A skill is reachable only after the graph resolves the dependencies that precede it. The graph
spans 8 strata from mathematical foundations to applications and is checked to be acyclic, connected from foundations to applications, and
free of dangling prerequisites. This follows the basic premise of prerequisite-sensitive curriculum design: ordering is part of the learning
object [ Bloom, 1956, Liang et al. , 2018]. The result is also interpretable through knowledge-space and learning-space theory, but only as
an authored closure condition over course content; this release does not infer an individual’s knowledge state or implement an adaptive
diagnostic learning-space engine [ Doignon and Falmagne , 1985, Falmagne and Doignon , 2011].
The evaluation is correspondingly bounded by four questions. First, does the curriculum graph satisfy its declared structural and assessment
invariants? Second, can quantitative examples be regenerated from tested kernels without unbacked or hand-entered result values? Third,
can citation roles, manuscript claims, figures, and generated artifacts be audited as one evidence surface? Fourth, does the export preserve
SkillTree’s ownership of learning paths, quizzes, progress, and scoring? These questions evaluate the artifact and its delivery contract;
they do not test whether the curriculum is optimal, whether an Active-Inference model is empirically true, or whether learners improve.
Every quantitative claim in a skill’s content is produced by a tested computational kernel and inserted through a claim:NAME token. The
platform refuses to publish a skill whose claims are not backed by code, and a linter flags any bare result number that slips into prose
outside a claim token or LaTeX math. Thus the worked example that teaches entropy shows 0.693147 nats for a fair coin and 4.60517
nats for a one-in-a-hundred event because the kernel computed those values.
Finally, the curriculum is built for delivery. It targets SkillT ree, an open-source training and gamification platform whose public
documentation describes Project -> Subject -> Skill composition, learning paths, and quiz-based knowledge checks [ National Security
Agency, 2020, SkillTree, 2026, National Security Agency , 2026b,a]. Active Skillference maps subjects, skills, prerequisite learning paths,
and quizzes onto that data model so the exported artifact can be inspected and, when the service is available, handed to a real progress-
tracking environment. The citation is a platform-provenance claim: it identifies the SkillTree substrate and its government open-source
origin, not an endorsement of this curriculum or a security assurance for any deployment.
The delivery layer also belongs to a wider community and open-science setting. The Active Inference Institute’s public website repository
describes a static resource hub with audience-specific pathways for newcomers, learners, researchers, developers, contributors, partners,
and supporters; public repositories and learning resources; and an explicit boundary for sanitizing and verifying public projections [ Active
Inference Institute , 2026e]. Active Skillference contributes one interoperable piece to that ecosystem: a provenance-bound curriculum and
SkillTree delivery contract that makes formal examples, source roles, prerequisites, and release checks inspectable. This is a contribution to
community-facing educational infrastructure, not a claim that the project represents the Institute, speaks for its contributors, or replaces
its broader open-science programs.
These design choices turn the paper’s contribution from a list of topics into a set of auditable contracts. Each contribution below is either
a checked structure, a provenance gate, a source-role audit, or a delivery path into the existing SkillTree runtime.
The contribution map is therefore tied to six evidence classes. Structural evidence covers graph, prerequisite, assessment, and artifact
invariants. Computed evidence covers values regenerated by tested kernels and resolved claim tokens. Simulation evidence covers
fixed-input demonstrations whose behavior is conditional on an explicit model and parameter regime. Literature evidence covers
historical lineage, formal definitions, pedagogical rationale, platform provenance, and critiques, each assigned a source role. Delivery
evidence covers export, REST-contract, dashboard, and publication surfaces. Scope-limit evidence records what the validators and
sources cannot establish: biological adequacy, learner-state inference, personalization, efficacy, retention, or transfer. No evidence class is
used as a substitute for another.
2.4 Contributions: graph validation, provenance binding, scholarship audit, and delivery
1. Structural evidence: a typed, validated prerequisite-DAG engine for curriculum design, including cycle detection, topological
ordering, depth levels, and connectivity checks.
2. Computed evidence: a content-provenance binding mechanism that ties every learner-facing number to a tested kernel and
fails the build otherwise.
3. Curriculum representation: an implemented Active Inference curriculum – 630 skills across all 8 strata – authored against
that engine.
4. Delivery evidence: a SkillT ree exporter and seeder that turns the curriculum into an importable, gamified training profile,
with live availability treated as an external dependency.
5. Literature evidence: a scholarship coverage audit that maps manuscript citations to source roles and generates a reproducible
coverage artifact.
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6. Scope-limit evidence: an explicit boundary between formal model demonstrations, implementation checks, delivery checks, and
future learner-outcome evaluation.
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3 Methods Overview: From Mathematical Kernels to a Validated Learning Artifact
The platform is layered so that each concern is testable in isolation and composes through fixed interfaces. The method therefore moves
from numerical sources of truth, to claim and citation gates, to the curriculum graph, and finally to the exported SkillTree and publication
surfaces that consume those artifacts.
The subsections below separate numerical provenance, scholarship coverage, curriculum structure, and delivery integration so each eviden-
tial claim can be checked against its own validator rather than blended into a single platform assertion.
3.1 Evidence classes and construction boundary
The construction uses a deliberately plural evidence contract. Structural validators check graph and artifact invariants; computational
validators regenerate numerical claims; fixed-input simulations illustrate model-specific behavior; scholarship validators check citation
resolution, source roles, and locator health; delivery checks inspect SkillTree export and dashboard handoff surfaces; and scope-limit
statements prevent any of those checks from being read as learner-outcome or theory-validation evidence. This separation is consistent
with reproducible-science and F AIR reporting principles, but the local implementation remains a repository-level artifact chain rather
than a general scientific provenance service [ Munafò et al. , 2017, Wilkinson et al. , 2016].
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4 Methods: Computational Kernels, Model Scope, and Quantitative Provenance
4.1 Computational kernels as the sole authority for quantitative examples
The src/kernels package implements the mathematics of Active Inference as pure, numerically stable, fully typed functions in natural-log
(nats) units:
• Information – surprise − ln 𝑝, Shannon entropy 𝐻(𝑝) = − ∑ 𝑥 𝑝(𝑥) ln 𝑝(𝑥), cross-entropy, Kullback-Leibler divergence KL (𝑝‖𝑞), and
mutual information, following the original and textbook information-theory treatments [ Shannon, 1948, Cover and Thomas , 2006].
• Distributions – normalization, a precision-parameterized softmax, and Dirichlet expectations; these are the small probability
utilities needed by the active-inference and variational examples [ Beal, 2003].
• Inference – exact discrete Bayesian posteriors and the variational free-energy identity 𝐹 [𝑞] = 𝔼 𝑞(𝑠)[ln 𝑞(𝑠) − ln 𝑝(𝑜, 𝑠)] =
𝐷KL(𝑞(𝑠)‖𝑝(𝑠)) − 𝔼𝑞(𝑠)[ln 𝑝(𝑜 ∣ 𝑠)] = 𝐷 KL(𝑞(𝑠)‖𝑝(𝑠 ∣ 𝑜)) − ln 𝑝(𝑜), together with the log-evidence and the evidence lower bound
[Bishop, 2006, Blei et al. , 2017, Buckley et al. , 2017].
• Active inference – expected free energy 𝐺(𝜋) with the convention that lower 𝐺 is preferred, policy selection 𝑞(𝜋) =
softmax(−𝛾𝐺(𝜋)) where 𝛾 is policy precision, and a DiscreteActiveInferenceAgent running a deterministic perception -> policy ->
action loop [ Friston et al. , 2015, 2017b, Da Costa et al. , 2020, Sajid et al. , 2021, Millidge et al. , 2021b, Champion et al. , 2026].
4.1.1 Scope of the implemented kernels
The implemented kernels intentionally stop at a small, discrete, one-step reference agent. Generalized free energy and action-oriented
model learning are important adjacent targets, but they remain scope markers for future kernels rather than hidden features of this release
[Parr and Friston , 2019, Tschantz et al. , 2020].
4.1.2 Evidence boundary and analytic checks
The resulting evidence is model-specific twice over. First, a tested function establishes that the implementation matches its stated
mathematical identity under the declared assumptions and conventions. Second, a fixed-input simulation shows how that selected model
behaves under its priors, likelihoods, preferences, precision, and policy set. It does not show that the same assumptions describe a
biological brain, a human learner, or all formulations of Active Inference. This distinction follows the separation between process-theory
construction, implementation choice, and empirical-status assessment in the Active-Inference literature [ Friston et al. , 2017a, Da Costa
et al. , 2020, Aguilera et al. , 2022, Hodson et al. , 2024].
Kernel correctness is asserted against analytic identities: the entropy of a uniform distribution over 𝑛 outcomes equals ln 𝑛; KL is non-
negative and zero only at equality; the free-energy bound is tight at the true posterior; and the expected-free- energy preference is
regime-dependent rather than unconditional. The reproducibility section records the exact coverage and mock-free gates that must run
before treating this build as release evidence.
4.2 Statistical boundary: deterministic validation without learner-outcome inference
Active Skillference does not report inferential statistics, learner outcomes, effect sizes, 𝑝-values, confidence intervals, or population-level
empirical claims. Its evidence has three narrower classes. First, deterministic enumeration validates graph shape, source-matrix coverage,
figure/artifact wiring, web math, roadmap state and SkillTree export counts. Second, computational kernels and simulation sweeps show
model-specific mathematical behavior under fixed inputs, such as noisy-sensor posterior sensitivity and the sample/commit expected-free-
energy boundary. Third, browser smoke checks and optional online source-health checks are delivery and currentness evidence. Those
checks can reveal stale routes, missing artifacts, failed MathJax rendering or unreachable source locators; they are not evidence that
learners improved or that a biological process theory is empirically confirmed.
The next two layers apply that evidence discipline to authored content. The first keeps numbers traceable to kernels; the second keeps
citations traceable to explicit support roles.
4.3 Provenance binding from kernel registry to learner-facing prose
Worked examples in src/kernels/examples.py run the kernels on fixed, documented inputs and register each result in a ClaimRegistry
under a stable name. Skill descriptions, assessment stems, correct numeric answer options, and manuscript prose reference these values
only through claim:NAME tokens, following the reproducible-reporting pattern of binding prose to executable analysis rather than hand-
maintained results [ Xie, 2015, Sandve et al. , 2013]. Three gates enforce integrity:
1. Resolution – every token must resolve to a registered claim, or rendering raises.
2. F ail-closed export – the SkillTree exporter calls the resolver; an unbacked token aborts the export with nothing written.
3. Bare-number linter – any decimal result with two or more fractional digits sitting in plain prose (not inside a claim token or $...$
math) is reported as a violation. Correct answer options are scanned as well, so a derived value cannot bypass provenance by hiding
in an answer key; wrong distractors remain free-form.
We use “provenance” in a PROV-inspired sense: the system records entities, activities, and agents sufficient for local traceability, but
it does not claim conformance to a full W3C PROV interchange implementation. In Active Skillference terms, the entities are claim
records, rendered skills, manuscript fragments, figures, and SkillTree exports; the activities are kernel execution, rendering, export, and
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validation; and the agents are the source files and scripts that produce each artifact. The PROV overview, data model, and constraints
recommendation supply the formal vocabulary for derivation, attribution, bundles, and validity checks, while this project implements
a project-local claim registry and release ledger rather than a general PROV service [ Groth and Moreau , Moreau and Missier , Cheney,
Missier, Moreau and De Nies ]. Workflow Run RO-Crate and the related Provenance Run Crate profile provide a useful interoperability
target for the same boundary: they package workflow-run metadata, inputs, outputs, parameters, and execution provenance as RO-Crate
records. Active Skillference’s claims.json, manuscript variables, figure registry, and release ledger play a narrower analogous role inside this
repository. The project does not emit a conforming RO-Crate, validate against those profiles, or expose a general provenance interchange
endpoint; the comparison is a roadmap-level mapping from the local artifact chain to a recognized workflow-provenance model [ Leo et al. ,
2024].
This mechanism sits in the lineage of literate and reproducible computational reporting: Knuth made code and explanation a joint
design object, Sweave/knitr-style workflows bind prose to executable analysis, and reproducible-research arguments treat rerunnable
computational claims as a minimum standard when independent replication is difficult [ Knuth, 1984, Xie, 2015, Peng, 2011, Sandve et al. ,
2013]. Active Skillference moves that binding to a registry and build gate that protects both the manuscript and the exported curriculum.
Provenance binding and kernel testing are complementary halves of the same pedagogical contract. Binding establishes that a value a
learner sees is a kernel’s output rather than a hand-typed figure; the deterministic kernel tests establish that the kernel computes the
intended quantity in the first place. A learner-facing number is therefore only as sound as the kernel behind it, which is why the integrity
story rests on the tested kernels and not on the binding alone: the binding makes a number traceable, and the tests are what make that
traceability worth anything. The export and render gates abort when a kernel token is missing or a result was typed into prose, so an
unbacked number cannot reach a learner unnoticed.
Numeric traceability is not enough for a scholarly curriculum. The prose also needs to show which cited sources support lineage, method,
delivery, critique, and scope claims.
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5 Methods: Evidence Roles, Curriculum Graph, and Stratum Design
5.1 Source-role matrix for citation coverage and source health
The manuscript uses a second, non-numeric integrity gate for citations. The static data/scholarship_sources.yaml file maps each cited
BibTeX key to a scholarship lane, source kind, locator, and support role. src/scholarship/coverage.py parses the authored manuscript
sections, validates those keys against docs/manuscript/references.bib , and rejects any cited key that lacks a matrix entry or authoritative
locator. It also rejects stale source-matrix rows and uncited BibTeX entries, because those orphans would otherwise let the coverage
artifact imply support that no manuscript sentence actually uses. The resulting JSON and figure are generated artifacts, not hand-
maintained summaries. The source-health check adds a second layer over this matrix: DOI and URL locators must be syntactically
authoritative, documentation/software rows must point to URLs, and each role must explain how the source supports the manuscript
rather than merely naming the source.
This gate does not decide whether a paper is true. It prevents weaker failure modes that ordinary citation checks miss: an undefined
BibTeX key, a citation with no stated role, a new source absent from the coverage matrix, or a source lane that disappears from the
manuscript without anyone noticing. The scholarship figure is therefore an audit view over the current manuscript, not a bibliography
inventory.
The matrix is also a situation map for the paper’s evidence boundary. Foundational and historical sources explain vocabulary and lineage;
Active-Inference sources define or compare model formulations; critique and empirical-status sources calibrate generality and application
language; pedagogy and assessment sources motivate ordering and future study design; platform and software sources establish delivery
and implementation context; and provenance, reproducibility, accessibility, and security sources describe artifact obligations. A source can
occupy more than one lane, but its role is declared at the claim boundary rather than inferred from prestige, recency, or citation count.
5.2 Curriculum graph model, DAG validation, and quality gates
5.2.1 Pedagogical quality constraints
src/curriculum defines typed Stratum, Subject, Skill, and AssessmentQuestion records, with field names mirroring SkillTree so export is a
thin map. The CurriculumGraph validates structural invariants – unique identifiers, no self-loops, no dangling prerequisites, no cycles, and
roots only in the two foundational strata – and provides deterministic topological order, per-skill depth levels, and a reachability check
from foundations to applications. Cycle detection uses the standard three-color depth-first-search pattern from graph algorithms [ Cormen
et al. , 2022].
Structural validity is necessary but not sufficient for learning quality, so the build also runs src/curriculum/quality.py. That gate rejects
shallow one-question skills, missing tags, terse subject or skill descriptions, answer-position bias, and backward prerequisites that would
force later-stratum concepts into earlier learning paths. Such cross-stratum references are retained as non-blocking related links rather than
exported as SkillTree dependencies. A semantic-prerequisite gate additionally rejects any skill that is assessed on a load-bearing concept
(KL divergence, Markov chains, the Laplace approximation, the ELBO) without the concept’s defining skill in its transitive prerequisites,
flagging pedagogical inversions while exempting forward references for motivation; a non-empty-scan guard fails the build if any guarded
concept stops appearing in the live corpus, so the gate cannot silently become vacuous.
The newest hardening layer expands continuous dynamics, state-space probability, rate-distortion information, continuous variational
inference, predictive-coding process theory, hierarchical active inference, reproducible simulation, and applied evaluation dossiers. Those
additions are grounded in filtering, information bottleneck, Helmholtz-machine, and generalized-filtering references while remaining peda-
gogical rather than production inference engines [ Särkkä, 2013, Tishby et al. , 1999, Dayan et al. , 1995, Friston et al. , 2010].
The same graph is educationally meaningful only in a narrower sense: it approximates a knowledge-space style closure condition over
authored skills, where later tasks should not be treated as feasible before their prerequisites, but it does not infer an individual learner’s
latent knowledge state [ Doignon and Falmagne , 1985, Falmagne and Doignon , 2011].
5.3 Stratum order as an inspectable historical and pedagogical design choice
The eight-stratum ordering – mathematics, probability, information theory, variational methods, the Free Energy Principle, Active Infer-
ence, computation, and applications – is an authored design decision, not a unique decomposition of the domain. An applications-first path
or a thematic split between generative-model statics and inference/action dynamics could serve different learners. This release chooses
the linear order because the graph can then enforce a simple local rule: concepts precede the concepts that depend on them [ Bloom, 1956,
1968, Liang et al. , 2018]. Historical sources motivate the categories and expose older problems of demonstration, perception, probability,
and ordered instruction, but they do not determine the graph edges or certify the selected order as historically inevitable [ Aristotle, 1901,
Ibn al Haytham , 1989, Quintilian, 95, Huygens, 1657].
Historical pedagogy sources support this as a bounded history of design concerns rather than as an efficacy claim: Bacon emphasizes
staged inquiry, Comenius ordered progression, Locke habit and practice, and Rousseau developmental sequencing [ Bacon, 1620, Comenius,
1657, Locke, 1693, Rousseau, 1762]. Knowledge-space and learning-space theory provide a formal comparison for prerequisite-constrained
mastery states, while curriculum-aware cognitive diagnosis work shows that a stronger learner-model claim would require response data
and an explicit model of learner state [ Doignon and Falmagne , 1985, Falmagne and Doignon , 2011, Fu and Fang , 2025]. Hugo’s Active
Inference account of teacher development, embodied cognition, and education-oriented Active Inference work motivate keeping the order
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revisable and uncertainty-sensitive, but this release does not evaluate teacher development, infer learner states, or implement an adaptive
diagnostic engine [ Hugo, 2026, Clark, 2016, Di Paolo et al. , 2024].
Microlearning and gamification sources motivate modular delivery while warning that implementation context, satisfaction, and measured
effectiveness can diverge [ Septiani and Rosmansyah , 2021, Monib et al. , 2025, Rof et al. , 2024]. Active-learning, epistemic-curiosity,
cognitive-engagement, retrieval-practice, and evidence-centered-assessment sources therefore inform the design of interaction and quiz
tasks; they do not substitute for data from this curriculum or establish durable competence [ Freeman et al. , 2014, Emin et al. , 2026, Chi
and Wylie, 2014, Roediger and Karpicke , 2006, Mislevy et al. , 2003]. Constructive alignment, mastery learning, formative assessment, and
feedback provide the local rule that objectives, activities, and quiz gates should be co-specified and used as revision checkpoints [ Biggs,
1996, Bloom, 1968, Black and Wiliam , 1998, Shute, 2008]. The resulting ordering is a parameter to inspect and revise, not a theoretical
or empirically tested claim about how the field must be structured.
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6 Methods: SkillTree Export, Dashboard Ownership, and Publication Surfaces
6.1 SkillTree export as a dependency-safe project, subject, skill, and quiz map
src/skilltree converts the validated, provenance-rendered graph into SkillTree’s Project -> Subject -> Skill model, emitting prerequisite
learning-path dependencies and a quiz per skill from its assessment questions. A REST client and seeder create the project on a
configured SkillTree instance, when available, in dependency-safe topological order: subjects, then quizzes, then skills, then dependencies.
Quiz definitions must exist before quiz-gated skills are created, and dependencies are assigned only after both endpoint skills exist. The
REST contract is verified against a local pytest-httpserver mirror of SkillTree’s documented endpoints, so the integration is tested without
requiring a running Java service.
6.2 Learner dashboard as a local lesson surface and read-only SkillTree handoff
The local learner dashboard is intentionally a shell rather than a competing learning runtime. It reads generated artifacts ( skilltree_p
roject.json, claims, stats, figures, and the rendered manuscript), probes SkillTree’s public status endpoint, and mounts the official @sk
illtree/skills-client-js learner display when a local trusted-client bridge is ready. The bridge keeps SkillTree credentials server-side:
browser code sees only the configured service URL, project id, and a dashboard-local token endpoint.
The curriculum explorer in the dashboard is therefore read-only. It renders the exported subjects, skills, quiz identifiers, and prerequisites
so authors can inspect the exact artifact that will be seeded. The Learn action opens a local modular lesson record for the selected skill
from course_material_index.json : concept, prerequisite context, downstream role, assessment summary, provenance note, related local
skills, and pinned external pointers. Open Subject exits to the official SkillTree subject learning path ( /subjects/{subjectId}), where
SkillTree lists subject progress, rank, and quiz affordances. Quiz deep-links to /subjects/{subjectId}/skills/{skillId}/quizzes/{quizId}.
This distinction matters operationally. Lesson summaries, search, artifact previews, and status summaries are project-owned conveniences;
progress, ranking, scoring, and completion remain SkillTree-owned learner state.
6.3 Publication surfaces for cover, page-two atlas, figures, and artifact audits
The first and second pages are also generated, not hand-composed. scripts/run_kernels.py writes active_skillference_cover.png from
the live curriculum graph, claim registry and roadmap metrics, and writes active_skillference_skill_atlas.png from every authored skill
name in the curriculum graph. The cover audit checks that book.cover.image resolves to the cover PNG; the front-matter visual audit
checks that front_matter.page_two_visual.image resolves to the atlas PNG. Both audits require a nonblank image, dashboard exposure
as non-figure artifacts, and exclusion from figure_registry.json. This keeps the public entry surfaces aligned with the same artifact layer
that drives the results figures while preserving the stricter cross-reference contract for numbered manuscript figures.
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7 Results Overview: Structural, Provenance, Scholarship, and Delivery Validation
The results report validator-observed properties of the artifact: graph structure, assessment-depth checks, citation-role audits, kernel-
linked teaching examples, generated figure and manuscript artifacts, and SkillTree/dashboard export behavior. They do not report
learner-effectiveness evidence.
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8 Results: Curriculum Graph and Assessment Audits
8.1 Graph validity, prerequisite depth, and assessment-quality gates
The assembled curriculum is a valid directed acyclic graph of 630 skills in 111 subjects joined by 1199 prerequisite edges , spanning
all 8 strata with a maximum prerequisite depth of 75 and two foundational roots (set theory and the probability axioms). Validation
passes: no cycles, no dangling prerequisites, and every applications skill is reachable from a foundational root. fig. 1 lays the graph out by
stratum and depth, uses wrapped stratum-count labels and root annotations, and reports the maximum-depth summary.
The learner-quality gate then checks whether that valid graph is also usable as a micro-learning catalog. Each skill has at least 4 failable
assessment items including a transfer-oriented multiple-choice check. Answer position and answer length are then audited per answer-count
bucket, in every bucket carrying enough single-choice items to read. Each option slot has to be correct at a rate inside a band around
chance, between 0.10 and 0.45 of the bucket, so no slot either dominates or sits unused. The length check is two-sided: the rate at which
the correct option is the strictly longest answer and the rate at which it is the strictly shortest answer must both land between 0.12 and
0.42 of the bucket. Bounding only the long end would relocate the cue rather than remove it, because an answer that is never the longest
teaches a learner to pick the shortest one. Multiple-choice correctness is kept from collapsing onto a template: no option index sits in
the correct set for more than 0.60 of multiple-choice items, and no single correct subset covers more than 0.30 of them. Objective strings
may repeat, but only within a bound — at most four skills may share one objective before the gate reports templated filler. No single
name-normalized answer body is reused across more than 0.12 of skills, with the declared shared rubric exempted as described below. Skill
tags include both stratum and topic context, and backward cross-stratum references are kept out of the exported prerequisite path.
The one deliberately shared instrument is disclosed rather than hidden. Each skill carries bespoke domain assessment items; most skills
additionally carry a shared transfer-and-misuse rubric, a uniform metacognitive instrument applied across the curriculum and marked as
such. The answer-reuse gate exempts that declared rubric, requires every skill to retain at least one bespoke item, and bounds every other
name-normalized answer body to at most 0.12 of skills. The exemption is what makes that bound readable rather than misleading: each
per-family review rubric spans a whole stratum-by-role family, a footprint above a tenth of the curriculum and therefore above the bound
the gate enforces on authored answers. We name the instrument and the exemption instead of letting a shared rubric sit quietly under a
threshold, and the bespoke-item requirement keeps the rubric scaffolding rather than a substitute for per-skill assessment.
These checks establish structural assessment discipline, not psychometric validity. They show that quiz-gated completion is not thin,
duplicated, or position-biased under the local validators; they do not estimate item reliability, construct validity, learner ability, retention,
transfer, or differential item functioning.
The headline depth of 75 is not a single conceptual chain. It is dominated by the per-stratum mastery and open-science review ladders, which
sequence skills for completeness rather than by genuine conceptual prerequisite. Excluding only those review and meta scaffold families and
keeping every genuine concept family (mathematics through applications, plus the dynamical, geometric, hierarchical, implementation and
cognitive-security families), the substantive concept chain a learner must climb is 33 deep; the difference is review and mastery sequencing,
not conceptual dependency. We report both so the headline number is not read as the depth of the conceptual curriculum.
The full DAG is the most faithful view, but it is intentionally dense: learners need to see both the individual prerequisite chains and
the curriculum’s aggregate traffic pattern. fig. 2 adds that second view. Rows are prerequisite strata, columns are dependent strata, and
the right-hand bar gives the node count in each stratum. Diagonal cells show within-stratum reinforcement; upper-triangular cells show
forward conceptual transitions. A populated lower-triangular cell would mean a backward learning-path dependency, which the quality
gate rejects before export.
The same graph also needs diagnostic views for authors. fig. 3 orders high-outdegree skills by how much downstream curriculum they
unlock, so review can focus on nodes where a weak explanation would create wide learner risk. fig. 4 complements that global view
with a representative long dependency path. The figure keeps the route line readable while a checkpoint ledger names the start, stratum
transitions, and endpoint, so a reader can inspect one concrete route without zooming the full DAG or relying on angled micro-labels.
fig. 5 audits the semantic-prerequisite gate, which checks that no skill is assessed on one of the gate’s tracked load-bearing concepts unless
a skill defining that concept sits in its transitive prerequisite closure. The gate reads assessment text only, and it considers only definers at
or before the using skill’s own stratum, so a forward reference to a later concept is not counted as a missing prerequisite. For each concept
the figure reports how many skills reach the defining skill through their prerequisite closure and how many are assessed on the concept,
so an inversion in which a learner met a concept before they could have learned it would be visible rather than buried in the dense graph.
The quiz surface is itself part of the curriculum result rather than a decorative add-on. fig. 6 summarizes the assessment-depth invariant
by stratum. Every skill now carries the same 4-question floor, so completion is not certified by a single recognition item or by a thin
two-question check.
The latest expansion also needs a balance check. fig. 7 summarizes skills, subjects, assessment questions and incoming prerequisite edges
by stratum. This makes a different false-certification path visible: the graph can be larger while still overloading one pedagogical layer or
leaving another underdeveloped.
Having established the graph surface, the next result is source support. These audits ask whether the manuscript’s scholarly claims are
connected to the roles its citations are supposed to play.
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Figure 1: The Active Inference curriculum as a prerequisite DAG generated from the current curriculum graph after structural checks.
Each point is one skill, color encodes the pedagogical stratum, horizontal position follows the stratum order, vertical position follows
longest prerequisite depth, and gray segments encode exported learning-path dependencies. Wrapped stratum chips report live skill totals,
the two foundational roots are labeled, and the inset reports the live skill, subject, edge, and maximum-depth counts. This figure is an
overview for structure and reachability; it does not by itself certify prose quality, assessment quality, or scholarship support.
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Figure 2: Prerequisite-edge flow across strata generated from every exported learning-path edge. Rows are prerequisite strata, columns are
dependent strata, cell labels count edges, and the side bar reports how many skills live in each stratum. Diagonal cells show within-stratum
reinforcement, upper-triangular cells show forward conceptual transitions, and a populated lower-triangular cell would expose a backward
prerequisite that should be rejected before export.
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Figure 3: Graph bottlenecks in the exported prerequisite network generated from downstream dependent counts. Bars rank the top
fan-out skills, y-axis labels pair skill ids with readable names, hatching and stratum chips identify pedagogical layer, and annotations
report dependent count and prerequisite depth for each high-impact node. The figure is a review-priority surface: it identifies where a
weak explanation, stale citation, or thin quiz would propagate to many later skills even though the graph remains valid.
Figure 4: Representative skill-neighborhood path generated from the longest-path diagnostic in the current graph. The line orders skills
by prerequisite depth, stratum bands and vertical position encode pedagogical layer, and a checkpoint ledger names the start, every
stratum transition, and the endpoint. This view deliberately trades global completeness for readability: it lets authors audit one concrete
foundation-to-application pathway without relying on the dense full-DAG overview or tiny rotated labels.
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Figure 5: Semantic-prerequisite coverage generated from the live graph and the quality gate’s concept map. For each load-bearing concept
the open marker and stem report skills whose prerequisite closure reaches the concept’s defining skill, while the filled marker and direct
label report the assessed subset inside that reachable set. The subtitle and badge report the gate verdict so a passing curriculum shows
zero skills tested on a concept they could not yet have learned.
Figure 6: Quiz assessment depth by stratum generated from the skill assessment tuples. The bar panel counts quiz-bearing skills per
pedagogical stratum, and the accompanying audit table reports, per stratum, the minimum questions per skill, below-floor counts, questions
per skill, and a PASS/REVIEW verdict against the configured 4-question failable floor. The intended check is not average quiz length but
minimum learner friction: any below-floor mass would show that a skill can be completed without the configured 4-question floor.
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Figure 7: Curriculum stratum-balance audit generated from current graph statistics after structural checks. Three panels report the same
underlying build: the left panel shows skills and subjects per pedagogical stratum, the middle panel shows assessment-question volume
and incoming prerequisite edges, and the right panel shows companion per-unit (per-skill and per-subject) normalized measures. Reading
the panels together shows whether expansion is distributed across the learning path or whether one layer is absorbing disproportionate
topic, quiz, or dependency load. The figure bounds the claim to catalog structure; it does not measure learner outcomes.
9 Results: Scholarship Coverage and Evidence-Status Audits
9.1 Citation coverage, locator health, and evidence-class auditing
The manuscript’s source coverage is also treated as a generated result. Every authored citation is parsed from the section files, matched
against references.bib, and mapped through data/scholarship_sources.yaml into a source lane and support role before the build writes
figures. The current audit covers 188 cited sources; that number is generated from the manuscript and source matrix, not maintained as
a prose claim. fig. 8 shows both sides of the contract: the left panel compares cited sources with the matrixed source pool for each lane,
while the right panel shows which manuscript sections carry each lane. The audit is deliberately orthogonal to the numeric claim registry:
one gate protects computed quantities; the other protects the scholarly support surface.
The open-science education layer is also explicit. Active Skillference cites the Active Inference Institute’s public education ecosystem,
including the open-source course infrastructure, the START curriculum-generation project, and the Institute website’s open-science educa-
tional mission [ Active Inference Institute , 2026b,d,c]. The generated local Learn material also links every exported skill to commit-pinned
external material pointers, including the Active Inference Institute cognitive knowledge base [ Active Inference Institute , 2026a]. fig. 9
shows both surfaces together: cited public sources remain visible, and the course-material source rows expose how the cognitive knowledge
base enters the generated skill records.
The source matrix is then checked a second way: source locators must be syntactically specific, documentation entries must use URL
locators, and each support role must be substantive enough to explain why the source is present. fig. 10 reports the resulting offline source-
health audit and provides a place for optional online DOI/URL reachability checks to surface when release review requires currentness
evidence.
The validation surface is intentionally visible rather than left as a hidden CI detail. fig. 11 summarizes which gates bind generated JSON,
negative controls, manuscript text, dashboard artifacts, release documentation, and SkillTree handoffs. Its count panel is rebuilt from
the same graph, claim registry, scholarship audit and roadmap state that produce the curriculum and manuscript variables. The result
is a falsifiable release claim: if a count, citation, figure, dashboard artifact, release inventory row, or roadmap item drifts away from its
generator, the validator fails before the manuscript or SkillTree export can be presented as current.
fig. 12 makes the negative controls behind that release view explicit. It separates missing registry rows, missing files, missing manuscript
labels, missing dashboard artifacts, missing roadmap metrics, and stale manifests so the validation layer is judged by concrete false-
certification classes rather than by the presence of a polished figure alone.
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Figure 8: Scholarship coverage audit generated from manuscript citations, BibTeX keys, and the source-role matrix. The left panel
compares cited versus matrixed sources by scholarship lane; the right panel maps manuscript sections to citation lanes. This figure makes
support distribution inspectable: a large bibliography is not treated as sufficient unless sources are cited in the sections and roles where
they support specific claims.
Figure 9: Active Inference Institute open-science learning sources across scholarship support and generated course-material pointers.
Summary cards report local material records, external pointers, cognitive knowledge-base pointers, and topic-matched KB skills; ledger
rows distinguish the course repository, cognitive knowledge-base source, Institute homepage, website repository, and START source with
locator and role text. The website repository is a community/resource-hub pointer rather than a course or knowledge-base record. The
figure records pinned links and cited support surfaces; it does not claim endorsement, completeness, or copied ownership of the public
education ecosystem.
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Figure 10: Scholarship source-health audit generated from locator syntax, lane assignment, and support-role checks. Status cards separate
matrixed source count, offline validation issues, and online-check mode, while the ledger rows distinguish local metadata health from
optional network reachability. The figure separates two different claims: a source can be present in the matrix and still require locator,
role, or currentness repair before release.
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Figure 11: Validation-surface matrix for Active Skillference. Each cell is derived from the on-disk presence of the validator or test file
that binds that artifact surface: a cell reads “bound” only when every file named in its binding exists, so deleting a gate flips its cell
to “n/a” rather than leaving a stale “bound” claim. Row and column labels state bound-cell totals, and each cell carries text, so the
figure remains readable without heatmap color. The side panel reports live evidence counts for skills, subjects, dependencies, assessment
questions, claims, cited sources, and registered figures, read directly from the graph, claim registry, scholarship audit, and figure inventory.
The interpretation boundary is important: a bound cell means a named validator file exists for that artifact surface (not that the gate is
guaranteed sound), while an unbound cell marks a surface that is out of scope for that gate.
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Figure 12: Validation-intent mismatch matrix for Active Skillference. Rows enumerate concrete false-certification classes: a rendered
PNG omitted from the registry, a registered figure without a readable file, a registered label omitted from the manuscript, a registered
figure omitted from the dashboard artifact roster, a missing roadmap live metric, and a stale generated manifest. Columns show whether
the source-of-truth, manuscript gate, or dashboard gate is expected to reject each mismatch before release. The right panel records the
planned registered-figure count and which optional audit surfaces were enabled for the render.
The same contract is now represented inside the curriculum rather than only enforced at release time. The visual-evidence slice adds a
prerequisite-gated authoring path for question contracts, data grain, encoding strength, annotation boundaries, static/mobile readability,
and negative-control validation. fig. 13 shows that path as generated curriculum evidence: each rung names the live skill that carries the
review step, while the registry and dashboard expose the same structured chart-contract fields. This is a deterministic artifact-integrity
claim, not a claim that the visual curriculum has been externally tested with learners.
The scientific-claim layer adds one more audit view. fig. 14 is generated from manuscript claim-support anchors and the support matrix in d
ata/manuscript_claim_support.yaml. Its rows separate computed, descriptive-count, simulation, literature, scope-limit, delivery and visual
claims; its columns show whether each class is backed by claim tokens, manuscript variables, citations, figures, validators or limitation
markers. This is not a statistical result. It is a release integrity check that prevents strong scientific wording from passing without a
declared evidence class.
The platform evidence above is intentionally separated from the small mathematical examples. Those examples are still central teaching
artifacts, but their scientific meaning depends on the supplement’s model assumptions and regime boundaries.
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Figure 13: Visual-evidence contract ladder generated from the live curriculum graph and figure registry metadata. Six numbered rungs
move from question contract through data grain, encoding strength, annotation boundary, static/mobile survival, and negative-control
validation. Each rung names the skill identifier that carries the review obligation and reports whether the skill is registered in the
graph. The figure documents the authoring and validation lifecycle for visual evidence; it does not claim learner effectiveness or external
deployment performance.
Figure 14: Scientific evidence matrix for manuscript claims generated from claim-support anchors and
data/manuscript_claim_support.yaml. Rows are claim-support types, columns are evidence classes, cell counts show how many
anchored claims use each evidence class, and the side panel counts claims per type. The purpose is epistemic classification rather than
statistical inference: computed, descriptive, simulation, literature, scope-limit, delivery, and visual claims are forced to declare the kind
of support they actually have.
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10 Results: Kernel-Backed Teaching Examples and Model Boundaries
10.1 Kernel-generated mathematical examples and model boundaries
The main result is not that this repository contains two illustrative Active Inference plots; it is that the plots are generated from the
same kernel registry that binds the curriculum and manuscript. The variational free-energy sweep fig. 22 and the expected-free-energy
policy comparison fig. 23 are therefore treated as supplemental scientific walkthroughs rather than headline platform evidence. They
remain central teaching artifacts, but their interpretation belongs with the derivation, model assumptions, and regime boundaries in the
supplement. This separation keeps Results focused on what the platform demonstrates: structure, provenance, scholarship, artifact wiring,
and SkillTree handoff all regenerate together.
The final result is delivery: the same validator-backed artifact that supports the paper also becomes the SkillTree import payload and
dashboard inspection surface.
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11 Results: SkillTree Export and Learner-Surface Handoff
11.1 SkillTree export, learner-surface handoff, and generated front matter
All registered claims are produced by tested kernels; every claim token in the 630 skills resolves, and the bare-number linter reports zero
violations across descriptions, assessment prompts, and correct numeric answer options. The scholarship audit similarly resolves every
authored citation against the bibliography and source-role matrix. The provenance-rendered graph exports to a SkillTree project of 111
subjects, 630 skills, 1199 learning-path dependencies, 630 quizzes, and 2520 assessment questions, with no unresolved tokens in the output.
The source reaches the configured src/ coverage gate with no mocks; exact current coverage is reported by the test run rather than asserted
in prose.
The dashboard delivery surface exposes the same contract. Its status payload cross-checks SkillTree export counts against build stats,
tracks the claim ledger and figure artifacts, exposes the manuscript artifact audit, mounts the official learner display when SkillTree is
online, and gives each skill a direct handoff into that display’s subject and quiz-gated skill pages. This avoids the common false demo in
which a polished local UI merely copies course text: the local page can inspect and preview artifacts, but the learning and quiz actions
are routed to the seeded SkillTree project.
Prose cannot settle that distinction, so the delivery surface is also recorded directly. fig. 15 shows the landing view reporting exported
counts, live SkillTree dependency parity, and artifact presence together. fig. 16 shows SkillTree’s own client library rendering the seeded
project inside the local page, with its native points, level, progress rings, and ranking; that is the observation the export-lands-in-the-
platform reading needs, and no exported JSON file can supply it. fig. 17 shows the generated lesson record for one skill beside its
commit-pinned upstream pointers, so the material boundary is visible rather than asserted. fig. 18 shows the captioned prerequisite-flow
table that carries the heaviest subject-to-subject relations for readers who cannot use the lane drawing, which is a ranked summary of
that structure rather than a complete substitute for every edge, and fig. 19 shows deployment-mode readiness beside the browser-visible
runtime payload, where the admin password and client secret are absent. These five images are recordings rather than renders: unlike
every other figure here they cannot be rebuilt from project data, so they are version-controlled, marked in the figure registry as captured
evidence, and stamped with the dashboard URL, viewport, live counts, capture timestamp, per-image digest, and full-stack mode of the
run that produced them. Being recordings, they are also the one family exempt from the check that every registered figure was emitted
by the render pipeline; the exemption is recorded in the registry rather than left implicit, and their integrity rests on the digest and
provenance record instead.
The front matter follows the same rule. The cover image is generated from the live graph, claim registry and roadmap metrics, and the
page-two named-skill atlas is generated from every authored skill name in the curriculum graph. Both images are checked by publication-
asset audits before rendering. The cover uses a square layout so it can serve both PDF title-page and web artifact contexts; the atlas
uses a wide page-two layout so the skill-name vocabulary remains readable by showing a reduced weighted-term sample and representative
dependency checkpoints rather than a complete printed index. Neither asset is a numbered scientific figure: the dashboard exposes them
as cover-category artifacts, while figure_registry.json remains reserved for manuscript figures that are declared and cross-referenced in
the text. This prevents polished front matter from becoming a false-certification path outside the artifact contract.
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Figure 15: Learner dashboard landing view captured from the running local stack. The metric row reports exported skills, subjects,
quizzes, maximum prerequisite depth, registered claims, and artifact presence, while the dependency metric shows live SkillTree parity
against the export rather than the exported count alone. The evidence list below restates those machine-checked conditions in words:
export counts, live graph parity, local auth readiness, course-material coverage, and figure-contract completeness are surfaced together on
one page instead of being scattered across separate reports. The list continues past the fold; the frame shows the first ten of its rows.
The artifact tally counts this evidence family among the tracked artifacts, so it is a completeness check on the roster rather than an
independent audit of these images.
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Figure 16: The Official Display tab captured while the embedded SkillTree learner client is mounted against the seeded project. The
overview header, overall points, level and level progress rings, the achieved-skills bar reading zero of the full exported skill count, the
personal rank card, and the point-history panel are all rendered by SkillTree’s own client library inside the project dashboard, which is
the demonstration that the export lands in the platform runtime rather than in a project reimplementation of it. Every progress value
reads zero and the point-history panel shows its first-point empty state, because the capture uses a freshly seeded local identity that has
completed nothing.
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Figure 17: The Curriculum tab captured with the generated lesson record for the variational free energy skill opened through its Learn
action. The panel shows the concept context, prerequisite position, downstream role, assessment summary, and the commit-pinned external
source pointers generated for that skill, alongside the handoff buttons that return the learner to the official SkillTree subject and quiz
routes. It is the delivery counterpart to the course-material provenance rule: local lessons link out to pinned upstream sources instead of
copying them.
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Figure 18: The accessible prerequisite-flow table in the Graph tab, scrolled past the subject lane canvas that sits above it in the same
panel. Table and canvas are built from the same server-computed flow summaries, so the prerequisite structure reaches a keyboard or
screen-reader user as captioned rows with real header cells rather than only as a drawing. Each row is one directed subject-to-subject
relation: a prerequisite subject, the dependent subject that requires it, and how many skill dependencies run between them, ordered by
weight.
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Figure 19: The browser-visible runtime payload in the Runtime tab, scrolled past the deployment-mode matrix to the local development
block. The default password field is an empty string and its availability flag is false, so the page reports the local username and the
recovery commands while withholding the credential itself; no client secret and no bearer token appear anywhere in the payload. The
same view carries the SkillTree reachability block and the live dependency parity block reporting complete parity with no missing or extra
edges, which is the auth-boundary evidence in the form the browser actually receives it.
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12 Discussion and Conclusion: What the Artifact Establishes—and What It Does
Not
12.1 Principal findings: a claim-calibrated curriculum artifact
Active Skillference establishes a bounded engineering result: a contested formal subject can be represented as an inspectable prerequisite
graph whose structural, quantitative, scholarly, and delivery contracts are checked separately. The curriculum graph is a validated DAG;
learner-facing computed values are resolved from tested kernels; citations are assigned source roles; and generated figures, manuscript
references, and SkillTree payloads are audited as connected artifacts. These are properties of this checkout and its build process, not
evidence that the selected order is optimal for every learner or that the underlying scientific framework is empirically confirmed.
12.2 Interpretation: infrastructure rather than intelligence or personalization
The mathematical examples and the curriculum infrastructure operate at different levels. The kernels implement small, model-dependent
demonstrations of inference and policy evaluation. The curriculum layer orders and assesses concepts. The provenance layer records how
values and sources enter the learner-facing artifact. None of these layers is an intelligent tutor, a general-purpose reasoning system, or
an adaptive learner model. In particular, the term agent in the mathematical examples refers to a formal model with specified states,
observations, preferences, and policies; it should not be read as a claim about consciousness, general intelligence, autonomy, or human-like
learning.
The related-work literature supports this separation. Active Inference formulations can connect belief updating, action selection, learning,
and epistemic behavior, but their predictions depend on the generative model, preferences, approximation, and policy set [ Rao and Ballard ,
1999, Friston, 2013, Ramstead et al. , 2018, Friston, 2019]. The supplement therefore keeps the VFE/EFE derivations, sign conventions,
and preference assumptions visible instead of treating one computed policy comparison as a universal account of intelligent behavior.
Learner-state models such as knowledge tracing and Q-matrix cognitive diagnosis belong to a separate adaptive-learning evidence layer
and are cited here as comparison boundaries, not as components of this artifact [ Piech et al. , 2015, Tao et al. , 2024].
12.3 Implications: future adaptive learning systems and open-science interoperability
The delivery contract is useful precisely because it does not hide ownership boundaries. The exporter creates a SkillTree-compatible
Project -> Subject -> Skill profile with learning-path dependencies and quiz associations. The local dashboard exposes generated artifacts
and opens official subject and quiz routes, while SkillTree remains the owner of progress, ranking, scoring, completion, and authentication.
This makes the repository a candidate foundation for a later adaptive study, not an adaptive system in the present release.
A genuinely adaptive extension would require learner response data, a validated model of learner state, a transparent rule for selecting the
next skill, item-level measurement, privacy and consent controls, and prospective evaluation against a preregistered comparator [ Doignon
and Falmagne , 1985, Fu and Fang , 2025, Design-Based Research Collective , 2003, Liu and Khalil , 2023, Piech et al. , 2015, Tao et al. ,
2024]. Knowledge tracing and Q-matrix cognitive diagnosis are comparison literatures for those stronger commitments, not components of
the present export. Provenance and graph validation would remain necessary infrastructure, but they would not by themselves establish
personalization quality or learning effectiveness.
The community contribution should be read at the same bounded level. The Active Inference Institute’s public website repository presents
a public resource hub for learning, research, participation, and contribution, with public repositories and a documented boundary between
internal inputs and verified public projections [ Active Inference Institute , 2026e]. Active Skillference contributes a complementary artifact
to that open-science environment: it packages a prerequisite graph, kernel-backed examples, source-role audit, and SkillTree handoff into
a rerunnable educational release. The relationship is interoperable and community-facing, not institutional: this repository does not claim
to be an Institute product, an official curriculum, or an endorsement of any contributor or public pathway.
12.4 Limitations and planned empirical evaluation
The release state is machine-readable. As of 2026-07-31, tasks.yaml records 78 roadmap tasks: 74 completed milestones and 4 open items
across metadata, skilltree, validation. TODO.md is rendered from the same source. fig. 20 shows the current distribution of open versus
completed work by surface and release class, keeping release-readiness language tied to the validator state that drives the roadmap.
The limitations are therefore substantive rather than cosmetic. The kernels cover fixed, small discrete teaching cases and bounded
predictive-coding or planning demonstrations; their conclusions depend on the stated model, preferences, inputs, and approximation
regime. The runtime depends on SkillTree for live progress and quiz state, and this release contains no learner-response data, learner-state
inference, retention measure, transfer measure, efficacy estimate, or differential-benefit analysis. The planned pilot described in sec. 13
and the fuller boundary analysis in sec. 19 define the evidence needed before those claims can be made.
12.5 Conclusion: a bounded, reproducible contribution
The strongest conclusion is narrower and more useful: Active Skillference makes the construction, provenance, and delivery of a formal
Active Inference curriculum more inspectable, reproducible, and auditable. It establishes an infrastructure artifact and an open-science-
compatible handoff surface; it does not establish intelligent tutoring, personalization, learner improvement, or empirical confirmation of
the Free Energy Principle.
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Figure 20: Roadmap status by artifact surface generated from tasks.yaml and the live metrics stored in the roadmap source data. The first
panel counts roadmap items by status, the middle panel groups remaining open or blocked work by surface and release class with direct
EXT/GIT segment labels, hatching, and surface totals, and the final panel reports current curriculum, assessment, scholarship, and figure
metrics. The figure separates external blockers from git-authorization hygiene using machine-readable roadmap data, so release-readiness
text must agree with the same source.
13 Evaluation Setup: Deterministic Artifact Tests and Planned Learner Study
13.1 Deterministic model suite and fixed inputs
All quantitative results derive from small, fully specified discrete models evaluated by the kernels with a fixed seed ( 20260609). Within
the supported Python, NumPy and Matplotlib environment, the model inputs and graph ordering are deterministic; this is a software-
environment reproducibility claim, not a cross-hardware bitwise proof.
• Noisy-sensor model (Bayes and VFE results): two latent states, present and absent, with prior 𝐷 = [0.01, 0.99] and likelihood
rows [0.99, 0.05]and [0.01, 0.95]over positive and negative observations. These model parameters are registered as kernel claims,
so the prose cannot drift from src/kernels/examples.py. This model emits the posterior, evidence probability, negative log-evidence,
and free-energy sweep.
• Informative-sensor model (EFE decomposition): likelihood rows [0.9, 0.1]and [0.1, 0.9]with observation preference 𝐶 = [0.9, 0.1].
This isolates epistemic value under uncertain versus certain beliefs.
• T wo-action agent: a neutral-preference sample/commit model with prior 𝐷 = [0.5, 0.5]. The sample action preserves the uncertain
belief; the commit action collapses belief mass toward one state. A sensor-precision sweep over this same model identifies the
sample/commit switch.
• Information examples : a fair coin, a fair six-sided die, a biased 0.9/0.1 coin, a one-in-a-hundred event, and a correlated joint
distribution with entries [0.4, 0.1, 0.1, 0.4].
13.2 Curriculum export and runtime ownership boundary
The curriculum spans the eight strata declared in docs/manuscript/config.yaml. The maximal-enumeration floor is 630 skills (achieved: 630),
and the current catalog adds practice-and-translation material across all strata. That material specifies opportunities to rehearse model
checking, debugging, policy design, implementation review, and application evaluation; it does not establish that learners will acquire those
abilities. Skills default to 100 points, one occurrence to completion, and Quiz self-reporting so completion is assessment-gated. SkillTree
export targets project id activeInference.
13.3 Reproduction, delivery, and release checks
The generation pipeline is deterministic and runs from the project root with no Docker daemon and no browser. Only validate_course_m
aterial_sources.py --online leaves the machine, and only to re-resolve the cited course-material sources:
uv run python scripts/build_curriculum.py # validate the DAG, emit stats
uv run python scripts/run_kernels.py # claims, scholarship, figures, artifact audit
uv run python scripts/build_course_materials.py
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uv run python scripts/validate_course_material_sources.py --online
uv run python scripts/validate_manuscript_artifacts.py
uv run python scripts/validate_docker_digests.py
uv run python scripts/validate_release_readiness_inventory.py
uv run python scripts/export_skilltree.py # provenance-gated SkillTree JSON
uv run pytest --cov=src --cov-fail-under=90 # coverage gate
validate_docker_digests.py belongs in that offline list because it reads docker-compose.yml as a file rather than talking to a daemon:
it fails when any Compose image is named by a mutable tag instead of a sha256 digest, which is what keeps the live rehearsal below
reproducible instead of tag-dependent. Its --file flag points the same check at another Compose file.
The remaining checks are live. All of them need a reachable Docker daemon, and the browser-facing ones additionally need the chrome-de
vtools-axi Chrome driver. They fail closed rather than downgrading to the artifact-only dashboard:
./scripts/run_skilltree.sh # docker compose up + seed
./scripts/run_learning_dashboard.sh # SkillTree + seed + local dashboard
uv run python scripts/reconcile_live_dependencies.py --check-only
uv run python scripts/validate_live_skilltree_dashboard.py --require-live-admin --require-live-dependencies --require-figure-contracts
uv run python scripts/validate_skilltree_form_login.py
uv run python scripts/validate_dashboard_console.py --require-browser --require-dashboard
uv run python scripts/validate_full_app_stack.py --app-ready --skip-regenerate
uv run python scripts/validate_full_app_stack.py --skip-regenerate
run_learning_dashboard.sh regenerates the curriculum, claims, course-material index, figure registry, manuscript artifact audit, figures,
and SkillTree export before seeding, so the learner surface is inspected as an actual course interface rather than a static preview. With
local auth enabled, the dashboard serves http://127.0.0.1:8765/api/skilltree/token as a trusted-client bridge for the official SkillTree
display; it never embeds the project client secret in static JavaScript. The dashboard can also run in artifact-only mode with uv run pyth
on scripts/serve_learner_dashboard.py --check , which verifies that the server, route map, and artifact inventory load without requiring
Docker.
The live gates divide by what they are willing to assert. reconcile_live_dependencies.py --check-only reports live prerequisite parity
against output/data/skilltree_project.json without writing anything; dropping --check-only writes the missing edges into the running
service, with --attempts-per-edge , --max-edges, and --max-runtime-seconds bounding a chunked local run. validate_live_skilltree_d
ashboard.py checks the SkillTree and dashboard status endpoints, export-count agreement, artifact completeness, and the trusted-client
token exchange by default; its --require-live-admin, --require-live-dependencies, and --require-figure-contracts flags escalate that to
live admin skill-count parity, exact live dependency parity, and figure-registry-to-dashboard contract agreement. validate_skilltree_form
_login.py drives the raw SkillTree FORM login in a real browser through to the administrator surface and redacts local credentials from its
JSON summary. validate_dashboard_console.py with --require-browser --require-dashboard fails on any dashboard console error other
than the allow-listed upstream Inception status probe; without those flags it reports a skip when the browser or dashboard is absent.
validate_full_app_stack.py composes the live sequence end to end: Compose bring-up, seeding, dashboard start, dependency reconcili-
ation, the live dashboard checks, the form login, and the console gate. --app-ready skips dependency reconciliation and labels its JSON
summary app-ready; the flagless run additionally requires every exported prerequisite edge to be present in the live service and labels the
summary full-parity. --skip-regenerate reuses existing output/ artifacts instead of rerunning the generation sequence, --skip-browser
drops the browser stage, and --keep-running leaves the dashboard up for manual inspection. Each successful run writes a redacted JSON
summary under the local state directory. That evidence is per-run and host-local: it records what one rehearsal observed on one machine,
not a public deployment.
13.4 Planned learner-outcome study (design only)
The next learner-outcome step is preregistered in experiment_plan.yaml as design-only work, not as a result. The planned pilot uses adult
volunteer consent, data minimization, and deidentified study ids before any SkillTree-owned progress export is analyzed. Learner progress,
quiz attempts, completion, and ranking remain owned by SkillTree; the local dashboard is not treated as a learner-truth database.
The planned measurement sequence is baseline concept items, an assigned SkillTree learning path, immediate post-items, delayed-retention
items, and an optional transfer prompt scored by a frozen rubric. Active-learning, cognitive-engagement, retrieval-practice, and evidence-
centered assessment sources motivate that design envelope, but they do not substitute for data from this curriculum [ Freeman et al. , 2014,
Chi and Wylie , 2014, Roediger and Karpicke , 2006, Mislevy et al. , 2003, Black and Wiliam , 1998, Shute, 2008, Monib et al. , 2025, Rof
et al. , 2024, Emin et al. , 2026].
Design-based research is the intended study posture because the intervention is a curriculum artifact, a SkillTree delivery configuration,
and a measurement workflow that must be studied in context and iterated before stronger outcome language would be appropriate
[Design-Based Research Collective , 2003, Barab and Squire , 2004]. Learning-analytics reviews and intervention meta-analyses motivate
the outcome-evaluation path but also make privacy, data protection, and instrumentation validity part of the study boundary [ Liu et al. ,
2025, Hernández-Campos et al. , 2025, Liu and Khalil , 2023, Sclater and Bailey , 2015]. The protocol will use pilot-and-feasibility reporting
and an intervention-description checklist so that implementation readiness is reported separately from definitive learner outcomes [ Eldridge
et al. , 2016, Hoffmann et al. , 2014]. The analysis plan is within-learner baseline-to-post change with delayed retention reported separately
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after the sample size, item families, exclusions, and rubric are frozen. This manuscript does not report those learner data and makes no
efficacy, retention, or transfer claim.
13.4.1 Browser-facing validation path
For end-to-end browser validation, open the dashboard, confirm the official SkillTree panel mounts the activeInference project, switch
to the curriculum panel, and open a skill’s Learn, Open Subject , and Quiz actions. Learn opens the local modular lesson record
and should not emit a direct /subjects/{subjectId}/skills/{skillId} SkillTree target. Open Subject reopens the dashboard with
skillsClientDisplayPath=/subjects/{subjectId} so the subject learning path, skill list, progress, and rank are exercised inside the offi-
cial SkillTree display. Quiz opens skillsClientDisplayPath=/subjects/{subjectId}/skills/{skillId}/quizzes/{quizId}, so quiz loading,
scoring rules, and completion remain SkillTree-owned rather than dashboard-owned.
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14 Reproducibility and Release Evidence: Regenerating the Artifact Chain
Every artifact in this work is regeneratable from source with fixed seeds; nothing in output/ is authoritative, and nothing learner-facing is
hand-typed.
14.1 Deterministic kernel outputs, stable graph ordering, and rerunnable counts
The kernels use natural-log units and either no randomness or an explicitly seeded NumPy generator; the worked examples register
rounded values so two runs of build_claim_registry() are byte-identical. The curriculum graph’s topological order processes ready nodes
in sorted-id order, so the exported SkillTree JSON is stable across runs.
14.2 Verification gates for code, prose, artifacts, and rendered surfaces
The gate tables treat the curriculum build as scientific computing: reported values should be traceable to workflow steps, custom code
should be tested and versioned, and research claims should be reported with enough transparency that readers can distinguish reproducible
local artifacts from future replication or learner-outcome evidence [ Sandve et al. , 2013, Wilson et al. , 2014, Munafò et al. , 2017].
The gates split into two families. The first runs anywhere the repository can be checked out, with no Docker daemon and no browser.
Offline gate Command Evidence
Tests + coverage uv run pytest --cov=src --cov-fail-under=
90
enforced aggregate floor: 90%; use
command output as the current coverage
number, and read the module caveat below
Kernel coverage within the coverage run kernel package coverage is checked inside
the same suite rather than asserted from
prose
Lint uvx ruff check src scripts tests clean required
Format uvx ruff format --check src scripts tests clean required
Types uv run mypy src no issues required
No mocks rg "unittest|MagicMock|@patch|autospec"
tests/
no mock factories should be present
Provenance scripts/run_kernels.py and scripts/export
_skilltree.py
every claim token resolves and the
bare-number linter reports zero violations
Scholarship every authored citation maps to a BibTeX
entry and source-role lane
enforced before figures
Figure artifacts generated figure registry, manuscript
labels/cross-references, PNG information
content, captions, alt text, and dashboard
figure artifacts agree
enforced by run_kernels.py and validate_m
anuscript_artifacts.py
Image pinning uv run python scripts/validate_docker_di
gests.py
every Compose image reference carries a
sha256 digest; the file is parsed, no daemon
is contacted
Release inventory uv run python scripts/validate_release_r
eadiness_inventory.py
checked against live git status, untracked
files, ignored outputs and blocker probes
Manuscript render parent template renderer citation and figure references must resolve
The second family is live. Every row below needs a reachable Docker daemon, and the browser rows additionally need the chrome-devt
ools-axi Chrome driver. These gates fail closed rather than downgrading to the artifact-only dashboard, which is why they are listed
separately instead of folded into the table above.
Live gate Command Evidence
Live dependency parity uv run python scripts/reconcile_live_dep
endencies.py --check-only
live prerequisite edges compared with the
exported project JSON without writing
edges
Live dashboard wiring uv run python scripts/validate_live_skil
ltree_dashboard.py --require-live-admin
--require-live-dependencies --require-fi
gure-contracts
live admin skill-count parity, exact live
dependency parity, and figure-contract
agreement
Raw SkillTree login uv run python scripts/validate_skilltree
_form_login.py
a real browser drives the FORM login through
to the administrator surface; the JSON
summary is redacted
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Live gate Command Evidence
Dashboard console uv run python scripts/validate_dashboard
_console.py --require-browser --require-
dashboard
no console errors except the allow-listed
upstream Inception status probe
App-ready stack uv run python scripts/validate_full_app_
stack.py --app-ready --skip-regenerate
Compose bring-up, seed, dashboard, login,
and console pass; the JSON summary is
labeled app-ready
Full-parity stack uv run python scripts/validate_full_app_
stack.py --skip-regenerate
the same sequence plus complete live
dependency reconciliation; the JSON
summary is labeled full-parity
14.2.1 Coverage floor caveat
The coverage floor is enforced on the aggregate, and the aggregate is not the same claim as every module meeting it. Four modules
currently sit below the floor while the suite as a whole clears it: src/integration/full_app_stack.py , src/skilltree/form_login_valid
ation.py, src/scholarship/source_health_network.py , and src/dashboard/payload.py . Each one is a boundary to something the offline
suite cannot reach — a Docker daemon, a browser subprocess, live HTTP, or the local-auth server path — and each is exercised instead
by the live gates in the second table. Two defects show why that exception is a routing decision rather than an excuse: a dashboard
content-security-policy directive that blanked the official SkillTree panel was invisible to every offline test and surfaced only in the browser
console gate, and a reconciler abort-on-slow-write surfaced only during a complete live dependency run. Read the aggregate number as a
property of the suite, not as evidence of uniform module coverage.
14.3 Release-evidence ledger and publication blockers
14.3.1 Release inventory comparison
Release evidence is recorded as a local, machine-checked ledger rather than as a clean public-release assertion. During review, the checkout
may intentionally contain dirty or untracked files; docs/release_readiness_inventory.md is the authoritative inventory for that state and
is regenerated with uv run python scripts/validate_release_readiness_inventory.py --write .
The default validator then compares the document with live git status --short , git ls-files -o --exclude-standard , git diff --stat ,
ignored generated-output examples, and the current external blocker probes. Those probes report three distinct positions, and collapsing
them into one blocked-or-ready summary would misstate all three.
docker info reaches the Docker/Colima daemon, so the Docker-dependent live rehearsal is no longer blocked. It has been run: valid
ate_full_app_stack.py --skip-regenerate completed against the pinned SkillTree image in full-parity mode, with the live dependency
graph matching the export edge for edge, the live admin skill count matching the exported catalog, every registered artifact and figure
contract agreeing, the raw FORM login reaching the administrator surface, and the browser console gate clean. What remains is to rerun
that sequence as a release handoff rather than as a development rehearsal, on the commit that is actually released.
git ls-remote against the public ActiveInferenceInstitute/Active_Skillference target now succeeds, and that is precisely why the
validator distinguishes reachability from publication: the target exists and holds no refs. An exit-code-only check would read that empty
placeholder as ready. The inventory therefore reports it as reachable but empty, and DOI, archive-deposit, and public-repository metadata
fields stay blank until refs actually land there. Nothing in this repository is published at that address today.
The independent cross-vendor audit is closed in tasks.yaml, not pending. Its recorded scope is a point-in-time, read-only static review of
the dashboard, local-auth, and full-stack sources and their tests, and within that scope it returned no unresolved critical or high findings.
It did not cover dependency and supply-chain exposure, secrets in git history, or dynamic runtime testing, so it closes a specific review
task rather than establishing operational assurance. Branch hygiene remains open while the checkout still carries modified tracked paths.
Reproducible-workflow guidance motivates the rerunnable command ledger, versioned code and transparent artifact state [ Sandve et al. ,
2013, Wilson et al. , 2014], while F AIR and transparency-oriented open-science standards motivate the metadata and evidence-ledger shape
[Wilkinson et al. , 2016, Nosek et al. , 2015, Munafò et al. , 2017]. These sources do not turn a dirty local checkout into a public archive.
Software-citation and machine-readable metadata guidance define the repository-level citation files and release metadata surfaces [ Smith
et al. , 2016, Citation File Format Project , 2026, CodeMeta Project , 2026], while GitHub and Zenodo documentation keep DOI language
tied to a real public target and deposit [ GitHub Docs , 2026, Zenodo, 2026]. SPDX, OSI and REUSE references bound the MIT and third-
party-notice signposting [ SPDX Workgroup , 2026, Open Source Initiative , 2026, Free Software Foundation Europe , 2024]; they do not
provide legal advice, relicense third-party materials, or remove the need for separate deployment review. A release record should include
the base commit hash, whether the worktree was dirty, the generated inventory, the commands in the gate table above, and the artifact
manifests in output/data/ and ../figures/figure_registry.json; it should not promote this local evidence into a DOI or public-repository
claim until the external blockers are resolved.
The release inventory also carries a machine-checked Key Artifact Hashes section. During validation, it records SHA-256 digests and
byte sizes for output/data/claims.json , output/data/manuscript_variables.json , output/data/skilltree_project.json , ../figures/figu
re_registry.json, and every registered figure PNG currently named by the figure registry. The same validator rejects stale hash rows,
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so a regenerated artifact cannot silently drift from the manuscript’s local reproducibility evidence. This is still local evidence: without a
release tag, public archive, and clean external blocker probes, it remains a checkout ledger rather than an immutable publication bundle.
14.4 Provenance chain from kernel function to learner-facing value
The chain from kernel to learner is auditable end to end: a kernel function computes a value -> examples.py registers it as a named Claim
with its source function -> a skill or manuscript section references it as claim:NAME -> the exporter or render hook resolves it, failing
closed if it cannot -> the value appears in the SkillTree skill description, the manuscript, and output/data/claims.json. Changing a kernel
changes the claim, the rendered content, and the figures together. The separate manuscript artifact audit closes the non-numeric drift
path: a figure cannot be registered without a manuscript declaration, referenced without a declaration, missing from disk, or absent from
the dashboard figure inventory.
14.5 Disposable generated artifacts and output inventory
scripts/run_kernels.py writes output/data/claims.json (the provenance ledger), output/data/manuscript_variables.json (claims plus
curriculum counts), output/data/scholarship_coverage.json (the citation lane audit), output/data/scholarship_gap_report.json (the
scholarship frontier-source and claim-boundary audit), output/data/manuscript_artifacts.json (the figure-surface audit), output/data/co
ver_asset_audit.json (the first-page cover audit), output/data/front_matter_visual_audit.json (the page-two atlas audit), output/data/
course_material_index.json (local modular lesson records), and output/data/course_material_audit.json (the lesson-record completeness
audit), plus 21 registered figures plus the generated cover and named-skill atlas PNGs under ../figures/. scripts/export_skilltree.
py writes output/data/skilltree_project.json , the import-ready SkillTree project. scripts/z_generate_manuscript_variables.py is the
parent-template render hook: it runs the kernel/provenance gate and writes resolved manuscript copies under output/manuscript/ before
PDF, HTML, slide, DOCX, or EPUB rendering.
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15 Scope and Related Work: Model, Platform, and Outcome Boundaries
This section separates platform boundaries, Active Inference lineage, delivery/security/reporting context, and limitations so each evidence
class is stated next to the sources and caveats that bound it.
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16 Scope: Curriculum Platform, Runtime Ownership, and Non-Claims
16.1 Curriculum platform scope: not an inference engine or hosted LMS
Active Skillference is curriculum-authoring and delivery infrastructure, not a new inference engine. Its kernels are pedagogical, correctness-
first reference implementations chosen for legibility and testability, not for the performance or breadth of production toolboxes such as
pymdp for discrete state spaces, concise discrete-time tutorial/code treatments aligned with that ecosystem [ Heins et al. , 2022, van Oostrum
et al. , 2025], or SPM/DEM for continuous dynamic systems [ Friston et al. , 2007, 2008]. The contribution described here is how the
curriculum is built and verified : a validated prerequisite DAG with content-provenance binding, exported into an existing micro-learning
runtime. The repository is not an intelligent tutor, a learner-state estimator, or a general-purpose adaptive system.
The curriculum is deliberately broad (630 skills across all eight strata) and designed so that growth is a data operation: adding skills
or worked examples extends the catalog without changing the engine. Its new practice layer is still pedagogical rather than clinical
or operational advice: domain-fit, ethics, misuse, and evaluation skills train learners to ask better modeling questions, not to deploy
autonomous interventions without local review. Out of scope for this release are hosting and authentication of a production SkillTree
deployment, learner analytics, and a textbook-length treatment of every individual skill. The local dashboard narrows that boundary: it
is enough to show that the export can be inspected locally and, when a SkillTree runtime is available, opened through subject and quiz
routes; it is not a hosted LMS or analytics product.
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17 Related Work: Historical Foundations, Active Inference Evidence Status, and
Information Sources
17.1 Active Inference and FEP lineage: historical context and curriculum scope
17.1.1 F ormal lineage and model commitments
The FEP is commonly situated in a perception-as-inference lineage that includes Helmholtz, Bayesian-brain accounts of uncertainty, and
tutorial treatments that translate the mathematics into worked learning sequences [ von Helmholtz, 1867, Knill and Pouget , 2004, Bogacz,
2017, Sprevak and Smith , 2023]. Its modern variational formulation is developed in Friston’s early and review papers [ Friston et al. , 2006,
Friston, 2010], and recent overview work usefully maps the breadth of related research without removing the need for local modeling
commitments [Friston et al. , 2023, Zhang and Xu , 2024]. Mathematical reviews and textbooks situate the principle as a variational bound
on surprise and a process theory for perception, action, and learning [ Buckley et al. , 2017, Parr et al. , 2022, Smith et al. , 2022, Pezzulo
et al. , 2024]. Associative-learning work makes attention, prediction-error, blocking, and surprise phenomena explicit inside an Active
Inference account, which is useful pedagogical context but not direct evidence for this curriculum’s outcomes [ Anokhin et al. , 2024]. The
discrete POMDP formulation with 𝐴, 𝐵, 𝐶, and 𝐷 matrices is the direct basis for the kernels here [ Friston et al. , 2017a, Da Costa et al. ,
2020], while epistemic value and curiosity are treated by the expected-free-energy literature [ Friston et al. , 2015, 2017b, Millidge et al. ,
2021b, Sajid et al. , 2021, Champion et al. , 2026]. Recent variational-inference and message-passing treatments make those formulation
choices explicit by separating epistemic-prior, entropy-correction, and planning-correction assumptions rather than treating EFE as one
unqualified objective [ Nuijten et al. , 2026a,b, 2025]. The planning-frontier argument therefore leans on those EFE planning and message-
passing papers rather than on informal Lazy Dynamics talk material, because the manuscript needs source-backed claims about objectives,
factorization, and assumptions rather than a weak conference-talk framing.
17.2 Ancient, medieval, and early-modern foundations of inference, perception, and sequenced learn-
ing
This is a selective history of problems and design concerns, not a single-lineage account of Active Inference. The sources below are used
to motivate an inspectable prerequisite order and to distinguish historical analogies from modern formal commitments. They do not show
that modern FEP, Bayesian inference, variational inference, adaptive tutoring, or Active Inference existed before their later mathematical
and computational formulations. The historical source set spans demonstration, optics, method, probability, association, education, and
structured cognition [ Aristotle, 1901, Ibn al Haytham , 1989, Quintilian, 95, Bacon, 1620, Descartes, 1628, Huygens, 1657, Bernoulli, 1713,
de Moivre , 1718, Bayes and Price , 1763, Laplace, 1774, Newton, 1704, Berkeley, 1709, Locke, 1690, Hume, 1739, 1748, Hartley, 1749,
Condillac, 1754, Comenius, 1657, Locke, 1693, Rousseau, 1762, Kant, 1781].
17.2.1 Demonstration, vision, and ordered instruction before modern probability
Ancient demonstration and explanation provide one early vocabulary for separating reasons, evidence, and teachable order [ Aristotle, 1901].
Quintilian’s Institutio Oratoria presents a staged rhetorical education involving elementary studies, practice, revision, and progression
through increasingly demanding forms of performance [ Quintilian, 95]. Its relevance here is historical: it shows that ordered instruction
and repeated practice were explicit design concerns, not that ancient rhetoric implemented a learner model or adaptive tutor.
The medieval Arabic optical tradition adds a distinct perception strand. Ibn al-Haytham’s Book of Optics , represented here through a
critical edition and translation, treats vision through geometrical analysis, observation, and experimentally constrained accounts of seeing
[Ibn al Haytham , 1989]. That tradition belongs in the history of perception and evidence, but it is not a predictive-coding theory, a
Bayesian brain model, or evidence for the biological adequacy of the kernels in this release.
17.2.2 Early-modern method, probability , and experimental perception
Early-modern method sources connect ordered reasoning to observation without collapsing their differences. Bacon’s induction and
experimental program, and Descartes’s rules for moving through clear intuitions, deductions, and probable conjectures, make method and
sequence explicit [ Bacon, 1620, Descartes, 1628]. Huygens’s work on expectation in games of chance provides an early published probability
treatment, while Bernoulli, de Moivre, Bayes, and Laplace extend the mathematical history of repeated observation and inverse-probability
reasoning [ Huygens, 1657, Bernoulli, 1713, de Moivre , 1718, Bayes and Price , 1763, Laplace, 1774]. These sources support a history of
calculation and inference under uncertainty, not a claim that their concepts were already variational or computational in the modern sense.
Newton’s Opticks adds a controlled-experiment and sensory-observation strand: optical phenomena are varied, separated, and analyzed
through explicit apparatus and mathematical argument [ Newton, 1704]. Berkeley’s account of vision similarly treats visual perception as
involving learned relations among sensory cues, but neither source should be translated directly into modern generative models or neural
predictive coding [ Berkeley, 1709].
17.2.3 Eighteenth-century association, education, and structured cognition
Eighteenth-century sources then widen the educational and psychological context. Locke, Hume, Hartley, and Condillac discuss experience,
association, habit, sensation, and reflection in different philosophical registers [ Locke, 1690, Hume, 1739, 1748, Hartley, 1749, Condillac,
1754]. Comenius, Locke’s education writings, and Rousseau’s Émile make staged progression, practice, motivation, and learner-sensitive
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ordering visible as historical pedagogical concerns [ Comenius, 1657, Locke, 1693, Rousseau, 1762]. Kant provides a different account of
structured cognition and recognition [ Kant, 1781]. Together these sources justify treating sequence, representation, and evidence status
as design questions; they do not establish modern cognitive science, personalized learning, or empirical efficacy.
17.3 Evidence-status critiques and empirical boundaries
The same literature also warns against collapsing every layer of the project into a single grand claim. A variational identity, a process theory,
a simulation recipe, and an empirical neuroscience hypothesis have different evidential burdens. Technical critiques of the FEP’s particular-
physics assumptions and subsequent Bayesian-mechanics framing therefore belong in the curriculum as boundary-setting material, not
as adversarial footnotes [ Aguilera et al. , 2022, 2023]. Recent empirical reviews similarly motivate the platform’s wording discipline: the
curriculum can teach predictive coding and Active Inference as formal frameworks while still marking where direct empirical support
is incomplete or domain-specific [ Hodson et al. , 2024]. Psychology-and-psychiatry reviews add the same caution at the behavioral-
modeling level: Active Inference is productive for model fitting and explanation, but its theory-level empirical adequacy still requires direct
comparison with alternatives [ Badcock and Davey , 2024]. Gershman’s critique is useful here because it asks what distinctive predictions
the FEP contributes, which is exactly the kind of scope boundary this manuscript treats as a claim-support obligation [ Gershman, 2019].
The perception literature supplies a complementary, narrower evidence map: Bayesian vision and predictive-processing accounts connect
generative hypotheses to perceptual inference, while cortical-microcircuit, efficient-coding, perceptual-decision, and neurophysiological
reviews expose model-specific assumptions and unresolved empirical questions [ Yuille and Kersten , 2006, Clark, 2013, Hohwy, 2013,
Kersten et al. , 2004, Bastos et al. , 2012, Spratling, 2017, Summerfield and de Lange , 2014, Wei and Stocker , 2015, Millidge et al. , 2021a,
Smith et al. , 2020, Seth and Hohwy , 2020]. These sources sharpen the perception-as-inference lane without turning predictive processing
into a settled theory of the brain or this curriculum into a neuroscience intervention.
17.4 Information, variational, filtering, and application sources
17.4.1 Information and filtering sources
The information-theoretic and variational strata follow standard treatments of entropy, KL divergence, variational inference, graphical-
model message passing, the ELBO/free energy relationship, and rate-distortion or bottleneck views of task-relevant compression [ Shannon,
1948, Beal, 2003, Cover and Thomas , 2006, Tishby et al. , 1999, Bishop, 2006, Jordan et al. , 1999, Winn and Bishop , 2005, Blei et al. ,
2017]. State-space filtering, continuous-time predictive coding, and neural message passing are represented in the curriculum, with deeper
executable worked examples left for later releases; the relevant literature includes Bayesian filtering, Helmholtz machines, hierarchical
predictive coding, the graphical-brain account, DEM, generalized filtering, and generalized free energy [ Särkkä, 2013, Dayan et al. , 1995,
Rao and Ballard , 1999, Friston, 2008, Friston et al. , 2017c, 2008, 2010, Parr and Friston , 2019, Kataoka and Doya , 2026, Tschantz et al. ,
2023]. Probability, information, and Bayesian-analysis textbooks provide adjacent mathematical foundations for these implementations
and help distinguish a reusable inference method from any one Active-Inference formulation [ MacKay, 2003, Jaynes, 2003, Murphy, 2012,
Gelman et al. , 2013, Murphy, 2022].
Broader applications to embodied cognition, biological self-organization, collective behavior, action-oriented model learning, and insti-
tutional sensemaking motivate the applications and practice strata [ Clark, 2016, Friston, 2013, Tschantz et al. , 2020, Ramstead et al. ,
2018, Friston, 2019]. Application-facing work spans cognitive control, hierarchical motivation, computational consciousness, clinical neu-
roscience, control-as-inference comparisons, robotics, and model-based cognitive flexibility [ Pezzulo, 2012, Pezzulo et al. , 2018, Vilas et al. ,
2022, Smith et al. , 2021, Pezzulo et al. , 2023, Imohiosen et al. , 2020, Millidge et al. , 2020, Schwartenbeck et al. , 2015, Lanillos et al. , 2021,
Catal et al. , 2021, Queißer et al. , 2021, Sales et al. , 2019]. These are application and comparison surfaces, not evidence that the present dis-
crete kernels scale to robotics, clinical decision support, or human cognitive change. Predictive-learning neuroimaging evidence strengthens
the plausibility of predictive learning as a representational mechanism, but it remains neural-task evidence rather than evidence that this
microlearning curriculum changes learner representations [ Greco et al. , 2024]. Current frontier sources sharpen the roadmap boundary.
Reactive message passing and RxInfer show a software and algorithmic path for scalable Bayesian message passing; coupled free energy
points toward robust variational geometry; active predictive coding and divide-and-conquer predictive coding extend the neural-inference
and scalability context. Those sources are cited as scope-extending scholarship, not as implemented kernels, production-scale evidence, or
learner-outcome evidence in this release [ Bagaev and de Vries , 2023, Bagaev et al. , 2023, Nelson et al. , 2025, Rao et al. , 2024, Sennesh
et al. , 2024]. Recent experimental work also grounds the FEP outside purely theoretical exposition [ Isomura et al. , 2023].
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18 Related Work: SkillTree Delivery, Security Posture, Reporting, and Open Science
18.1 SkillTree delivery as prerequisite learning and gamified practice context
SkillTree supplies the delivery substrate: Project, Subject, Skill, learning-path dependencies, quizzes, and report endpoints [ National
Security Agency , 2020, SkillTree, 2026, National Security Agency , 2026b]. Active Skillference uses those concepts directly rather than
inventing a new learning platform. The graph layer adds prerequisite validation and connectivity checks because the learning object is
the ordered dependency structure, not only the text attached to each node [ Bloom, 1956, 1968, Liang et al. , 2018, Doignon and Falmagne ,
1985, Falmagne and Doignon , 2011].
Gamification is relevant but not itself the contribution; SkillTree provides the gamified runtime, and the project contributes the verified
content and graph [ Dicheva et al. , 2015]. Visual skill-tree and gamified microlearning sources provide useful delivery precedent for
making dependencies and progress inspectable, but they do not constitute learner-outcome evidence for this export [ Burau, 2022, Septiani
and Rosmansyah , 2021]. Constructive alignment and formative-assessment sources supply the education-facing language for this design:
objectives, activities, feedback, and quiz gates are treated as aligned checkpoints, not as outcome evidence [ Biggs, 1996, Bloom, 1968,
Black and Wiliam , 1998, Shute, 2008].
18.2 SkillTree provenance, deployment security, and access-control boundaries
SkillTree’s origin matters for the delivery argument. NSA describes SkillTree as an internally developed open-source training platform,
and the public skills-service repository and NSA open-source portal identify the service as public government-origin software rather
than a generic third-party widget [ National Security Agency , 2020, 2026b,a]. Active Skillference therefore treats SkillTree as a real delivery
substrate with explicit platform ownership boundaries: SkillTree owns learner progress, quiz state, authentication, ranking and reporting
semantics, while this project owns the curriculum graph, provenance gates, manuscript evidence and exported SkillTree payload.
That provenance raises the bar for deployment language. A national-security posture would still require organization-specific threat
modeling, zero-trust access decisions, supply-chain provenance, software-development process, artifact signing, SBOM visibility, and
adversary-technique coverage beyond this educational repository [ Rose et al. , 2020, National Institute of Standards and Technology , 2022,
SLSA Framework, 2026, Sigstore, 2026, Cybersecurity and Infrastructure Security Agency , 2026, MITRE, 2026]. Active Skillference can
support defensible training review because claims, figures, source roles, dashboard artifacts and SkillTree handoff routes are inspectable;
it does not certify operator competence, harden a production deployment, replace APT-aware threat modeling, or imply NSA approval of
the curriculum.
18.3 Reproducible reporting, accessible artifacts, and source-role auditing
The content-provenance binding is closest in spirit to literate programming and literate statistical reporting: code and explanation are
kept mutually inspectable, and R/knitr- style workflows bind prose to executable analysis so reported numbers can be regenerated [ Knuth,
1984, Xie, 2015]. Peng’s reproducible-research argument is the reporting boundary used here: when a claim is computational, readers
should be able to rerun the workflow that produced it [ Peng, 2011, Sandve et al. , 2013]. Scientific-computing practice adds the software-
maintenance side of the same argument: versioned code, automated checks, and tested workflows are part of the scholarly surface when the
artifact is executable [ Wilson et al. , 2014]. W3C PROV gives the formal provenance vocabulary for entities, activities, agents, derivations,
bundles, and validity constraints, while reproducible-science sources keep the claim boundary tied to transparent methods rather than trust
language [ Groth and Moreau , Moreau and Missier , Cheney, Missier, Moreau and De Nies , Munafò et al. , 2017]. F AIR and transparency
guidelines extend that boundary from rerunnable computation to metadata and open-science claim discipline [ Wilkinson et al. , 2016,
Nosek et al. , 2015]. Active Skillference differs by keeping a central claim registry, applying the same gate to curriculum content and
manuscript prose, and refusing export when any token is unbacked. The DAG engine uses standard graph algorithms for cycle detection
and topological ordering [ Cormen et al. , 2022]. Figures use an accessible categorical palette rather than a single-hue theme [ Wong, 2011].
The figure-registry validator also treats alt text and semantic caption metadata as reporting obligations, aligning the artifact layer with
ACM/SIGCHI, Section 508 and WCAG guidance that accessible figures and web content should carry meaningful alternatives rather than
empty labels [ ACM DIS 2023 Accessibility Chairs , 2023, Section508.gov, 2026, World Wide Web Consortium , 2023].
Release metadata and license notices are treated as delivery evidence, not as scientific or learner-efficacy support. Software citation
principles, Citation File Format and CodeMeta guide the citation and metadata surfaces [ Smith et al. , 2016, Citation File Format Project ,
2026, CodeMeta Project , 2026]. GitHub and Zenodo documentation bound the repository and DOI workflow to an actual public target
and deposit [ GitHub Docs , 2026, Zenodo, 2026]. SPDX, OSI and REUSE license-notice references support the MIT and third-party notice
signposting [ SPDX Workgroup , 2026, Open Source Initiative , 2026, Free Software Foundation Europe , 2024]. These references do not
provide legal advice, do not relicense third-party course materials or vendored browser bundles, and do not turn a static review checkout
into a production deployment.
The scholarship audit extends that reproducible-reporting idea from numbers to source coverage. Ordinary citation validators can show
that [@key] resolves, but not that the key has a stated role in the argument. The source matrix makes that role explicit and keeps
the bibliography aligned with the manuscript’s actual lanes. The frontier gap report adds a narrower check for newly emphasized source
areas: a pedagogy, assessment, formal provenance, workflow provenance, knowledge-space, cognitive-diagnosis, message-passing, predictive-
coding, design-based evaluation, open-science, security, accessibility, reproducibility, or course-material source must be cited and tied to
an explicit claim-support class and, where needed, a boundary row before it can count as a scholarship upgrade.
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18.4 Open-science education ecosystem, interoperability, and community handoff
Active Skillference also sits inside a broader open-science educational ecosystem. The Active Inference Institute publishes a public
resource hub with community pathways, verified public repositories, and a documented projection boundary; it also publishes public
course infrastructure, a START project for tailored Active Inference research and training curricula, and an institute website that frames
education as part of an open-science mission [ Active Inference Institute , 2026b,d,c,e].
The cognitive knowledge-base repository is treated as course-material provenance for generated Learn pointers, not as copied prose or as
an endorsement claim [ Active Inference Institute , 2026a]. This project does not replace those efforts; it contributes a provenance-bound
SkillTree export, quiz gate, and verifier suite that can interoperate with public learning paths. Recent educational Active Inference
scholarship also motivates prepared learning environments and active exploration as topics for future classroom evaluation, while this
release limits itself to the inspectable curriculum and delivery contract [ Di Paolo et al. , 2024].
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19 Limitations and Planned Evaluation: Delivery Dependencies, Kernel Scope, and
Learner Outcomes
19.1 Delivery dependencies, fixed-kernel scope, and learner-outcome boundaries
19.1.1 Delivery substrate and kernel scope
The local learner dashboard can be verified in artifact-only mode without Docker; live progress, ranking, and quiz completion still require
a SkillTree service. When that service is available, the dashboard mounts the official learner display and separates local lesson material
from official subject and quiz handoffs, but it does not provide a second source of learner truth. The REST contract is verified against a
pytest-httpserver mirror of SkillTree’s endpoints, and the live seed is documented and scripted.
The implemented kernels cover discrete pedagogical cases, reference variational examples, and bounded predictive-coding or planning
demonstrations; they do not provide a continuous, scalable, robust, or deep predictive-coding inference engine. Reactive message passing,
RxInfer, coupled free energy, active predictive coding, divide-and-conquer predictive coding, generalized filtering, generalized free energy,
hierarchical multi-step planning, and richer reinforcement-learning comparisons remain cited roadmap context rather than hidden imple-
mentation claims [ Bagaev and de Vries , 2023, Bagaev et al. , 2023, Nelson et al. , 2025, Rao et al. , 2024, Sennesh et al. , 2024, Friston
et al. , 2008, 2010, Parr and Friston , 2019]. The RL comparison matters because probabilistic-inference views of control overlap with, but
are not identical to, active-inference policy selection [ Levine, 2018]. Psychology-and-psychiatry Active Inference reviews likewise support
cautious application language rather than theory-level validation claims [ Badcock and Davey , 2024]. Finally, the platform’s contribution
is the verifiability and provenance of curriculum content, not a measured claim about learning outcomes: it does not evaluate teaching
effectiveness, retention, or transfer, and no efficacy study is reported here. Microlearning studies, epistemic-curiosity evidence, learning-
analytics reviews, and learning-analytics intervention meta-analyses motivate the proposed study design but do not substitute for data
from this curriculum, and learning analytics also introduces privacy and data-protection obligations before progress exports can be treated
as evidence [ Monib et al. , 2025, Rof et al. , 2024, Emin et al. , 2026, Liu et al. , 2025, Hernández-Campos et al. , 2025, Liu and Khalil ,
2023]. Whether this provenance discipline actually improves how people learn the framework remains future work, outside the scope of
what the code, citations, and validators establish. The preregistered pilot design in experiment_plan.yaml is therefore a setup artifact: it
defines consent, privacy, SkillTree progress-export, pre/post, delayed-retention, and analysis boundaries, and follows design-based research
framing for studying a curriculum artifact in context, but it is not learner-outcome evidence until real data are collected and added to the
claim-support record [ Design-Based Research Collective , 2003, Barab and Squire , 2004].
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20 Supplement: Model-Bounded Variational and Expected Free Energy Examples
This supplement carries the kernel-level mathematical walkthroughs that support the curriculum without crowding the main Results
section. The purpose is narrow: show how the claim registry, figures and prose jointly teach the two optimization objectives that organize
the course. Variational free energy is used for belief updating under a fixed generative model; expected free energy is used for policy
evaluation under uncertainty. Both examples are deliberately small enough to audit by rerunning the kernels.
Notation is fixed before the examples: 𝑠 is the latent state, 𝑜 is the observation, 𝑝(𝑠) is the prior, 𝑝(𝑜 ∣ 𝑠) is the likelihood, 𝑝(𝑜, 𝑠) is the
joint generative density, and 𝑞(𝑠) is the recognition density. The variational free-energy convention is 𝐹 [𝑞] = 𝔼 𝑞(𝑠)[ln 𝑞(𝑠) − ln 𝑝(𝑜, 𝑠)] =
𝐷KL(𝑞(𝑠)‖𝑝(𝑠 ∣ 𝑜)) − ln 𝑝(𝑜), so the bound is tight when 𝑞(𝑠) equals the exact posterior. For expected free energy, this supplement uses
the lower-is-better policy convention 𝑞(𝜋) = softmax(−𝛾𝐺(𝜋)) with policy precision 𝛾; the plotted terms are therefore tied to the stated
policy set and preference distribution, not to a universal sign convention [ Buckley et al. , 2017, Parr et al. , 2022, Millidge et al. , 2021b,
Champion et al. , 2026, Nuijten et al. , 2026a,b].
20.1 SkillTree bridge from derivations to quiz-gated subjects
The supplement is not an isolated appendix of equations. It is a bridge from kernel claims to quiz-gated SkillTree material: sub_free_energy
and sub_efe carry the core VFE/EFE concepts, the scientific-supplement subjects train bound and term interpretation, and the mastery
subjects turn those interpretations into route-handoff and quiz-authoring checks. fig. 21 is generated from the current curriculum graph after
structural checks and SkillTree route contract so the text, figure, and dashboard artifact all point to the same subject, skill, prerequisite-
edge, quiz, and route data. This bridge also keeps the VFE/EFE distinction scholarly rather than mnemonic: VFE is an evidence bound for
belief updating, while EFE is a policy-evaluation objective whose formulations and preference assumptions need explicit scope boundaries
[Buckley et al. , 2017, Parr et al. , 2022, Smith et al. , 2022, Millidge et al. , 2021b, Champion et al. , 2026, Nuijten et al. , 2026a,b].
Layer Skills
Prerequisite
edges Depth
Representative
IDs Learn route Quiz route
VFE core 7 16 total; 6
internal
7-10 vm_elbovm_percep
tion_descentvm_m
essage_passing
/subjects/su
b_free_energ
y
/subjects/su
b_free_energ
y/skills/vm_
message_pass
ing/quizzes/
vm_message_p
assing_quiz
EFE core 7 15 total; 7
internal
18-20 ai_expected_free
_energyai_explor
e_exploitai_free
_energy_of_futur
e
/subjects/su
b_efe
/subjects/su
b_efe/skills
/ai_free_ene
rgy_of_futur
e/quizzes/ai
_free_energy
_of_future_q
uiz
VFE
supplement
6 11 total; 5
internal
9-14 sci_variational_
01sci_variationa
l_04sci_variatio
nal_06
/subjects/su
b_scientific
_supplement_
variational
/subjects/su
b_scientific
_supplement_
variational/
skills/sci_v
ariational_0
6/quizzes/sc
i_variationa
l_06_quiz
EFE
supplement
6 11 total; 5
internal
19-24 sci_active_infer
ence_01sci_activ
e_inference_04sc
i_active_inferen
ce_06
/subjects/su
b_scientific
_supplement_
active_infer
ence
/subjects/su
b_scientific
_supplement_
active_infer
ence/skills/
sci_active_i
nference_06/
quizzes/sci_
active_infer
ence_06_quiz
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Layer Skills
Prerequisite
edges Depth
Representative
IDs Learn route Quiz route
VFE
mastery
8 17 total; 7
internal
36-43 mastery_variatio
nal_01mastery_va
riational_05mast
ery_variational_
08
/subjects/su
b_skilltree_
mastery_vari
ational
/subjects/su
b_skilltree_
mastery_vari
ational/skil
ls/mastery_v
ariational_0
8/quizzes/ma
stery_variat
ional_08_qui
z
EFE
mastery
8 17 total; 7
internal
52-59 mastery_active_i
nference_01maste
ry_active_infere
nce_05mastery_ac
tive_inference_0
8
/subjects/su
b_skilltree_
mastery_acti
ve_inference
/subjects/su
b_skilltree_
mastery_acti
ve_inference
/skills/mast
ery_active_i
nference_08/
quizzes/mast
ery_active_i
nference_08_
quiz
20.1.1 Bridge figure
Figure 21: VFE/EFE SkillTree bridge generated from the current curriculum graph after structural checks. Rows are real SkillTree
subjects, columns report skill counts, prerequisite-edge counts, graph-depth ranges, representative skill IDs, and learn-route and quiz-
route fragments that expose the subject identifiers; color marks separate VFE-centered rows from EFE-centered rows. The figure connects
the supplement’s mathematical plots to quiz-gated curriculum nodes and prevents the derivation from being presented as a standalone
artifact disconnected from the learning path.
20.2 Noisy-sensor posterior and the tight variational-free-energy bound
On a noisy-sensor model with a 1% base rate, 99% sensitivity, and 95% specificity, a positive observation yields a posterior of 0.166667 for
the latent condition. This is the same base-rate calibration problem that makes Bayesian reasoning difficult in natural-language judgment
48

## Page 50

tasks and motivates frequency-format teaching examples for Bayesian inference [ Cosmides and Tooby , 1996, Gigerenzer and Hoffrage ,
1995]. The evidence probability of the positive observation is 0.0594, so the observation surprise is 2.82346 nats.
Noisy-sensor parameter Paper value
Latent states present, absent
Observation index positive
Prior 𝐷 [0.01, 0.99]
Likelihood rows 𝐴𝑜,𝑠 positive: [0.99, 0.05]; negative: [0.01, 0.95]
VFE sweep coordinate 𝑞(present) over [0.001, 0.999], plus the exact posterior and 0.5
Evaluating variational free energy across beliefs confirms the central inequality: a uniform belief gives 3.11735 nats, while the true posterior
gives 2.82346 nats, exactly the negative log-evidence. fig. 22 shows the full one-dimensional sweep over 𝑞(present): the curve bottoms
out at the exact posterior and stays above the surprise line everywhere else. The plot should not be read as a universal geometry of all
recognition densities. It is a controlled diagnostic for one discrete model, one observation and one belief coordinate.
Figure 22: Variational free-energy sweep for the noisy-sensor model, generated from the same claim-registry kernels that provide the
manuscript tokens and taught through the VFE core, supplement, and mastery SkillTree subjects in the VFE/EFE SkillTree bridge. The
x-axis is the recognition belief 𝑞(present), the blue curve is 𝐹 (𝑞) in nats, the horizontal dashed line is − ln 𝑝(𝑜), the vertical dotted line is
the exact posterior, and the labeled points compare the looser uniform belief with the posterior value. The intended takeaway is bound
tightness at the posterior for this one discrete model and observation; the plot does not establish that the model is true or that the same
geometry holds for every recognition family.
20.3 Expected free energy: epistemic value and bounded policy comparison
Decomposing expected free energy on an informative sensor shows that an uncertain belief carries positive epistemic value (0.368064
nats) while a certain belief carries none (0 nats): once the state is known, an observation cannot reduce uncertainty. In the two-action
agent, preferences are neutral, so the pragmatic term is identical for both actions and the choice is decided by information gain. Under
the lower-is-better 𝐺(𝜋) convention above, the information-seeking sample action scores lower expected free energy (0.546837 nats) than
commit (0.613267 nats), because it carries higher epistemic value (0.146311 nats versus 0.079881 nats). This is the epistemic drive of
49

## Page 51

Active Inference: no separate curiosity bonus is required in the reference formulation used here [ Friston et al., 2015, 2017b, Da Costa et al. ,
2020, Sajid et al. , 2021]. Derivation and unification work is the reason the supplement keeps the formulation explicit instead of presenting
“expected free energy” as a single unqualified object [ Millidge et al. , 2021b, Champion et al. , 2026, Nuijten et al. , 2026a,b].
20.3.1 Preference convention and parameters
One convention should be made explicit here: the kernel normalizes the preference vector 𝐶 to a probability distribution before taking its
logarithm for the pragmatic term ( preferences = normalize(C) , then 𝔼𝑞(𝑜)[ln normalize(𝐶)]), so the values plotted above assume 𝐶 is a
preference distribution. This is a specific modeling choice rather than the only option: canonical Active Inference often treats 𝐶 directly
as a vector of log-preferences, which shifts the pragmatic term by an additive constant and can change its scale relative to the epistemic
term.
20.3.2 EFE parameter table
EFE sample/commit parameter Paper value
Sensor likelihood 𝐴 = [[𝑝, 1 − 𝑝], [1 − 𝑝, 𝑝]], with 𝑝 = 0.9
EFE decomposition preference 𝐶 = [0.9, 0.1] for the informative-sensor term example
Agent preference Neutral 𝐶 = [0.5, 0.5] for the sample/commit policy comparison
Prior before observation 𝐷 = [0.5, 0.5]
Actions sample uses identity dynamics; commit maps both prior states
toward state 0 with transition columns [0.95, 0.05]
Policy posterior 𝑞(𝜋) = softmax(−𝛾𝐺𝜋) over the two one-step policies
20.4 Sensor precision as a sample-versus-commit policy boundary
The expected-free-energy result is not a universal law. It is pinned to a specific moderate-precision regime: at sensor precision 0.9, sampling
wins; at the computed switch precision 0.95, sample and commit tie; above that point, the commit action can become lower in expected
free energy. fig. 23 shows both the sample/commit comparison and the precision sweep, making the regime boundary explicit instead of
hiding it in prose [ Millidge et al. , 2021b, Champion et al. , 2026, Nuijten et al. , 2026a,b].
Precision-sweep parameter Paper value
Sensor precision range 𝑝 ∈ [0.5, 0.99] with the computed switch inserted into the grid
Sweep samples 101 grid samples before inserting 𝑝 = 0.9 and the switch precision
Prior, preference, actions Same neutral two-action agent as the sample/commit comparison
Tie criterion The switch is the bisection point where 𝐺sample − 𝐺commit = 0
fig. 24 broadens the sensitivity view without changing the evidence class. The left panel varies the base rate in the same noisy-sensor
likelihood and shows how a positive observation’s posterior changes as the prior changes. The middle panel repeats the VFE bound profile
so the equality point remains tied to the exact posterior. The right panel repeats the EFE precision sweep and marks the computed
sample/commit boundary. These panels are deterministic kernel diagnostics: they identify where the teaching example is stable or
regime-limited, not whether a learner or a biological system will behave that way.
20.5 Exact inference on a chain with sum-product belief propagation
The message-passing curriculum is bound to a worked example rather than left as prose. Because the example factor graph is acyclic,
sum-product belief propagation returns the exact node marginals for a three-node sticky latent chain with a single informative reading
at the middle node. The observed node settles at 0.9, and the smoothing message lifts each unobserved neighbor from its prior to 0.74
– a lift of 0.24 that is produced purely by propagating the one reading along the transition factor. Because a chain is a tree, these
marginals are the same numbers a brute-force enumeration of the full joint would give; the test suite pins that equality, so the example is
an exact-inference check, not an approximation [ Kschischang et al. , 2001, Pearl, 1988, Parr et al. , 2022]. Factor-graph message passing is
also a live implementation path for EFE minimization, but this supplement uses the chain only to teach exact sum-product inference and
keeps policy-optimization message passing out of scope for this release [ Nuijten et al. , 2025].
20.5.1 Sum-product parameter table
Sum-product parameter Paper value
Chain length 3 binary latent nodes
Prior on node 0 [0.5, 0.5]
Sticky transition [[0.8, 0.2], [0.2, 0.8]]
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Sum-product parameter Paper value
Evidence potentials node 0: [1, 1]; node 1: [0.9, 0.1]; node 2: [1, 1]
20.6 Conjugate parameter learning from Dirichlet beliefs
Parameter learning is shown as a Dirichlet-conjugate update of one sensor column. A flat prior has mean 0.5; after a fixed observed count
vector the posterior mean sharpens to 0.75, and the expected log-parameter that variational message passing actually consumes is -0.30202
nats, computed through the digamma identity 𝔼[ln 𝑝𝑖] = 𝜓(𝛼 𝑖) − 𝜓(∑ 𝑗 𝛼𝑗). This is the learning half of active inference: beliefs about
model parameters move with evidence, not just beliefs about states [ Da Costa et al. , 2020].
20.7 Bayesian model reduction: whether data require added structure
Bayesian model reduction compares the evidence of a full model against a reduced model whose pruned state carries negligible prior mass.
After a positive reading on the noisy-sensor model the full-model log evidence is -2.82346 nats and the reduced-model log evidence is
-2.99571 nats, so the log Bayes factor is 0.172252 nats. The positive sign says the data support keeping the pruned structure: the reduced
model is rejected. Both terms are exact log evidences, so the reported factor is the difference of two computed numbers rather than a
hand-typed value [ Friston et al. , 2018, Parr et al. , 2022].
BMR parameter Paper value
Full prior 𝐷 = [0.01, 0.99]
Reduced prior the present state is set to 10−6 and the prior vector is renormalized
Observation and likelihood same positive observation and noisy-sensor likelihood as the VFE
example
Score log 𝑃 (𝑜 ∣ full) − log 𝑃 (𝑜 ∣ reduced)
20.8 Ambiguity as a state-dependent component of expected free energy
Expected free energy carries an ambiguity term – the expected entropy of the likelihood columns – that the main decomposition leaves
implicit. On an asymmetric sensor whose two states differ in how noisy their readings are, the crisp state has likelihood entropy 0.325083
nats, while a uniform belief averages to 0.509115 nats. Ambiguity is therefore a property of where the agent believes it is, which is why
reducing it is part of what drives exploration [ Parr et al. , 2022, Sajid et al. , 2021].
20.9 Multi-step planning and model-dependent look-ahead value
The one-step agent above is generalized to a planner that scores every length- 𝐻 action sequence by its cumulative expected free energy and
forms a posterior over policies, 𝑞(𝜋) = softmax(−𝛾𝐺𝜋); at 𝐻 = 1 the per-policy values reduce exactly to the one-step result, a consistency
invariant the tests pin. Whether planning beats a myopic agent is then a measured question, not an assumption, and the answer is regime
dependent.
In a decoupled sample/commit model the optimal multi-step plan coincides with the strongest greedy policy, so the look-ahead gain is
exactly 0 nats – an honest null. In a coupled investment model, where a state that is myopically attractive blocks access to a higher-value
goal, the planner pays a worse first step to reach the goal: the greedy policy scores 3.02826 nats, the optimal plan scores 2.20728 nats,
and the look-ahead gain is 0.820981 nats with the policy posterior concentrating 0.915376 of its mass on the winning plan (fig. 25). The
comparator is the strongest myopic policy, not a weakened straw baseline, so the gain is a fair number; the first-principles lesson is that
look-ahead pays only when immediate and long-run value diverge [ Parr et al. , 2022, Da Costa et al. , 2020].
Planning parameter Decoupled sample/commit model Coupled investment model
States 2 latent states start, comfort, goal, climb
Observation model informative sensor with 𝑝 = 0.9 identity likelihood
Preference vector 𝐶 = [0.9, 0.1] 𝐶 = [0.03, 0.22, 0.55, 0.20]
Initial prior 𝐷 = [0.5, 0.5] 𝐷 = [1, 0, 0, 0]
Actions sample identity; commit collapse toward
state 0
comfort sends start to comfort; climb sends
start to climb and climb to goal
Horizon and policy precision 𝐻 = 2 , 𝛾 = 4 𝐻 = 2 , 𝛾 = 4
20.10 Continuous predictive coding as model-specific prediction-error descent
The discrete examples above update beliefs in one shot; the continuous-state face of the Free Energy Principle instead settles a latent
estimate by descending the free energy along precision-weighted prediction errors. On a single linear-Gaussian level – a vague unit-variance
51

## Page 53

Figure 23: Expected free-energy decomposition for the sample/commit task, generated from the discrete active-inference kernel and
taught through the EFE core, supplement, and mastery SkillTree subjects in the VFE/EFE SkillTree bridge. The left panel compares
total expected free energy and epistemic value for the two policies at sensor precision 0.9; the right panel sweeps sensor precision and
marks the computed sample/commit switch. The visual encoding keeps policy score and epistemic value separate so the reader can see
why sampling is preferred in the reference regime, while the right panel prevents the teaching example from being overgeneralized beyond
the stated policy set and preferences.
Figure 24: Kernel sensitivity profiles for the supplemental examples, generated from the noisy-sensor posterior, VFE sweep, and EFE
precision sweep, then positioned in the SkillTree bridge as a route from mathematical identity to quiz-gated interpretation. The left
panel varies the base rate and shows posterior sensitivity after a positive observation, the middle panel plots the VFE bound profile with
posterior and evidence references, and the right panel sweeps expected free energy for sample and commit policies across sensor precision
while marking the computed policy switch. These panels identify deterministic regime boundaries for teaching examples; they are not
empirical estimates, confidence intervals, or learner outcome statistics.
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Figure 25: Multi-step planning look-ahead, generated from the discrete active-inference planning kernel. The left panel contrasts the
measured look-ahead gain (optimal multi-step plan versus the strongest greedy myopic policy) in a decoupled model, where the gain is
zero, and a coupled investment model, where paying a worse first step to unlock the goal yields a positive gain. The right panel shows
the cumulative expected free energy of every enumerated policy in the coupled model, with the greedy and optimal policies marked. The
panel demonstrates that look-ahead value is a measured, model-dependent quantity, not a universal property of planning.
prior at zero meeting a more precise reading of 2 – the estimate relaxes from the prior mean to 1.33333, the exact analytic posterior mean,
where the sensory prediction error (0.666667) and the prior prediction error (1.33333) balance once each is weighted by its precision.
fig. 26 shows the estimate converging to that fixed point while, under the stated objective and step size used in this simulation, the free
energy decreases monotonically to its analytic minimum. The result is exact for the linear-Gaussian case the kernel tests pin against a
closed form; it is a controlled teaching example, not a claim that prediction-error descent is exact for nonlinear or hierarchical models
[Kalman, 1960, Särkkä, 2013, Buckley et al. , 2017, Parr et al. , 2022, Kataoka and Doya , 2026].
Predictive-coding parameter Paper value
Prior mean and variance 𝜇0 = 0, 𝜎2
0 = 1
Observation and variance 𝑦 = 2, 𝜎2
𝑦 = 0.5
Descent rate and steps rate 0.1, maximum 400 steps
Analytic target Gaussian posterior mean from the linear filtering update
Figure 26: Continuous predictive-coding descent generated from the linear-Gaussian kernel. The left panel traces the latent estimate
from the prior mean to the analytic precision-weighted posterior (dashed) as it balances the sensory and prior prediction errors; the right
panel shows the free energy descending monotonically to its analytic minimum. The figure supports a bounded claim that prediction-error
descent implements exact Gaussian inference for this one model, not a universal convergence guarantee for all generative models.
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21 Code and Data A vailability
The artifact described here is a single repository containing the kernels, the curriculum catalog, the SkillTree exporter, the learner
dashboard, the manuscript sources, and every validator named in sec. 14. Authored code and content are MIT licensed with Daniel Ari
Friedman as the copyright holder, recorded in LICENSE and mirrored in the citation, CodeMeta, and Zenodo metadata files. Vendored
runtime dependencies and their separate notices are listed in THIRD_PARTY_NOTICES.md . External Active Inference Institute course and
knowledge-base material is referenced by repository, path, and commit rather than copied, so upstream licensing stays with upstream.
The code is public at github.com/ActiveInferenceInstitute/Active_Skillference, released as v1.0.0 and archived on Zenodo at
doi.org/10.5281/zenodo.21865644. The identifier appears on the cover page, in the citation file, in the CodeMeta and Zenodo
records, and in the archived copy of this document, all naming the same deposit. That agreement is enforced rather than assumed: a test
cross-checks each surface against the recorded deposit and fails on any divergence, which is what keeps a plausible but wrong identifier
out of a citation. Version-specific identifiers stay blank because no versioned deposit beyond this one exists, and the same test holds them
blank.
Generated outputs are not committed, because they are rebuildable: running the kernels regenerates the claim ledger, manuscript variables,
scholarship audits, figures, lesson records, and the SkillTree export from source. The one exception, described in sec. 11, is the captured
evidence. A recording of a running system is not a function of project data and cannot be rebuilt from a checkout, so those images are kept
in the version-controlled tree rather than the generated one, alongside a provenance record that names the dashboard address, viewport,
live counts, full-stack mode, capture timestamp, and a digest per image for the session that produced them. That record travels with
the repository, so it becomes checkable at the same moment the code does. Every other image in this paper regenerates; those five are
archived precisely because they do not.
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22 References: Bibliography and Source-Role Audit
Bibliography lives in docs/manuscript/references.bib and is read by Pandoc during PDF render. The build pipeline invokes Pandoc with
--natbib, so every [@key] citation in the manuscript is rewritten to the appropriate \cite{}/\citep{}/\citet{} LaTeX command and
resolved against the bib file.
To validate that references.bib is syntactically clean and contains the required fields per entry type:
uv run python -m infrastructure.reference.citation.cli validate \
docs/manuscript/references.bib --strict
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*Extraction method: pypdf*