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The applications trail

The applications change the object of learning while preserving the structural grammar.

Chapter and domain	Structural focus	Control and admission lens
11 — causal discovery	visible intervention geometry and causal graphs; Lie-bracket nonclosure or unstable extraction	audit the model class and identification assumptions
12 — neural architecture	Kan-extended representation routes; disagreement between direct and extended base or tangent behavior	repair the route or architecture, then test finite behavior
13 — neural adaptation	ordered families of low-rank adapters; order-sensitive interaction	route, merge, separate, or retrain under an explicit admission test
14 — skills	sections over a Lie algebroid; anchor, bracket, or feasibility failure	repair the skill or workflow while preserving declared constraints
15 — reinforcement learning	Bellman and policy factorizations; base or tangent inconsistency	use decision quotients, effective sample size, quarantine, and recovery
16 — preference learning	relational preferences; cycles or a nonintegrable preference field	retain a relational fallback and guard any reward update
17 — relational learning	charts, joins, and shared manifold geometry; incidence, cocycle, or apex failure	make sparse, decision-preserving chart repairs
18 — distributed learning	local predictive sections; compatibility or effectivity failure	audit descent and apply transverse correction only when admitted
19 — argumentation	typed Toulmin roles and source-bound sections; role, qualifier, rebuttal, or gluing failure	localize edits while preserving source provenance
20 — foundation models	foundry artifacts and validation contracts; cross-agent or cross-stage failure	separate construction, scrutiny, and final admission

Chapters 11–15 emphasize causal, architectural, skill, and sequential decision systems. Chapters 16–20 emphasize relational evidence, distributed reasoning, argumentation, and foundation-model foundries. Chapter 21 then compresses their recurring choices into a pattern language. Chapter 22 asks how the framework might learn its own probes, covers, declarations, and theories, and how it might participate in scientific discovery rather than only repair a fixed model.