Causation (constrained composition)

Causation, in the τ-framework, is admissible factorization of morphisms in the coherence kernel. f causes g iff g decomposes as h ∘ f for some admissible h that respects the kernel's narrowing rules. Causation is therefore structural and categorical — not a Humean regularity, not a brute productive relation, not a counterfactual primitive.

Categorical invariant. f →_c g ⟺ ∃ h ∈ AdmMor(K_τ) : g = h ∘ f ∧ h satisfies OR1–OR6.

Primary registry anchor: VII.D32

Supporting items: VII.D37, VII.D31, VII.P06

1. VII.D37 — Six Ontic Requirements (OR1–OR6) — admissibility constraints on kernel morphisms.
2. VII.D31 — Law as admissible continuation — laws are the admissibility class for kernel morphisms.
3. VII.D32 — Causation as Constrained Composition — f causes g iff g admissibly factors through f.
4. VII.P06 — Temporal ordering from persistence — causal asymmetry inherits from the persistence-induced temporal order.

Lean modules referenced: TauLib.BookVII.Meta.Registers

Causation is instantiated whenever we say 'A brings about B', 'A produces B', or 'A is the reason B happened'. Striking a match causes flame; passing a law causes legal-state-change; proving a lemma causes the dependent theorem to follow; committing to a course causes downstream stance-change. All four are read as admissible factorizations in the relevant register's morphism class.

Examples:

• Empirical: 'striking the match caused the flame' — flame-state morphism factors admissibly through strike-state morphism.
• Normative: 'enacting the law caused the legal-state change' — legal-update morphism factors admissibly through enactment morphism in Reg_P.
• Proof-theoretic: 'the lemma caused the theorem to follow' — theorem-derivation factors admissibly through the lemma-derivation in Reg_D.
• Commitment-theoretic: 'the promise caused the obligation' — obligation-stance factors admissibly through promise-stance in Reg_C.

Register codomain: Cross-register (causation is defined wherever a register has admissible morphisms; the structure is uniform).

Manuscript reference: manuscript-sources/book-07/part02/ch24.tex

Status: Formalized

Module: TauLib.BookVII.Meta.Registers

Lean kind: def

Lean symbol: Tau.BookVII.Meta.Registers.CausationAsConstrainedComposition

Related glossary entries

• MG-O03-modality Modality (necessity, possibility, contingency)
• MG-O08-event Event (admissible morphism realization)
• MG-O11-time Time (persistence-induced order)

Referenced by

• MG-O03-modality Modality (necessity, possibility, contingency)
• MG-O08-event Event (admissible morphism realization)
• MG-O11-time Time (persistence-induced order)