OR2 Finite Signature

OR2 is the second of six original-rule narrowing principles (the Six Ontic Requirements, VII.D37). It states the finite-signature requirement: any τ-categorical reality candidate must be generated by finitely many generators subject to finitely many axioms — no infinite primitive specification is admissible at the foundational layer.

Categorical invariant. Finite-signature narrowing: ∀ candidate ontic structure τ', τ' is fully specified by a finite generator set G and a finite axiom set A. Infinite axiom schemes are admissible only as derived consequences, not as primitives.

Primary registry anchor: VII.D37

Supporting items: VII.L31, VII.P08, VII.T14

1. I.K0 — Universe Postulate
2. VII.D37 — Six Ontic Requirements (OR1-OR6) — OR2 is finite signature
3. VII.L31 — OR1+OR2 Narrowing — finitely generated, locally small, full Yoneda category
4. VII.P08 — Each Requirement Independently Necessary — dropping OR2 admits ZFC and other infinitely-axiomatized systems

OR2 is instantiated whenever a foundation is rejected — or graded down — for requiring infinitely many primitive specifications. The intuition: a finite being should be able to grasp the complete specification of reality's substrate. τ delivers this with five generators and seven axioms; ZFC's replacement scheme and PA's induction scheme do not.

Examples:

• Foundations of mathematics: rejecting ZFC's infinite replacement scheme as a primitive specification (admitting it only as a derived consequence)
• Physics: preferring a Lagrangian with finitely many terms over a non-renormalizable theory with infinitely many counterterms
• Philosophy: the surveyability constraint (Wittgenstein, finitists) — a foundation must, in principle, be graspable by a finite intellect
• τ-framework: the entirety of Books I-VI emerges as consequence of five generators + seven axioms

Register codomain: Proof (diagrammatic — finite-signature is a proof-theoretic / surveyability constraint that lives in Reg_D's codomain)

Manuscript reference: manuscript-sources/book-07/part02/ch29.tex

Status: Planned

Related glossary entries

• MG-P01-or1-self-coherence OR1 Self-Coherence
• MG-P07-six-ontic-requirements Six Ontic Requirements (OR1-OR6)
• MG-R03-diagrammatic-register Diagrammatic Register (Reg_D)

Referenced by

• MG-P07-six-ontic-requirements Six Ontic Requirements (OR1-OR6)
• MG-P10-independent-necessity Independent Necessity of Each Ontic Requirement