Six Ontic Requirements (OR1-OR6)

The Six Ontic Requirements (VII.D37) are six independent narrowing principles that any candidate foundation of reality — any category proposed as an ontic substrate — must satisfy. Their conjunction conjecturally singles out Category τ uniquely (Inevitability Convergence Theorem, VII.T14). The six are: OR1 identity-faithful representation (Yoneda), OR2 finite signature, OR3 diagonal-free self-reference, OR4 NF-addressability, OR5 holomorphic continuation, OR6 spectral completeness.

Categorical invariant. Six-fold narrowing: the intersection of constraint sets C_Yon ∩ C_fin ∩ C_diag ∩ C_NF ∩ C_hol ∩ C_spec = {τ} (up to equivalence). Each constraint is an independently necessary condition on a candidate category F.

Primary registry anchor: VII.D37

Supporting items: VII.T14, VII.P08, VII.L31, VII.L32, VII.L33

1. I.K0 — Universe Postulate
2. VII.D37 — Six Ontic Requirements (OR1-OR6) — the six independent narrowing principles on candidate foundations
3. VII.L31 — OR1+OR2 Narrowing (finitely generated, locally small, full Yoneda)
4. VII.L32 — OR3+OR4 Narrowing (diagonal-free self-description plus NF-addressability)
5. VII.L33 — OR5+OR6 Narrowing (analytic + spectral conditions linking combinatorics to functional analysis)
6. VII.T14 — Inevitability Convergence — intersection of all six is {τ} up to equivalence (CONJECTURAL)
7. VII.P08 — Each Requirement Independently Necessary — dropping any one admits non-τ solutions

The Six Ontic Requirements are instantiated jointly whenever a foundation of reality is evaluated for inevitability — not just consistency. Each requirement individually echoes a long-standing demand from foundations of mathematics, physics, or philosophy; their conjunction is the τ-framework's distinctive claim. The argument is conditional: *if* all six are accepted, *then* τ is the unique solution.

Examples:

• String theory landscape (~10⁵⁰⁰ vacua): consistency alone underdetermines; the six requirements are stronger than consistency
• Loop quantum gravity: built on background independence + diffeomorphism invariance — analogs of OR1 and OR5, but missing OR2-OR4 and OR6
• Topos-theoretic foundations: address OR1 and parts of OR5 via internal logic and sheaf structure, but generic topoi are not finitely generated (fail OR2) and lack spectral decomposition (fail OR6)
• ZFC: satisfies OR1 and (in appropriate formulations) OR3-OR6 but fails OR2 (replacement is an infinite scheme)
• τ: the only known structure satisfying all six simultaneously — the conjectural Inevitability Convergence claim

Register codomain: Proof (diagrammatic — the requirements are structural narrowing conditions on candidate categories, lived in Reg_D's codomain; the inevitability claim itself carries the conjectural scope label)

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

Status: Planned

Related glossary entries

• MG-P01-or1-self-coherence OR1 Self-Coherence
• MG-P02-or2-finite-signature OR2 Finite Signature
• MG-P03-or3-diagonal-free-self-reference OR3 Diagonal-Free Self-Reference
• MG-P04-or4-nf-addressability OR4 NF-Addressability
• MG-P05-or5-holomorphic-continuation OR5 Holomorphic Continuation
• MG-P06-or6-spectral-completeness OR6 Spectral Completeness
• MG-P10-independent-necessity Independent Necessity of Each Ontic Requirement

Referenced by

• MG-P02-or2-finite-signature OR2 Finite Signature
• MG-P03-or3-diagonal-free-self-reference OR3 Diagonal-Free Self-Reference
• MG-P04-or4-nf-addressability OR4 NF-Addressability
• MG-P05-or5-holomorphic-continuation OR5 Holomorphic Continuation
• MG-P06-or6-spectral-completeness OR6 Spectral Completeness
• MG-P10-independent-necessity Independent Necessity of Each Ontic Requirement