Independent Necessity of Each Ontic Requirement

Independent Necessity (VII.P08) is the principle that each of the Six Ontic Requirements is independently necessary: dropping any one OR_i while retaining the other five admits non-τ candidate foundations. The six are not redundant — each excludes a genuine class of alternatives. The principle backs the inevitability argument by ruling out compressibility of the requirement set.

Categorical invariant. Independence narrowing: ∀ i ∈ {1,...,6}, the constraint set ⋂_{j≠i} C_{OR_j} ⊋ {τ}. Equivalently: each OR_i contributes a non-trivial dimension to the narrowing intersection.

Primary registry anchor: VII.P08

Supporting items: VII.D37, VII.T14

1. I.K0 — Universe Postulate
2. VII.D37 — Six Ontic Requirements (OR1-OR6) defined
3. VII.P08 — Each Requirement Independently Necessary — for each i, dropping OR_i admits non-τ solutions (counterexamples sketched)
4. VII.T14 — Inevitability Convergence — independence is structurally required for the convergence to {τ} to be non-trivial

Independent Necessity is instantiated whenever an alternative foundation is exhibited that satisfies five of the six requirements but fails one — demonstrating that the missing requirement is not a derivable consequence of the others. The counterexamples in VII.P08 are: any category with non-trivial automorphisms (drops OR1); ZFC (drops OR2); naive set theory (drops OR3); AC-dependent constructions (drops OR4); purely combinatorial categories (drops OR5); non-self-adjoint operator algebras (drops OR6).

Examples:

• Drop OR1: any category with non-trivial automorphisms (haecceity-admitting) satisfies OR2-OR6 but not OR1
• Drop OR2: ZFC satisfies OR1, OR3-OR6 in appropriate formulations but its signature is not finite
• Drop OR3: naive set theory with unrestricted comprehension is inconsistent but formally satisfies OR1, OR2, OR4-OR6
• Drop OR4: AC-dependent constructions (e.g., choice-function-defined sets) satisfy the others but lack canonical addresses
• Drop OR5: purely combinatorial categories (no analytic structure) satisfy OR1-OR4 and OR6 but lack lawfulness
• Drop OR6: non-self-adjoint operator algebras have analytic structure but no full spectral decomposition

Register codomain: Proof (diagrammatic — independence is a structural / counterexample-based proposition over the requirement set, lives in Reg_D's codomain)

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

Status: Planned

Related glossary entries

• MG-P07-six-ontic-requirements Six Ontic Requirements (OR1-OR6)
• 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

Referenced by

• 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-P07-six-ontic-requirements Six Ontic Requirements (OR1-OR6)