OR3 Diagonal-Free Self-Reference

OR3 is the third of six original-rule narrowing principles (the Six Ontic Requirements, VII.D37). It states the diagonal-free self-reference requirement: any τ-categorical reality candidate must support internal self-description without generating diagonal paradoxes (Russell, Burali-Forti, Cantor) — and must do so structurally, not by ad hoc comprehension restrictions.

Categorical invariant. Diagonal-free narrowing: ∀ candidate ontic structure τ', τ' supports a self-description morphism τ' → τ' such that no diagonal construction yields a paradox. The mechanism must structurally eliminate the diagonal — not merely block it via restricted comprehension.

Primary registry anchor: VII.D37

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

1. I.K0 — Universe Postulate
2. VII.D37 — Six Ontic Requirements (OR1-OR6) — OR3 is diagonal-free self-reference
3. VII.L32 — OR3+OR4 Narrowing — diagonal-free self-description plus NF-addressability
4. VII.P08 — Each Requirement Independently Necessary — dropping OR3 admits naive set theory with unrestricted comprehension

OR3 is instantiated whenever a self-referential construction is reformulated to dissolve a paradox structurally rather than block it by fiat. Examples: τ's NF-address system (no universal set; addresses are never self-containing); type theory's universe hierarchy (eliminating Type:Type); rejection of naive set theory's unrestricted comprehension as a foundational primitive.

Examples:

• τ-framework: the NF-address system — every entity has a unique address, but no address contains itself; the diagonal is structurally absent
• Type theory: stratified universes (Type_0 : Type_1 : Type_2 : ...) — the diagonal Type:Type is structurally eliminated
• Russell's paradox: the foundation must explain *why* the diagonal cannot be formed, not merely forbid it (ZFC blocks via separation; τ eliminates via NF-addressing)
• Computer science: rejecting fixed-point combinators in total / strongly normalizing systems

Register codomain: Proof (diagrammatic — diagonal-free self-reference is a structural / proof-theoretic constraint on the self-description morphism, 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-P04-or4-nf-addressability OR4 NF-Addressability
• MG-P07-six-ontic-requirements Six Ontic Requirements (OR1-OR6)

Referenced by

• MG-P04-or4-nf-addressability OR4 NF-Addressability
• MG-P07-six-ontic-requirements Six Ontic Requirements (OR1-OR6)
• MG-P10-independent-necessity Independent Necessity of Each Ontic Requirement