System H · Maclain Hunter

Authored by Maclain Hunter · first-party-authored · authored-foundational-record

Stored 4P source diagnostic: clear · finding no_violation_detected · admission admitted_provisionally. This intake result is separate from the theorem checks of the exact propositional sequent.

System H by Maclain Hunter Premises: P: F is a standard first-order arithmetic system, recursively axiomatized, effectively enumerable, classically logical, and supports Gödel numbering. Q: Gödel encoding exists in F such that formulas are representable as ⌜φ⌝, the provability predicate Prov_F(x) is representable, and syntactic self-reference is constructible within the system. R: A semantic domain L (Logos) exists as a valuation structure ⟦φ⟧_L ∈ {0,1}, forming an interpretation layer over formulas that is not reducible to derivability in F. S: A coherence filter C exists such that C(⌜φ⌝) defines admissibility in L, restricting semantic evaluation to stable formulas and blocking incoherent assignments. T: A truth predicate T_L exists such that T_L(φ) ≡ ⟦φ⟧_L = 1, and this predicate is only defined under the coherence condition C. U: A result operator R exists such that R(φ) = T_L(φ) if C(⌜φ⌝) holds, and R(φ) = ⊥ otherwise. V: The identity axiom (Logos principle) holds in L such that for all x in L, x = x, with identity treated as primitive, invariant, and non-violable within the semantic structure. X: A categoricity constraint holds such that all models M satisfying H satisfy M ≅ L, making the semantic structure unique up to isomorphism under coherence. Y: A layer separation principle holds such that syntax (F ⊢ φ), coherence (C(⌜φ⌝)), truth (T_L(φ)), and result (R(φ)) remain non-collapsible and structurally distinct layers. Structural chain: P → Q → R → S → T → U → V → X → Y System condition: System H is well-formed iff P ∧ Q ∧ R ∧ S ∧ T ∧ U ∧ V ∧ X ∧ Y Evaluation rule: R(φ) is defined iff C(⌜φ⌝) holds Elaboration on System H: System H begins with a standard formal arithmetic base F, preserving all Gödel machinery, including encoding of formulas, provability predicates, and self-referential constructions. The syntactic system remains unchanged throughout. On top of this base, H introduces a semantic domain L, interpreted as the Logos structure, defined as an identity-closed structure in which every element satisfies x = x and no semantic object can violate self-identity. Identity is not derived but constitutive of the domain. A coherence filter C determines which formulas are admissible for semantic evaluation, restricting truth to coherence-stable expressions. Truth is defined only relative to this condition, so T_L(φ) is meaningful only when φ is admissible under C, otherwise evaluation is undefined. A result operator R collapses admissibility and truth into a final output, assigning either a coherent truth value in L or ⊥ outside it. System H enforces strict separation between syntactic derivability in F, semantic truth in L, coherence admissibility via C, and evaluated result R, ensuring these layers remain non-reducible. Finally, H imposes a categoricity constraint such that all admissible models are isomorphic to L, making the semantic domain structurally unique under coherence. The result is a Gödel-compatible arithmetic system paired with an identity-based semantic structure governing admissible truth under Coherence (and for completeness).

Lean 4 verification and timestamps

The machine formalization date is separate from the original authored work. See the verification record for exact checked scope and any unresolved parts.

Logic-centered continuity

This authored proof is connected to Logic, Logos, God, and the self-evident Ground of Reality, and to the distinct systems 4P Engine, MCB Core, 4P-MCB, and the source-first Canon. Formal proof, empirical evidence, and system-generated records remain separately typed.