Lean verification registry · Authored proofs · The Logos Architecture
Source recorded — formalization still required
Axiom-free: not established · 0 checked theorem statements.
Lean establishes the listed formal statements and dependencies. The exact authored source, Ground, known premise meanings, application validation and empirical evidence remain separately typed. No unencoded section is certified by a neighboring theorem.
Original source author: Maclain Hunter. Machine formalization: Replit Agent. No original wording was rewritten.
Source registered: 2026-10-03T01:29:39.312Z (not original authorship).
Formalized: 2026-10-03T10:37:05.082Z.
Checked: 2026-10-03T10:41:50.823Z.
Independent timestamp: 2026-10-03T10:42:55.000Z.
Authored source SHA-256: 4b81cd82480df8f561c9cd48dae5d241e46717c2104895c192e0c299ce8e9c0e
Lean source · Verification JSON · Exact archived source JSON · Compiler report · Current authored source page · Signed release and verification instructions
Lean (version 4.7.0, commit v4.7.0, Release) Not executed: no complete licensed formal sequent recovered.
No complete formal sequent has yet been recovered. This source is recorded, not marked Lean-verified.
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).