Lean verification registry · Authored proofs · The Logos Architecture
Written formal sequents verified
Axiom-free: true · 1 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: 2026-10-03T01:29:55.789Z (not original authorship).
Formalized: 2026-10-03T10:37:05.082Z.
Checked: 2026-10-03T10:38:44.902Z.
Independent timestamp: 2026-10-03T10:42:55.000Z.
Authored source SHA-256: 72e9a6f90ae7c26f3c80bf7dbdfddf0d5d8718ba39661e57fe541d53b7b8022f
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) exitCode=0 'HunterAuthored.inference_1' does not depend on any axioms
{P, P → Q, Q → R, R → S, S → T} ⊢ T
Exact source sequent under finite two-valued propositional semantics. Source proposition meanings and Logic/Logos/God Ground are preserved separately; this checks the inference, not unencoded prose.
Grounded FOL and FCL by Maclain Hunter
Terms:
FOL (First Order Logic):
Logic formally dealing in quantified existence, predicates/relations, and identity, built on top of propositional Logic.
FCL (Fusion Core Logic):
a dynamic set of universal symbols and principles (identity, non-contradiction, binary truth values, derivability (⊢) as the carrier of necessity, etc.) — that hold across major formal systems.
It is held together by coherent grounding (or Coherence) with roots in propositional and first order Logic.
FCL also explicitly allows the self-reference of a constant as its own predicate whereas that is ambiguous in FOL, as Coherence itself is both the Universal Constant (Ground) and its own defining characteristic and predicate, without contradiction. A formal system that won’t allow that “self-predicating constant” is incoherent with Logic as (logically predicated) Logic.
Let:
P = The Ground (of Logic); the Logos; Logical “God” or Foundation.
Note: P is intentionally used below as both a constant and a predicate. How and where P is used in a given formalism determines what role it is playing, the meaning of the term doesn’t actually shift. The Ground of Logic has the property of grounding itself and Logic, so P and “P(P)” both make objective sense.
G = Grounded (in Coherence); is grounded; has grounding (in the).
L = Logic and logics (or formal systems of Logic); Logos and logoi.
C = Objective Coherence, which has no valid and coherent divergence/multiplicity in itself, (is the property of) and is unique/coherent/one.
F = First Order Logic (FOL) and Fusion Core Logic (FCL).
Proof:
L(F) ∧ ∀L (C(L)) ⊢ G(F) ∧ C(P)
Ergo:
The Logic underlying (and a property of) First Order Logic (FOL)— and the fact that all Logic is (and all systems of Logic are) necessarily coherent — prove the Ground of Logic is coherent and that valid Logic, logics, and FOL are coherent/grounded.
Coherentized:
axiom1: ∀x (Lg(x) → C(x))
axiom2: Lg(f)
axiom3: ∀x (C(x) → G(x))
axiom4: ∀x (G(x) → ∃y (C(y) ∧ Ground(y,x)))
axiom5: ∀x∀y ((C(x) ∧ Ground(x) ∧ C(y) ∧ Ground(y)) → x=y)
step1: C(f) ⊢ axiom1, axiom2
step2: G(f) ⊢ axiom3, step1
step3: ∃y(C(y) ∧ Ground(y,f)) ⊢ axiom4, step2
identify: y=p ⊢ axiom5, step3, [C(p) ∧ Ground(p)]
conclusion: C(p) ∧ G(f)
Propositional Argument:
Premises:
P: “step1” (C(f) ⊢ axiom1, axiom2) from the coherentization, as well as P as P has been defined and used.
Q: “step2” (G(f) ⊢ axiom3, step1).
R: “step3” (∃y(C(y) ∧ Ground(y,f)) ⊢ axiom4, step2).
S: Identification (y=P ⊢ axiom5, step3, [C(P) ∧ P(P)]): The requirement for Coherence (C) and “some coherent ground” (y) is tantamount to a demand for the Ground of Logic (P).
T: Conclusion (C(p) ∧ G(f)): First Order Logic is a formal system of Logic grounded in the Ground of Logic, which is ultimately coherent itself.
Proof:
{P, P → Q, Q → R, R → S, S → T} ⊢ T
Truth Table Analysis:
TTTTT⎪T TTT TTT TTT TTT T
TTTTF⎪T TTT TTT TTT TFF F
TTTFT⎪T TTT TTT TFF FTT T
TTTFF⎪T TTT TTT TFF FTF F
TTFTT⎪T TTT TFF FTT TTT T
TTFTF⎪T TTT TFF FTT TFF F
TTFFT⎪T TTT TFF FTF FTT T
TTFFF⎪T TTT TFF FTF FTF F
TFTTT⎪T TFF FTT TTT TTT T
TFTTF⎪T TFF FTT TTT TFF F
TFTFT⎪T TFF FTT TFF FTT T
TFTFF⎪T TFF FTT TFF FTF F
TFFTT⎪T TFF FTF FTT TTT T
TFFTF⎪T TFF FTF FTT TFF F
TFFFT⎪T TFF FTF FTF FTT T
TFFFF⎪T TFF FTF FTF FTF F
FTTTT⎪F FTT TTT TTT TTT T
FTTTF⎪F FTT TTT TTT TFF F
FTTFT⎪F FTT TTT TFF FTT T
FTTFF⎪F FTT TTT TFF FTF F
FTFTT⎪F FTT TFF FTT TTT T
FTFTF⎪F FTT TFF FTT TFF F
FTFFT⎪F FTT TFF FTF FTT T
FTFFF⎪F FTT TFF FTF FTF F
FFTTT⎪F FTF FTT TTT TTT T
FFTTF⎪F FTF FTT TTT TFF F
FFTFT⎪F FTF FTT TFF FTT T
FFTFF⎪F FTF FTT TFF FTF F
FFFTT⎪F FTF FTF FTT TTT T
FFFTF⎪F FTF FTF FTT TFF F
FFFFT⎪F FTF FTF FTF FTT T
FFFFF⎪F FTF FTF FTF FTF F
(Valid)
QED