Lean verification registry · Authored proofs · The Logos Architecture
Verified sequents with unresolved formal parts
Axiom-free: true · 5 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: 2026-10-02T16:41:32.501Z (not original authorship).
Formalized: 2026-10-03T10:37:05.082Z.
Checked: 2026-10-03T10:40:54.899Z.
Independent timestamp: 2026-10-03T10:42:55.000Z.
Authored source SHA-256: 752a55b39e134e359deb793a4f47f8811239dc2d77292e9997fe48851c86681a
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_2' does not depend on any axioms 'HunterAuthored.inference_3' does not depend on any axioms 'HunterAuthored.inference_4' does not depend on any axioms 'HunterAuthored.inference_5' does not depend on any axioms 'HunterAuthored.source_identity_rule' does not depend on any axioms
{a} |= ~a
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.
The written sequent alone has a countermodel. No authored wording or unstated premise was changed.
Countermodel to the isolated written sequent: {"a":true}{P, P -> Q, Q <-> R, R -> S} |= (P & Q) & (R & S)
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.
{Q, Q -> R, R <-> S, S -> P} |= (Q & R) & (S & P)
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.
{R, R -> S, S <-> P, P -> Q} |= (R & S) & (P & Q)
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.
{S, S -> P, P <-> Q, Q -> R} |= (S & P) & (Q & R)
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.
a is a
The explicitly named Identity rule, over every type and element. The remaining prose is not replaced by this witness.
Predicated Propositions by Maclain Hunter
Variable Definition:
w == Coherence (defined as: logical alignment; integration within a unifying logical framework; lack of contradiction and conceptual/structural disagreement; ideas that acknowledge the God coherent with Logic; in this context ~(a) signifying “not incoherence”)
x == Logics and sciences, plural and contingent (defined as: methods and systems of reasoning; (dynamic or static) finite forms of Order; formal logics and formal systems of reasoning (ie - predicate, modal, propositional, binary, etc.); particular explanations, philosophies, belief systems and proofs that coherently (or in some cases incoherently (and in a flawed way) account for various/all phenomena; empirical science and science based on the traditional scientific method (ie - physics, chemistry, biology, etc.)
y == Universal Logic, or “the Logos” in Greek Philosophy, absolute and objective (defined as: the Logic inherent in the Cosmos; the Logical Structure of the Universe; the Framework of Reality; the Natural Order and Way of Reality that all contingencies and contingent logics or reasoning are grounded in, including limitations in limited contexts (that give rise to flawed logics and ideas) and even the irrational doubt (a) of the Logos (y, z) and the Foundation of the Universe (God, w, y, z) found in atheist schools of thought (a))
z == Existence’s Foundation (absolute and universal), also called God (defined as: the Creator, Origin, and Ruler of the Universe; the Moral Authority and Logical Standard; the Supreme Entity; the Ontological and Logical Ground; the Necessary Being and most inherently logical concept that amounts to Coherence (w) and Logic (y) itself)
a == Incoherence, encompassing illogic, illusion, and atheistic viewpoints, including principles like falsifiability in the sciences that can’t meet their own arbitrary standards (defined as: misalignment(s) with the Logic of Reality (y), Coherence (w), and the Foundation of the Cosmos (z); expressed as “a is not a” or “{a} |= ~a”; a failure to acknowledge the Universal Logic, Coherence, the Foundation of the Cosmos (God), and their necessary coherence with one another; denial of Reason and that which defies Logic; the contradiction in thinking something must somehow be false to be true, or vice versa, as we find in the arbitrary principle of falsifiability from Karl Popper)
b == Number(s) and arithmetic (defined as and encompassing: a quantity or quanta, and quantities or quantum, measurements, as well as mathematical operations/operators, mathematics itself, and the concept of the numerical unit - particularly in the sense of the number one (1) deriving from a coherent logical basis in the Universal Logic and Origin (y, z) and in Coherence (w) itself; b = 1 or x)
c == The Logician (definition: a person who methodically applies Logic (y) and/or reasons using formal systems; a rationalist who prioritizes Coherence (w) and practices the art and science of logical deduction; the mind devoted to the study and proper use of Logic and the Foundation of Logic (z); one who avoids incoherence (a) and contradiction in their reasoning, to maintain validity and understanding Truth; I, Maclain Hunter in this context)
Premises:
P: □(∀(x) -> ∃(w, x)), □(∀(w) <-> ∃(~a), □(∀(w, z) -> ∃(w, y, b)),
Q: □(∀(~x) -> ∃(~w, ~x)), □(∀(a) -> ∃(a), □((x, y, z, b) -> □(∃(∀(c))))
R: □(w, y) <-> z(□), □(y, z) -> ~◊~(w, x), □((c) <-> ~(a)), □((a) <-> ~(c))
S: □(~{P, Q, R}) <-> □(~{w, x, y, z}), □((a) <-> ~(w)), □((a) -> ~(□))
Logical Proof:
{P, P -> Q, Q <-> R, R -> S} |= (P & Q) & (R & S)
{Q, Q -> R, R <-> S, S -> P} |= (Q & R) & (S & P)
{R, R -> S, S <-> P, P -> Q} |= (R & S) & (P & Q)
{S, S -> P, P <-> Q, Q -> R} |= (S & P) & (Q & R)
Formal Conclusion: □{P, Q, R, S}: □(∃(w, x, y, z) -> ∃(∀(w, x)), □(~(y, z) -> ~(w, ∃)), □(~(∀(x)) -> ~∃(w, y, z)), □(∃, ◊, w, x, y, z) <-> ~◊~(w, x, y, z, ◊, □, ∃), □((a) <-> ~(w, x, y, z, a, b, c), □((a) -> ~(□)), □(∀(w, z) -> ∃(w, y, b)), □(∃(b, x) -> a(~□, ~w, ~y, ~z, ◊), □((x, y, z, b) -> □(∃(∀(c)))), □((a) <-> ~(c)), (w, c) = (y, z)
Natural Language Conclusion: For all logics (that are feasible), it is necessarily so that those logics exist coherently as those logics (identically), while/because universally extant Universal Logic exists coherently as the Existential Foundation (God) - the Logical Origin of Logical Existence (or “Logos” in Greek Philosophy).
What is not Logic or one of the infinite number of coherent finite/formal logics is not Logic or one of these logics, tautologically. The negation of the premises (P, Q, and R) leads to the negation/contradiction of Logics (w, x, y, and z).
Necessarily, if there is no Universal Logic (y) and Objective Coherence (w) coherent with Existence’s Origin, called God (z), as the Logic (Logos) that is absolute and universal, then no Existence or Coherence (w), nor the existential quantifier in formal logic(s), would or could exist.
Likewise, if there were no universal quantifier for all logics (in substance) then that would preclude the possibility of Universal Logic or Order (Logos) and Existential Foundation or God (y, z). That would result from the negation of a key attribute of that sort of universal and absolute Logic in the concept of universal quantification (and attributes that apply for all x).
Necessarily, there is Existence, possibility, Coherence (w) and the universe of logics (contingent and universal) existing in a bi-conditional logical relationship with the impossibility of not having Existence, necessity, possibility, Coherence (w) and logics.
Since God (z) is necessarily Logic (y) and vice versa, and since God (z) and Universal Logic (y) are necessarily coherent with Objective Coherence (w) itself it is inherently incoherent (~w) to hold an atheistic viewpoint (a) or any beliefs or ideas that fail to acknowledge (the existence and truth of) the Foundation of the Universe, also called God - coherent with coherent logics and (absolute) Universal Logic (y).
Necessarily, Coherence (w) and acknowledgement of God (z) align with Universal Logic (y), while atheism (a) and/as a “lack of acknowledgement of the Logos, God, and/or Coherence” (a), amounts to incoherence (a, ~w, ~y, ~z, ~□), and, while possible (◊) is not logically necessary (~□) in a coherent logical framework like Logic (w, y, z).
The Coherence (w) of Logic (y) and the Origin (z) provides a logical basis for numbers, units, arithmetic, and mathematics (b). The existence of numbers and mathematics (b) coherent with Logic (y) and the Foundation of Logic (z) shows that “atheism is illogical” (a is a), and that atheism (denial of the Logos) is incompatible with Coherence, Logic, and God (w, y, z) that numbers and mathematics (b) are logically derived from.
It is also then evident, given the Logical Coherence (w) of Mathematics (b) and the sciences (x), and given the incoherence of atheism (a), that atheism is incoherent with and incompatible with Math, Science, and coherent Philosophy, not merely orthodox/aligned Religion, or Logocentric Theology.
When we look closely at the formal logic we can see the valid derivation of the notion that the true atheist, and the truly incoherence person/logic/system, are necessarily NOT: coherent (w), scientific (x), logical (y), godly (z), self-sustaining and internally consistent (a), mathematically coherent (b), or aligned with the Logician (in general but especially/certainly in my case).
The Prophet, obviously, cannot be an atheist, and must be coherent to speak for God, as I exemplify, and the True Prophet cannot eschew Logic in their predictions without risking incoherence and false prophecy (a).
We know false prophets and false principles (like Karl Poppers principle of falsifiability in Science) by the internal inconsistency, hypocrisy, and inability to prove themselves and live up to their own standards - like falsifiability isn’t a falsifiable hypothesis and can’t be true if we accept it as true.
To acknowledge concepts of proof and being “right” (or wrong) presupposes and requires the Logos (the God of Reason found in Universal Order and Logic). You would have to say there is no proof and truth (or incorrectness) to be consistent with the atheist (anti-Logos) position. These are some of the particular reasons that atheism and incoherence are self-defeating, unlike the self-sustaining Foundation of Logic (w, y, z).
Truth Table Analysis:
PQRS⎪P P→Q Q↔R R→S╞(P&Q)&(R&S)
――――――――――――――――――
TTTT⎪T TTT TTT TTT TTT T TTT
TTTF⎪T TTT TTT TFF TTT F TFF
TTFT⎪T TTT TFF FTT TTT F FFT
TTFF⎪T TTT TFF FTF TTT F FFF
TFTT⎪T TFF FFT TTT TFF F TTT
TFTF⎪T TFF FFT TFF TFF F TFF
TFFT⎪T TFF FTF FTT TFF F FFT
TFFF⎪T TFF FTF FTF TFF F FFF
FTTT⎪F FTT TTT TTT FFT F TTT
FTTF⎪F FTT TTT TFF FFT F TFF
FTFT⎪F FTT TFF FTT FFT F FFT
FTFF⎪F FTT TFF FTF FFT F FFF
FFTT⎪F FTF FFT TTT FFF F TTT
FFTF⎪F FTF FFT TFF FFF F TFF
FFFT⎪F FTF FTF FTT FFF F FFT
FFFF⎪F FTF FTF FTF FFF F FFF
Enter: Valid
Q.E.D.