Lean verification registry · Authored proofs · The Logos Architecture
Verified sequents with unresolved formal parts
Axiom-free: true · 6 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: 2026-10-02T14:38:29.622Z (not original authorship).
Formalized: 2026-10-03T10:37:05.082Z.
Checked: 2026-10-03T10:42:29.891Z.
Independent timestamp: 2026-10-03T10:42:55.000Z.
Authored source SHA-256: 9a721430954eafb053405f7db703d2158b367d1000a39c0598cf1e3e3b8f913a
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_4' does not depend on any axioms 'HunterAuthored.inference_5' does not depend on any axioms 'HunterAuthored.inference_7' does not depend on any axioms 'HunterAuthored.inference_8' does not depend on any axioms 'HunterAuthored.source_identity_rule' does not depend on any axioms
{1, 1 <-> 2, 2 <-> 3, 4 <-> 5, 5 <-> 6, 6 <-> 7}
|= (((1 & 2) & (3 & 4) & 5) & 6) & 7
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: {"1":true,"2":true,"3":true,"4":false,"5":false,"6":false,"7":false}{P, P <-> P, P <-> ~Q} |= 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.
{P, P -> Q} |= P and it leads to logical contradiction. It amounts to saying Logic is “not Logic” which is not Logic (Valid Order and Reasoning). To assume God (Logos) is not God (Logic) would mean that God’s output (Reality - All Existence) is not Reality (what Logic entails).
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.
Unsupported formal syntax near ". It amounts to saying Logic is “not Logic” w"
{P <-> Q, Q <-> R} |= (P <-> Q) & (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.
{P, P -> Q, Q -> R, R -> S, S -> T} |= ((P & Q) & (R & S)) & 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.
{LL <-> LS} |= LL & LS
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: {"LL":false,"LS":false}{P, P -> Q, Q -> R, R -> S, S -> T, T -> U, U -> V} |= (((P & Q) & (R & S) & T) & U) & V
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.
{P1, P2, P3, P4} |= (P1 & P2) & (P3 & P4)
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.
Law of Identity
The explicitly named Identity rule, over every type and element. The remaining prose is not replaced by this witness.
The Rule(s) of Logic and Logic Applied to Logic, Science, and Proof by Maclain Hunter:
Part 1:
The Rules of Logic Applied to the Concept(s) of God, Logic, Logos, and Reason:
Laws:
1: Law of Identity: A is A: applied to God/Logic (Logos) this means God/Logic (Logos) is God/Logic (Logos) (logically) and that this is a tautology, true by definition.
2: Law of Non-Contradiction: A can’t be “not A” validly: applied to God/Logic (Logos) this means it’s invalid to claim that God/Logic (Logos) is somehow “not God/Logic (Logos)”. This reinforces the Universal Truth coherent with God/Logic (Logos) and the tautology that God/Logic (Logos) is indeed God/Logic (Logos), as the application of the Law of Identity—to and coherent with God/Logic (Logos)—has already made certain.
3: Law of the Excluded Middle: A is A and A is not A or A is and A is not cannot both be true and valid in the same context at the same time: applied to God/Logic (Logos) this means that it must be true that God/Logic (Logos) is God/Logic (Logos) and that “God/Logic (Logos) is” while it cannot also be true that God/Logic (Logos) is not God/Logic (Logos) and that “God/Logic (Logos) is not”. Any proposed fact/truth/statement that contradicts or denies Logos (God a.k.a.,Logic) is illogical and counterfactual/untrue.
4: Principle of Sufficient Reason: Things must have Cause and Rational Support (Reason/Logos) and must coherent with Reason (Logos) to be valid, true, real, and logical: applied to God/Logic (Logos)—which is a term for the Origin of the Universe, the True God (the “Real Origin of Reality”) and the Perfect/Universal Method of Reasoning as it exists according to its own inherently valid structure beyond any illogic or misconceptions—God must be fully aligned with Reason and this “Principle of Sufficient Cause” as “Being” (and the Supreme Being), the “Moral Standard” (Moral Authority), and the “Universal Mind and Person” (Collective Consciousness) must also be.
5: Modus Ponens: P implies Q and P is true therefore Q must also be true: applied to God/Logic (Logos): because God/Logic (Logos) implies the Origin of God/Logic (Logos) the existence of any instance or all instances of Logic, God, or Logos (Reason) means God/Logic (Logos) is true (aligned with fact and Reality).
6: Modus Tollens: If P then Q. Not Q, therefore not P: applied to God/Logic (Logos): God/Logic (Logos) in any instance requires the Origin of God/Logic (Logos) therefore if we did not have any Origin of God/Logic (Logos) there would be no God/Logic (Logos) and no proof or Rationality (Reason). This is formal logical reasoning (logical proof) so we can be sure that contradiction arises from denial of the Logos (the Source of Proof).
7: DeMorgans Law(s): The negation of A and B is the same as not A or not B, while the negation of A or B is the same as not A and not B: applied to God, Logic, Reason, and Logos: the negation of Logic and God is the same as “not Logic or not God”, while the negation of Logic or God is the same as “not Logic and not God”. To use a single term (Logos) for A and B we can say that the negation of Logos and Logos is the same as the negation of Logos or Logos, while the negation of Logos or Logos is the same as the negation of Logos and Logos.
Formal Argument:
Using the Rules of Logic above, and their application to God, Logic, Reason, and Logos (as explained above) as premises:
{1, 1 <-> 2, 2 <-> 3, 4 <-> 5, 5 <-> 6, 6 <-> 7}
|= (((1 & 2) & (3 & 4) & 5) & 6) & 7
Conclusion: (((1 & 2) & (3 & 4) & 5) & 6) & 7:
I conclude that God/Logic (Logos) (Law singular and universal) encompasses and is capable of integrating all logical rules (laws plural and/or particular).
These rules applied to God/Logic (Logos) and themselves lead to the necessary conclusion that God (the Origin of Existence) and Logic (the Method of Reasoning) must be and are demonstrated to be coherent (one).
This coherent God and Logic (Origin and Reason) is necessarily coherent with and logically equivalent to the Logical Structure of the Universe (Logos).
Whatever someone wants to call the thing that supports/governs the Universe (God), and the thing that allows us to know about the Universe (Logic), they must be logically coherent to not be logically incoherent.
Common terminology was used here (like Logic and God) but the substance of the conclusion is not ultimately a matter of subjective acceptance of specific terms. Rather it is a matter of the logical coherence and structure that allows for both Reality and thought/knowledge.
Part 2:
Maclain Hunter on the Rule(s) of Logic:
The Traditional Rules of Logic are three, and some thinkers add even more, but I understand that this is divergent and incoherent, not unified/coherent. If someone asked me what the Rules of Logic are, I might answer:
The traditional rules are:
1) Identity: it must be so that A is A
2) Non-Contradiction: it cannot be so that A is not A.
3) Excluded Middle: a statement is either true or false with no middle ground, as even a partly true statement is ultimately false.
BUT…
That is “incoherent” (not unified with, through, or as the Logic of Reality or “Logos”) and the question of what the “Rules” (plural) of Logic are overlooks that Logic (or “Logos”) is its (own) ONE and ONLY Law - its own Origin and Ruler, and the “Origin and Ruler (First Cause - God: defined as the First Principle and Cause of Logical Existence) of ALL Things”.
The Rule of Logic is: Logic
The Law of Reason is that: the Law of Reason must be (fully coherent with) and cannot contradict the Law of Reason.
This can be further described by a single description like saying:
The Law of Reason (which by Law can never contradict the Law of Reason (Logos) and must be the Law of Reason) is that Truth and Logical Validity fit and must align with Logic’s A is A form while falseness and invalidity fit and suffer from “A is not A” (contradictory) form - with a spectrum of possible intermediate logical positions, that contingent concepts can have, in between and relative to (independent and non-contingent) Pure Truth (and A is A form) - synonymous with Logos - and pure falseness (which is complete contradiction and “A is not A” form) - synonymous with nonexistence and that which cannot be logical/extant.
Application of the Rule of Logic (Law of Reason) also known as “the Logos”:
To apply the Rule of Logic in critical analysis it is a matter of analyzing, in a context sensitive manner, and accounting for the fact that x = y when both x and y equal c (as c equaling c satisfies the need for an A is A form), whether statements and ideas/connections match that A is A form and meet that requirement (or at least cannot be found in violation of it). Universal/Objective Truth may only match A is A form while “subjective truth” can be either “A is A” form or “A is not A” form, or indeed anything on the spectrum in between, but Objective (Universal) Truth (synonymous with Logos and Logic itself) encompasses and supersedes all subjective truths not fully aligned aligned with the Objective Logical Truth (Logic, also known as “God”).
Self-Existence and Self-Validation of the Rule of Logic (Logos):
The Rule of Logic is the Rule of Logic according to the Rule of Logic - which says in part but consistently that A must be A - and the Rule of Logic (Logos) not being the Rule of Logic (Logos) cannot be true or valid according to the Rule of Logic (Logos) - which also, as the other side of the same coin, says it cannot be so that A is “not A” (universally).
Proof:
Premises:
P: the Rule of Logic (A is A and Truth/Logic is Truth/Logic) which entails that God is and must be God.
Q: not the Rule of Logic (A is not A and Truth/Logic is not Truth/Logic) which entails that God is not and does not need to be God.
Argument:
{P, P <-> P, P <-> ~Q} |= P & ~Q
Proof of Contradiction:
Assume (P & Q) or {P, P -> Q} |= P and it leads to logical contradiction. It amounts to saying Logic is “not Logic” which is not Logic (Valid Order and Reasoning). To assume God (Logos) is not God (Logic) would mean that God’s output (Reality - All Existence) is not Reality (what Logic entails).
Conclusion:
P & ~Q must be so while P & Q or P then Q cannot be so, according to Logic and P. God is God as Logic is Logic. I am that I am and cannot be “not me”. Truly it is part of the Rule of Logic and what it encompasses/entails that Logic is God since the Rule of Logic is Logic and is what makes Logic and All Things including Logic, the First Cause, and the True God (God) be Logic and All Things including Logic, the First Cause, and the True God (God). To separate God and Logic or deny Logic is God is to violate the Rule of Logic and embrace illogic, unresolved paradox, contradiction, and (rational) lawlessness. This is a Logos (Logic) aligned Objective Truth (Logic), as the True God (Logos) is a Logic (Logos) aligned God, and that supersedes, as a matter of Logic and the application of the Rule of Logic, all subjective opinions or views that may be contrary to the Rule of Logic or only partially aligned with Reason as Law (the Rule of Logic).
Part 3:
Logic Applied to Logic and Science:
Logic Applied to Logic (LL):
Premises:
P: Identity: Logic is Logic
Q: Non-Contradiction: Logic can’t be “not Logic”
R: Excluded Middle: Logic is true and can’t be true and false at once.
Proof:
{P <-> Q, Q <-> R} |= (P <-> Q) & (Q <-> R)
Conclusion: {P, Q, R}:
Logic is must be Logic and must be true and cannot be “not Logic” or no true, according to Logic and the proper application of Logic to Logic itself.
Truth Table:
valid
↓ ↓ ↓
PQR⎪P↔Q Q↔R╞(P↔Q)&(Q↔R)
――――――――――――――――――――――――
TTT⎪TTT TTT TTT T TTT
TTF⎪TTT TFF TTT F TFF
TFT⎪TFF FFT TFF F FFT
TFF⎪TFF FTF TFF F FTF
FTT⎪FFT TTT FFT F TTT
FTF⎪FFT TFF FFT F TFF
FFT⎪FTF FFT FTF F FFT
FFF⎪FTF FTF FTF T FTF
Logic Applied to Science (LS):
Premises:
P: Question: This is the first step of the traditional scientific process that involves researching a problem or problems that the scientific method might be meaningfully applied to. This step is well aligned with Logic and breaks none of the three traditional Rules of Logic.
Q: Form Hypothesis: This second step aligns with Logic well also and involves the scientist predicting what the expected outcome of the experimentation will be when applied to a particular question. This step helps to later understand whether predictions made with logical reasoning end up matching empirically observed Reality which can be useful in helping us to align our personal reasoning (“our logics”) with the Order inherent in Reality (“THE Logic” - the Logos - Objective Reason).
R: Testing/Experimentation: Design test(s) in logically coherent fashion and perform the test(s) in a valid and consistent fashion. This step encompasses all feasible “logical valid testing methods” and gives broad leeway to scientific experimentation to achieve logically valid and scientifically rigorous results. The key thing that makes testing and experimentation valid and reliable is the combination of logical coherence and scientific rigor, which includes the notion that the more iterative one’s testing is the more reliable their results can be considered to be.
S: Analyze Data and Report Conclusions: Gather and apply Logic to the experimental data to notice logical inconsistencies or flaws in the process, as well as to determine whether the results confirmed the hypothesis (and why or why not). These results and the relevant data supporting conclusions can and should be used to determine and report the conclusion(s) in a formal manner. This step is also logically relevant and coherent at this stage of the process, as it is the final step in gaining knowledge through a logically grounded scientific process. This step, like the others in the traditional scientific method aligns well with and doesn’t not contradict itself (internal consistency) or Logic (logical consistency).
T: Assume the well ordered application of the scientific method as outlined in premises P, Q, R, and S was logically invalid and could not help us (scientists, logicians, etc.) gain knowledge. This assumption leads to logical contradiction where logically coherent science contradicts Logic - an untenable and unscientific position to hold, refuted by the usefulness of Science and all logically and scientifically valid results in Logic and Science.
Formal Argument (Proof):
{P, P -> Q, Q -> R, R -> S, S -> T} |= ((P & Q) & (R & S)) & T
Conclusion:
The steps of the traditional scientific method, represented by the atomic sentences {P, Q, R, S, T} flow logically one into the next, and each satisfy and do not violate Logic (as validated logically above), and this allows for coherent Science which is the logical derivation of knowledge through coherent empirical observation of controlled testing. If logically coherent Science were not a logically valid and scientific means of understanding Reality that would contradict Logic - thus we know it cannot be so and that logically coherent Science is reliable, logical, and valid. Logic remains the “Logical Standard” in Science just as the scientific method is the scientific method within Logic.
Truth Table:
valid
↓ ↓ ↓ ↓ ↓ ↓
PQRST⎪P P→Q Q→R R→S S→T╞((P&Q)&(R&S))&T
――――――――――――――――――――――――――――――――――――――――
TTTTT⎪T TTT TTT TTT TTT TTT T TTT TT
TTTTF⎪T TTT TTT TTT TFF TTT T TTT FF
TTTFT⎪T TTT TTT TFF FTT TTT F TFF FT
TTTFF⎪T TTT TTT TFF FTF TTT F TFF FF
TTFTT⎪T TTT TFF FTT TTT TTT F FFT FT
TTFTF⎪T TTT TFF FTT TFF TTT F FFT FF
TTFFT⎪T TTT TFF FTF FTT TTT F FFF FT
TTFFF⎪T TTT TFF FTF FTF TTT F FFF FF
TFTTT⎪T TFF FTT TTT TTT TFF F TTT FT
TFTTF⎪T TFF FTT TTT TFF TFF F TTT FF
TFTFT⎪T TFF FTT TFF FTT TFF F TFF FT
TFTFF⎪T TFF FTT TFF FTF TFF F TFF FF
TFFTT⎪T TFF FTF FTT TTT TFF F FFT FT
TFFTF⎪T TFF FTF FTT TFF TFF F FFT FF
TFFFT⎪T TFF FTF FTF FTT TFF F FFF FT
TFFFF⎪T TFF FTF FTF FTF TFF F FFF FF
FTTTT⎪F FTT TTT TTT TTT FFT F TTT FT
FTTTF⎪F FTT TTT TTT TFF FFT F TTT FF
FTTFT⎪F FTT TTT TFF FTT FFT F TFF FT
FTTFF⎪F FTT TTT TFF FTF FFT F TFF FF
FTFTT⎪F FTT TFF FTT TTT FFT F FFT FT
FTFTF⎪F FTT TFF FTT TFF FFT F FFT FF
FTFFT⎪F FTT TFF FTF FTT FFT F FFF FT
FTFFF⎪F FTT TFF FTF FTF FFT F FFF FF
FFTTT⎪F FTF FTT TTT TTT FFF F TTT FT
FFTTF⎪F FTF FTT TTT TFF FFF F TTT FF
FFTFT⎪F FTF FTT TFF FTT FFF F TFF FT
FFTFF⎪F FTF FTT TFF FTF FFF F TFF FF
FFFTT⎪F FTF FTF FTT TTT FFF F FFT FT
FFFTF⎪F FTF FTF FTT TFF FFF F FFT FF
FFFFT⎪F FTF FTF FTF FTT FFF F FFF FT
FFFFF⎪F FTF FTF FTF FTF FFF F FFF FF
Meta-Proof (LL <-> LS):
LL: Logic applied to Logic and the Logic of Logic.
LS: Logic applied to Science and the Logic of Science.
{LL <-> LS} |= LL & LS
Conclusion:
LL & LS entails “Coherent Science and Logic”, validated formally by Logic. Logic remains the “Logical Standard” in Science just as the scientific method is the scientific method within Logic.
Part 4:
Proof of Proof’s Source in Logic:
Definitions:
Logic: the Method of Reasoning and Realization; the Order inherent in Proper Order and self-consistency.
God: the Origin and Ruler of the Cosmos; the Logos (Pure Reason - Universal Order and Logic) in rational theism and classical theology.
P: Proof exists.
Q: Assume P is not true, that leads to logical contradiction and wouldn’t be possible to prove (then).
R: Even if “nothing” (pure illogicality) were “the Source of Proof”, the Source of Proof would still be and have to be “something”.
R: The Source of Proof must be logically coherent or it is logically incoherent. An incoherent Source of Proof would be a logical contradiction.
S: The Source of Proof is Logic, and is fully coherent, as the alternative that the Source of Proof is “not Logic” and “not coherent/one with Logic” is illogical and untenable.
T: The Source of Proof is the Source of Rational Support and Rationality (Reality) in/underlying the Universe, as to assume the opposite or any true (substantive) alternative leads to logical contradiction.
U: All proof is entailed by and proves the existence of the Source of Proof (Logic - the First Cause). To have proof in hand without the Source of Proof objectively extant in and encompassing the (rationally supported) Cosmos would be a logical contradiction. This proof itself could not exist without the Source of Proof and is therefore empirical evidence of the Logic encompassing Reality (the First Cause).
V: The Source of Reality (the Origin of the Universe - God) is and must be coherent with Rational Law (Logic - Reason - Identity) to establish coherence at/as the Universal Foundation (First Cause), and in order for the rest of the Natural Order (Logic) to be logical and logically founded.
{P, P -> Q, Q -> R, R -> S, S -> T, T -> U, U -> V} |= (((P & Q) & (R & S) & T) & U) & V
Conclusion: (((P & Q) & (R & S) & T) & U) & V:
Logic is the Source of Proof, the First Cause, and God, and all proof immanently and intrinsically proves that, even if not every proof explicitly proves such things. That is true of all proof, including this proof itself. This proof is empirical evidence (and observable/observed proof) that the Logic of Reality (also understood as the First Cause) surely exists.
For this proof to exist without the First Cause of Logic (the Foundation of Logic and Reality - God) would be a logical contradiction. Logic and language demand God be the Foundation of Reality (by definition and to satisfy and be coherent with Logic), and that God and Logic must be coherent for both to be properly grounded and coherent.
Final Part: Meta-Proof:
P1: Part 1: [see above]
P2: Part 2: [see above]
P3: Part 3: [see above]
P4: Part 5 [see above]
Formal Argument (Proof):
{P1, P2, P3, P4} |= (P1 & P2) & (P3 & P4)
Conclusion: (P1 & P2) & (P3 & P4)
The combination of all the proofs and formal philosophical treatises above holds, as to deny any piece denies Logic and to acknowledge any piece acknowledges Logic. To acknowledge the very existence of this writing is to acknowledge what could not exist if the Logos did not exist. Any such engagement or acknowledgement means that the rest of the arguments and the Meta-Proof follows logically and that God is Logic - inherent in all logics and sciences, and all instances (and the general concept(s)) of evidence and proof. This proof could not even be invalid if there is no Logos (Logical Standard coherent with the Origin of Logic).