Lean verification registry · Authored proofs · The Logos Architecture
Written formal sequents verified
Axiom-free: true · 3 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: 2026-10-02T16:41:43.461Z (not original authorship).
Formalized: 2026-10-03T10:37:05.082Z.
Checked: 2026-10-03T10:42:27.911Z.
Independent timestamp: 2026-10-03T10:42:55.000Z.
Authored source SHA-256: c7fd9ee9e8abcd219b079284cb28403ac24cee528e0954dcac541b4a97ff763f
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 'HunterAuthored.inference_2' does not depend on any axioms 'HunterAuthored.source_identity_rule' does not depend on any axioms
{P, P <-> Q, Q <-> R} |= (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, a <-> b, b <-> c, c <-> d} |= (d -> c) -> (b -> 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.
Law of Identity
The explicitly named Identity rule, over every type and element. The remaining prose is not replaced by this witness.
The Redundancy of Proof for the Pythagorean Theorem by Maclain Hunter:
Premises:
Let P represent Logic and the Logical Order inherent in deriving trigonometry (a) from the Pythagorean Theorem (b), deriving the Pythagorean Theorem from syllogistic logical reasoning (lower order formal logic), and then deriving lower order syllogistic logic (c) from the Highest Order of Logic (Universal Logic) also called the Logos (d) or “Universal Logos” (containing all logoi and finite reasoning).
Let Q represent proof, as well as the contradiction inherent in the assumption that the validity and real-world success of trigonometry is not demonstration of the validity of the Pythagorean Theorem.
Let R represent the statement “the Pythagorean Theorem is valid and true, as demonstrated by its continuous practical and accurate function |= the Pythagorean Theorem is valid and true, as demonstrated by its continuous practical and accurate function”. Let R also represent the contradiction that arises from assuming that the Pythagorean Theorem is not inherently “proven” (aligned with Logos demonstrably) by being a valid geometric application of algebra in syllogistic form.
Assuming the negation of the set of premises {P, Q, R} leads to logical contradiction, therefore the set of premises is true.
Thus the reasoning holds that:
{P, P <-> Q, Q <-> R} |= (P & Q) & R
Conclusion: {P, Q, R}:
The Pythagorean Theorem is already self evident and is proven by its own valid and true identity, which is what any proof or logics must achieve (in any case) - so they are objectively consistent with Logic being Logic. It is absurd to think the Pythagorean Theorem lacks Coherence (Logic) and thus “lacks proof”.
It is self-evident in the same way a syllogism with true premises is. The Pythagorean Theorem can, in fact, only leas to a “wrong answer” when the values that are used are incorrect and don’t actually match the triangle at hand - which is not a flaw in the theorem but in the use of it.
The Logical Order of concepts a, b, c, and d, representing trigonometry, the Pythagorean Theorem, syllogisms, and Logic (Universal Logos) is as follows:
{a, a <-> b, b <-> c, c <-> d} |= (d -> c) -> (b -> a)
Truth Table Analysis:
valid
↓ ↓ ↓ ↓
PQR⎪P P↔Q Q↔R╞(P&Q)&R
――――――――――――――――――――――
TTT⎪T TTT TTT TTT TT
TTF⎪T TTT TFF TTT FF
TFT⎪T TFF FFT TFF FT
TFF⎪T TFF FTF TFF FF
FTT⎪F FFT TTT FFT FT
FTF⎪F FFT TFF FFT FF
FFT⎪F FTF FFT FFF FT
FFF⎪F FTF FTF FFF FF
Additional Note: If you don’t accept that the algebra of the Pythagorean Theorem is proven to not be invalid and arbitrary by its proper description of concepts in geometry, that have been shown to align, then identifying “proof” (alignment with Universal Logos) is surely not your forte. It is like thinking “2+2=4” needs external validation, or that the Law of Identity (which is what it is consistent with itself) needs external law. It’s not so.
Like the syllogism is a universally valid form of proving C based on A and B the Pythagorean Theorem is a more specific application of this same kind of Logic, tailored to the problem of figuring out a tangent based on sine and cosine, or any combination of those, using two (out of the sine, cosine, and tangent) to prove the remaining/unknown one.
Truly, to deny the Pythagorean Theorem which follows and gives example of valid syllogistic form would be to defy and undermine Logic in perhaps its most classical and sound form. Thus the Pythagorean Theorem can also be proven through contradiction by assuming that it is false and not possible to logically proof, and by inspecting the absurdity and negative implications for other aspects of Logic (like syllogistic form) that then arise - consistent with that line of reasoning.
It would make sense if Pythagoras had understood this and that it is why arbitrarily “external” proof of the theorem was not such a priority for him.
Q.E.D.