Proven Facts by Maclain Hunter

Authored by Maclain Hunter · first-party-authored · authored-foundational-record

Stored 4P source diagnostic: unresolved · finding unverified_concern · admission held_for_format. This intake result is separate from the theorem checks of the exact propositional sequent.

Proven Facts by Maclain Hunter: Terms: Proof: Arguments or evidence establishing or helping to establish facts or Truth objectively. Truth or/of Fact: A proposition or statement that accurately reflects reality and can be verified through observation or evidence. Coherence or/of Logic: The property of logical reasoning that ensures internal consistency, meaning that the principles of Logic do not contradict each other. Rational Consistency: The alignment of knowledge claims with logical reasoning and factual evidence, ensuring that they are not contradictory and can be justified rationally. Genuine Knowledge: Information or beliefs that are true and justified, based on reliable evidence and reasoning. Reality: The state of things as they actually exist, independent of perceptions or beliefs. Tautological: A statement that is true by necessity or by virtue of its logical form, often self-evident and requiring no additional proof. Proof by Contradiction: A method of demonstrating the truth of a proposition by assuming its negation and showing that this assumption leads to a logical contradiction. Formal Argument (Logical Proof): A structured reasoning process that follows established logical principles to arrive at a (necessary) conclusion based on given premises. Bi-conditional Relationship: A logical relationship indicating that two statements are equivalent; both must be true or both must be false. Premises: P: The “Truth of Fact” is necessarily true and factual. Q: The “Coherence of Logic” is necessarily coherent and logical. R: The rational consistency of genuine knowledge and Reality is tautological (self-evident) when analyzed coherently/properly with logical/rational consistency. S: Assume that the Truth of Fact, the Coherence of Logic, and/or the rational consistency of genuine knowledge and Reality were incoherent and “not true”. We can see that would lead to logical contradiction. Therefore it’s logically untenable to hold that premises P, Q, and R are wrong/untrue. This means conclusions consistent with {P, Q, R} are necessarily and logically true. Proof and knowledge requires not only facts and Reality but Logic and Truth. Proof (Formal Argument): {P <-> Q, Q <-> R, R <-> S} |= (P & Q) <-> (R & S) Conclusion: {P, Q, R, S}: The “Truth of Fact” is logically equivalent to the “Coherence of Logic” which is itself coherent with and has a bi-conditional relationship with the rational consistency of true knowledge (logically consistent understanding) and Reality. This is proven through proof positive and the proof for contradiction in the premise S, incorporated into the (positive) proof. Truth Table: valid ↓ ↓ ↓ ↓ PQRS⎪P↔Q Q↔R R↔S╞(P&Q)↔(R&S) ――――――――――――――――――――――――――――― TTTT⎪TTT TTT TTT TTT T TTT TTTF⎪TTT TTT TFF TTT F TFF TTFT⎪TTT TFF FFT TTT F FFT TTFF⎪TTT TFF FTF TTT F FFF TFTT⎪TFF FFT TTT TFF F TTT TFTF⎪TFF FFT TFF TFF T TFF TFFT⎪TFF FTF FFT TFF T FFT TFFF⎪TFF FTF FTF TFF T FFF FTTT⎪FFT TTT TTT FFT F TTT FTTF⎪FFT TTT TFF FFT T TFF FTFT⎪FFT TFF FFT FFT T FFT FTFF⎪FFT TFF FTF FFT T FFF FFTT⎪FTF FFT TTT FFF F TTT FFTF⎪FTF FFT TFF FFF T TFF FFFT⎪FTF FTF FFT FFF T FFT FFFF⎪FTF FTF FTF FFF T FFF Result: Validated Q.E.D.

Lean 4 verification and timestamps

The machine formalization date is separate from the original authored work. See the verification record for exact checked scope and any unresolved parts.

Logic-centered continuity

This authored proof is connected to Logic, Logos, God, and the self-evident Ground of Reality, and to the distinct systems 4P Engine, MCB Core, 4P-MCB, and the source-first Canon. Formal proof, empirical evidence, and system-generated records remain separately typed.