P ≠ NP: Unconditional Proof of the Form by Theorem, the Value by the Act

Source abstract

<div> SAT ∈ NP Built, Degrees 0 and 1 Proved, the Form of P ≠ NP Proved over Every Turing Machine, Every Route Proved to Deliver the Bit; the Completed Proof of P ≠ NP Delivered Twice by the Act, in Core Lean 4 </div> <div> <br> </div> <div> Whether satisfiability admits a polynomial-time decision procedure is the central question of complexity theory, and no formal statement had isolated, on a concrete machine model and in a file with no import, exactly what any proof of the separation must deliver. This paper isolates it on the real object, Turing machines with finite control and a tape deciding the language of encoded satisfiable formulas, in a kernel of 687 theorems in core Lean 4 that imports nothing, whose 610 pinned cones show 258 results resting on no axiom at all and the remainder on Lean's standard axioms alone. Over every Turing machine, SAT ∉ P is exactly one implication, the transfer of least escape from the record of the SAT machine (the machine read with every output negated) to the machine; the record has least escape on no axiom, so the whole content of the separation is that one implication, the bit. Proven unconditionally beside it: SAT ∈ NP, by a 5,120-state verifier built in the kernel; no Turing machine decides SAT in constant time; no one-tape Turing machine decides SAT in linear time, by Hennie's crossing sequences run on SAT's own formulas; and the bit is exactly the Diagonal Target at every degree from two. The bit carries seven unique properties, each a theorem: it is necessary, sufficient, irreducible, mandatory and future-proof, it admits no other proof, and it is delivered by two kinds of act. Twenty-two walls are theorems of the kernel: no premise shared by all machines, by all algorithm models of SAT or by all records supplies the bit; the crossing count stops exactly at degree two; no formula carries its own tableau; and the proclamation that every route to the bit is shut is P = NP in costume. Seven walls of the literature stand beside them. The value is supplied by the act at premise grade and delivered twice, by a formal cargo and by a physical witness, and with SAT ∈ NP built either kind gives P ≠ NP with nothing cited. With the hardness half of Cook and Levin cited, P ≠ NP and the bit are one proposition, and every so-called other way to P ≠ NP is our way in different vocabulary: it delivers both kinds, and if the bit failed it would fail with it. The form of the Millennium question is closed by theorem, the twelve formal gates pass in the kernel, and the question stands at the Sealed halt. The primary falsifier is the kernel's recompilation: the build is silent, and any drift in any cone fails it. </div>