P ≠ NP Closed to One Bit: The Unconditional Normal Form of the Separation of SAT on Turing Machines

An Unconditional Formal Proof in Core Lean 4 of the Trisduction Normal Form of the Separation, with Every Route Shown to Pass the Bit, the Uniform Route Barred by Theorem, and the Value P ≠ NP Delivered Twice at Premise Grade; the Full Kernel Attached Whether satisfiability admits a polynomial-time decision procedure is the open 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, and proves everything in Sections 4 to 14 with no posit, no custom axiom, no physics and no root: a kernel of 151 theorems in core Lean 4 that imports nothing, whose 93 pinned cones show 72 results resting on no axiom at all and 21 on Lean's three standard axioms, with every definition computable. On an abstract machine the separation is proved equivalent to a single implication, the transfer of least escape from the record of the machine (the machine read with every output negated) to the machine; the record has least escape unconditionally, so the whole content of the separation is that one implication, the exact bit. The bit is keyed: it holds in one coherent world and fails in another, now on worlds that decide the same formulas with the same predicate, and no property shared by all machines, no property shared by all algorithm models of SAT, and no principle true of every record can force it. Every route to the separation, from any premise, delivers exactly the bit; each stopped run certifies only its own pair; no stock of stops, finite or infinite, decides the machine; the involution exchanging the two worlds has no fixed point. The model is built, not assumed: the machines, their step semantics, the running time as the first halting step, a binary encoding of formulas with a decoder proved to invert it, SAT as a language, and the classes P and NP as definitions. Over every Turing machine, SAT ∉ P is exactly the bit, and with SAT ∈ NP, cited, the bit gives the sentence P ≠ NP. It proves, unconditionally and on the real machine, what the value is made of, what cannot supply it, and what any proof of it is; and its closing section supplies the value, P ≠ NP, at premise grade on the one posit, delivered twice, by a formal cargo and by a physical witness, both refuted at once by one polynomial-time Turing machine for SAT. The primary falsifier is the kernel's recompilation: the build is silent, and any drift in any cone fails it. Files. P_ne_NP_Closed_to_One_Bit.pdf, the paper (Math Journal edition, 31 pages, 21 October 2026; Appendix A prints the kernel whole with its line numbers and Appendix B its cones). PNP_Normal_Form_Paper_v1_3_0_CC.lean, the kernel: core Lean 4.19.0 with no import, 1,691 lines, 151 theorems, 93 pinned cones (72 on no axiom, 21 on Lean's three standard axioms), a silent build with warnings made fatal; SHA-256 97af802524b5134e026a4dbff7b1c1273f4c3e263e5dcdc7d413a7a2998b0a6c. Source: https://github.com/1000sapients/Trisduction/blob/main/FINAL_BINDING/PNP%20Binding/PNP_Normal_Form_Paper_v1_3_0_CC.lean