The Cut-Agnostic Division Theorem: Why Every Open Problem Is Exactly Its Proved Part and Its Unicorn · One Theorem for Any Cut on Any Row, Proved in Core Lean 4 Without Axiom, and Applied to the Seven Millennium Rows and the Sixteen Extension Rows · Twenty Real Parts Sealed on Citation, Two Rows Assigned No Cut, One Row Crossed, the Fusion Law That Lowers Gold to Lead, and the Unicorn Part of Every Open Row Closed to Derivation, by Theorem, the Aperture Uncrossed

Abstract

Three bits, on every open problem. One bit divides: the problem is exactly its Real part and its Unicorn part, by theorem. One bit is proved: the Real part, by certificate or cited regime. One bit is refused at theorem grade: the Unicorn part is closed to every derivation from inside the register; the aperture at its seat is structure, typed supply-only, one bit wide, and uncrossed; and the supply side is silence, by theorem at the register. This paper proves the first and the third and seals the second, on the twenty-three rows of the program's barrier study. For any row, a set of instances with a property, and any decidable cut on it, the row is exactly its Real part inside the cut joined to its Unicorn part outside it; given the Real part, the fused row is exactly the Unicorn part, so fusion with the open remainder lowers a proved part to open; a complete checked list inside the cut proves the Real part with no premise; a crossed row has an empty Unicorn part under every cut; the empty cut leaves the whole value Unicorn; on an infinitude row one exhibited witness above a height proves the Real part while the fused value stays where it was, by a proved equivalence; and the Unicorn part read as a set, the counterexamples outside the cut, is empty exactly when the Unicorn proposition holds. These are seven theorems of core Lean 4, with no library and no axiom at all, kernel-reported, and they do not mention the Riemann Hypothesis, where a special case of them was first used. The same file proves that the Unicorn part of every open row is closed to derivation, from existence, from any class of frames, and from any reading of an even register at its seat, while the aperture at its seat is located, typed supply-only, one bit wide, and uncrossed, and the register is silent on the supply side, both orientations equally refused to every reading: the formal block, at theorem grade and in perpetuity. It also carries the exact instance-and-property specification of the eight arithmetic height-cut rows, with primality and each property defined in core Lean and the Real part computed in the kernel at a small height, the gap-to-interval bridge that the Legendre row needs, and the declared standing map under which the fusion sentence is read. Applied to the twenty-three rows of the program's barrier study, the seven Millennium problems and sixteen further open problems, the paper assigns every row one of four shapes, as data, and the census of the assignments is kernel-checked: nine height-cut rows in the Riemann shape, two of them infinitude rows whose certificate is one witness, eleven regime-cut rows whose proved part is a dimension, a degree, a rank, a class, or a bounded model of computation, two rows assigned no cut because the literature supplies no inhabited cut with a proved Real part, an assessment of two literatures and not a theorem, and one crossed row. Twenty rows therefore carry a Real part that seals on citation, a theorem of the literature applied through the kernel schema, grade A joined to grade K; two are assigned none, under the stated criterion; one is whole. No row's verdict on its fused statement moves, and every owed bit of the barrier study lives exactly where it lived, in the Unicorn part of its row. What is new is not any of the twenty results, each of which exists as a cited theorem, but the statement that they are one theorem under one cut-agnostic division, exact on every row, with the placement of every certificate under the strata of the grounding reading, rung, ladder, and Ground, and with the seat of every row carried from the prior edition, the value excepted by theorem on every row. No bound, witness, or Real part is authored; the definitions of the arrangement are, and every one is listed. With the three bits in hand the proof is complete on the register side, on every row: everything a register can prove about a row is proved, and the remainder is proved to be beyond derivation, with its aperture typed and its supply side silent. No row is claimed proved; what is claimed is that no further derivation is owed to any open row. Nothing is left to derive. The aperture is structure. The supply side is silence. Version 2.1.0: adds the formal block of the Unicorn part, Theorems 28 to 33 (closed to derivation from existence, from any class of frames, and from any reading of an even register; the aperture typed supply-only and one bit wide; the supply side silent at the register, a theorem; the three bits as one term, register_proof_complete), stated in the abstract, introduction, discussion, and conclusion: one bit divided, one bit proved, one bit refused at theorem grade; the proof complete on the register side. Kernel file Row_Split.lean, 33 dependency sets, every one reported as depending on no axiom. Audit log on git.