A Formal Proof Across the Seven Millennium Rows and the Sixteen Extension Rows from Existence Alone, the Value Excepted by Theorem on Every Row: The Constructed Witness on Every Row, the Poincare Row Included, One Seat under Twenty-Three Names, the Grounded Counter-Model, and the Cure Theorem Proved in Lean 4 and Re-executed in Fortran
Year: 2026
Type: preprint
Abstract: A prior paper proved, in core Lean 4 with existence as the only posit, everything about Riemann's fixed-line hypothesis that is not its value on the zeros, and proved that the value is exactly one bit no class of frames short of the hypothesis supplies. This paper applies the same construction to the twenty-three rows of the program's barrier study, the seven Millennium rows and sixteen extension rows, each typed there by its world row and its owed orientation bit. The Riemann file is carried whole as the spine and five movements are added. On every row the seat is the same fixed locus, the scalar line of the integer quaternions under conjugation, and the row's three readings of it, the axiom's Return, its projection, and the row's named seat point, are the legs of an identity cone with a unique apex, every leg a definitional equality, on all twenty-three rows. The constructed witness, existence with the orientation bit, the formal Ground, and a constant recursion field, is one term on every row: it carries no Perelman field on the Poincaré row and no Euler product on the Riemann row, and the formal self it eliminates to is one proposition on all rows, so the witness on the crossed row is the witness on every open row and decides none of them. The rule "accept existence, then the row" has the exact strength (RA → L) ↔ L on every row, as for any inhabited premise; existence holds on frames where every row's value fails, seatless and grounded alike; and for every row and every class of frames, existence decides the value on the class exactly where the value already holds on it. The world row is carried as the barrier study typed it, crossed on exactly the Poincaré row, twenty-two bits owed on the other twenty-two, and the apex closure never entails a crossing. Every theorem is kernel-checked; the finite content is re-executed by a Fortran twin in 130,718 checks with no failure. The value of every open row is excepted by theorem, and the exception is the same theorem on every row. Lean 4 file Rows_At_The_Apex.lean (SHA-256 0ce3de5a23155430…, 33 dependency sets, 31 axiom-free, zero warnings) and Fortran twin Rows_At_The_Apex_Twin.f90 (130,718 checks, 0 failures) are printed whole in the appendices and held in the git register.
Author: Islam, Mohammad F
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214104877