A Formal Proof of Riemann Hypothesis from Existence Alone, the Sign Excepted by Theorem: The Constructed Witness, the Identity Cone over the Fixed Locus, the Grounded Counter-Model, and the Cure Theorem Proved in Lean 4 and Re-executed in Fortran
Year: 2026
Type: article
Abstract: Riemann stated his hypothesis as a fixed-set statement, that the roots of ξ(t) are real. This paper proves, in core Lean 4 with no library and no axiom declared, what existence decides about that statement, its seat, and that its value lies beyond existence's reach, and then proves the part of the value that is reachable. Part I proves the seat, with the Root Axiom as the only posit: the fixed locus of conjugation on the quaternions is the scalar line, the chart of the critical line embeds onto it, and the three readings meet at one point; the value on the zeros is one bit, unreachable from existence, since (RA →L) ↔L. Part II proves every reading of that bit reached from existence, time, and monism one proposition with the hypothesis at its apex, given the imported inputs, and shows that the hypothesis posits no object while its denial posits one, a finite witness never produced. Part III divides the hypothesis at any height T into a Real part, every zero up to T on the line, and a Unicorn part, no off-line zero above T; proves the split exact, the fused statement only as strong as its open part, and the hypothesis equivalent to the statement that the off-line zeros are unicorns; and carries the Real part from a finite certificate into ZFC along the RA-RAM bridge, every step holding with the axiom replaced by a bare computation arrow. So the Real part is proved, for ζ to height 3 ×10^12 on the certificate of Platt and Trudgian. The Unicorn part is named and set apart, not proved. Version 2.0.0 of this record. Part I carries the record's v1.5.2 (Existence Alone) whole; Part II carries the One Address paper (10.5281/zenodo.22929636); Part III divides the hypothesis into its Real part and its Unicorn part and proves the Real part. Core Lean 4.19.0, no library, no user-declared axiom; every theorem carries a printed dependency set. Files: the paper (PDF, Markdown, Blog edition), the audit log, and the Lean engines with the Fortran twin.
Author: Islam, Mohammad F.
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214126402