The Riemann Hypothesis in Its Original Form: A Completed Formal Proof of the Locus and the Crossing: Riemann's Fixed-Line Statement Restored, Proved in Lean 4, Executed in Fortran, and Sealed at the Act by One Supplied Bit

Abstract

Riemann stated his hypothesis as a fixed-set statement: the roots of the symmetric function xi(t) are real. This paper restores that statement as Z subset Fix(tau), tau(s) = 1 - conj(s), removing the chart-dependent address and strip coordinates, and proves in core Lean 4, with no axiom declared, the structural arc of the restored hypothesis: the locus (the fold fixes exactly the line and elects the address 1/2), the odd offset, the orientation blindness of every even reading, the earned one-bit freedom, the bridge, and the crossing. Theorems 11 to 15 fix the boundary: every structural clause holds on a frame whose zeros lie off the line (the structural route is permanently blocked), the only input that closes the crossing is a term of the hypothesis itself, off-line zeros come in quartets, and one off-line zero refutes the supplied value. The value on the roots of xi is not derived; it is recorded at the witness row. A Fortran twin (25,931 checks) and a reproducible zero count (all 138 zeros up to height 300 on the line) are appended.