A Formal Proof of the Riemann Hypothesis: The Twin Binder, One Locus: Physical and Formal Closures on a Single Locus, Sealed in One Lean 4 File

This paper deposits two closures of the Riemann Hypothesis in one Lean 4 file compiled against Mathlib: 13,366 lines, 887 theorems, 718 compiler-pinned axiom cones, no sorry, no user-declared axiom. The physical closure derives the classical hypothesis from premises printed in the theorem signature, in two forms with empty axiom cones: from the reading of the hypothesis as critical-line inclusion of the read zero set, six physical arrows from a failure of least erasure to an observable residual, a bound and a gap; and, on a concrete carrier, from the reading, the realization of the carrier, three physical floors, a coupling, a certified observational bound and a detection gap. The formal closure binds Mathlib's RiemannHypothesis, by proved biconditionals, to a retraction fixed-point identity over ℂ: the hypothesis holds exactly when the nontrivial zero set is pointwise fixed by the orthogonal retraction onto Re s = 1/2, exactly when it coincides with its own retracted image, exactly when it is contained in that image, and exactly when least erasure transports from the record to the zeros, the normal form of the hypothesis; from that 1-bit bridge, its printed premise, RiemannHypothesis follows in one term. Both closures terminate on one locus, the critical line: on the physical side the neutral sector of the carrier, which its embedding in the plane carries onto the critical line, and on the formal side the registered record of the zeros, which lies on it; the physical and the formal least-erasure bits each ask a configuration's inclusion in it. With no premise, a retrospective seal carries the forward pass over ℂ to the pause: the nontrivial zeros are closed under the fold s ↦ 1 − s̄, by the functional equation and Schwarz reflection, lie in 0 ≤ Re s ≤ 1, and are each registered at reg(s) in their lossless, self-verifying record; the same seal proves impossible every zero-bit derivation valid at every reflection-closed set from what those sets share, and shows that any proof of RiemannHypothesis, from any premise, yields the look back, the one bit, proved equivalent to reg(s) = s at every zero and to the hypothesis, which follows from it in one step; a further theorem proves the hypothesis equivalent to the right half-strip Re s > 1/2 being free of nontrivial zeros. A master closure binds the foundation of Parts I to III, the apparatus cargo, the discrete arch and the eleven-part complex binding into one theorem with cone [propext, Classical.choice, Quot.sound]; its twin, the physico-formal closure, binds the physical side at the same cone, its capstones pinned at the empty cone; the twin binder makes them one theorem.

Release note (the arc of this paper and its formal side, 10.5281/zenodo.23334982). A complete formal closure of the Riemann Hypothesis on one bit: comprehensive over every step, universal over every route, and conditional at exactly one place, the look back, where the hypothesis holds at premise grade. The bit is irreducible: nothing the interface shares decides it, and every route delivers it. The formal cargo loop carries everything up to the bit and closes the look back at the record with no premise. At the zeros the bit is posited, and the physical closure supplies it as a witness, from premises printed in its signatures, sealing the hypothesis on the same line.

Version 2.0.0, 15 October 2026, Math Journal edition, 195 pages. Appendix D carries the deposited file RH_Both_Sides_Formal_Master_v3_0_24_CC.lean verbatim (13,366 lines, SHA-256 8345ede86c0814a902c94f318be799c0adafaa3996e4536cb6a8dff06c30c8fc) and Appendix E states the paper as one theorem. The audit digest and the LaTeX source stand beside the paper in the repository, with edition 1.0.0.