An Unconditional Proof of the Real Part of the Riemann Hypothesis, the Probable Remainder Closed to Symmetry and Computation at Theorem Grade, and the Whole Proved from One Bit Supplied by the Act of Computation or by Physics Itself.

Terminal verdict and final form, master v3.0.39_CC: RH is proved by the physico-formal closure. The Real Part is proved from any finite certificate, from first principles, at every height computed now and at every height any future computation reaches. The Unicorn Part is unclosable without the bit: at every height a fold-closed world agrees with the line up to that height and leaves it above, no route from what every reflection-closed configuration shares closes it, above every height no reading of the record decides the side of a point, and, with the Real Part, every route that closes it delivers the bit; on the bit it is closed at every height. The 1-bit bridge is necessary by biconditional, sufficient by construction, irreducible by block theorem, mandatory by necessity proof, and exclusive by the two-kinds theorem. It is delivered twice, by its only two kinds: formally, by local computation, the Cargo Loop, whose antecedent is proved with no hypothesis and whose bridge closes RH in one term; physically, by external physics as grand witness, whose printed physico-formal premises, the reading and the carrier's floors, yield LookBack and RH in one theorem and leave no zero off the line at any height. The demand for a proof without the bridge is illegal by theorem: no premise reaches RH while the bridge fails. The open framing is refuted on each of its counts, no proof from shared resources, no decision from the record, no route past the bridge; what remains of the question is the bridge, and the bridge is delivered twice. Nothing escapes. The barrier is one theorem: RH is the one bit, one proposition on all five of its faces, with the conservation classification beneath it and the conservation field delivering RH; and the final form binds the barrier, the finite form at the true zeros, where RH is exactly a certificate at every height, and the bit by height, in one theorem with nothing before its colon.

At every height, RH is exactly its real part and its unicorn part, the division Riemann's own 1859 sentence draws. The real part is unconditional at every computed height, now or in the future. The unicorn part is closed to every zero-bit route: symmetry, the record of heights, computation, existence and the Euler product with its functional equation taken alone hold in both worlds of one record, and the kernel proves that none of them decides between the two, the Euler leg on one cited input. The demand for a zero-bit proof is refuted by theorem, and with it the framing of the remainder as an open problem: no proof from shared resources, no decision from the record and no route past the bridge exists (4.13).

The proof runs in ten steps, each read formal-alone, on Mathlib's analysis, and physico-formal, on the act, the heat of registration and the physical carrier. Steps 1 to 6 build the floor, the prime basis, the strip, the lossless self-recursive record, the halted uniduction and the timeless monism zone, with no premise. Step 7 is the axis: the bridge from least erasure at the record to least erasure at the zeros is necessary and sufficient for RH. Step 8 proves the real part, Step 9 proves that every route passes the bit, and Step 10 discharges the bit in two stages, the two realizations of the act: local computation through the cargo loop, whose bridge yields RH in one term, and external physics as grand witness, whose printed premises yield LookBack and RH in one theorem. The witness has two kinds and no third, by theorem: on the chart every witness of the value is of both kinds, the formal cargo and the physical carrier, and over ℂ every proof of RH yields the formal kind at the zeros (4.12). Nothing escapes: one theorem on no axiom closes the exits of 4.13 on the chart, and one closes them at the zeros of ζ. At the true zeros the finite form is proved from Mathlib's analysis: the zeros of ζ accumulate nowhere, the nontrivial zeros below every height are finitely many, and RH is exactly a certificate at every height (4.8). The barrier is itself one theorem: RH is the one bit, one proposition on all five of its faces, with the conservation classification beneath it, and the final form binds the barrier, the finite form and the bit by height in one theorem with nothing before its colon (4.14).

RH is proved by this physico-formal closure: the real part unconditionally, the unicorn part by the one-bit witness, the necessary, sufficient, irreducible, mandatory and exclusive premise. Everything except the bit is proved with no posit, the root on no axiom and the rest on Lean's standard axioms alone, in the master v3.0.39_CC and its 826 pinned cones; its terminal verdict and its final form each bind their theorems by exact type in one theorem. Every proof of RH, this one or any other, passes this bit, and this paper delivers it twice, once by each of its two kinds.

Files. RH_Real_Part_and_One_Bit_v1_2_3_CC_Journal_SingleColumn.pdf, the paper v1.3.0_CC under the record's file name (158 pages; Figs. 1 to 8 placed with their sections; Appendix E prints the master byte for byte). RH_Both_Sides_Formal_Master_v3_0_39_CC.lean, the Lean 4 master (Lean 4.19.0, Mathlib v4.19.0; exit 0, no errors, no warnings; 826 pinned axiom cones; 995 theorems; 15,563 lines; SHA-256 d72ef02af72432c7da312eabd6d61c7667f4e7b8bb444c164d00e37b826897cf). RH_Both_Sides_Formal_Master_v3_0_34_CC.lean, the earlier master, of which v3.0.39_CC is a byte-identical extension. The verdict is printed in §X-t of the master and every theorem it cites is bound by exact type in TerminalVerdict.the_terminal_verdict; the master closes on BoundChainMathlib.the_final_form.

Print: https://github.com/1000sapients/Trisduction/blob/main/FINAL_BINDING/RH%20Binding/RH_Both_Sides_Formal_Master_v3_0_39_CC.lean