A Formal Proof of the Real Part of the Riemann Hypothesis

Riemann stated his hypothesis as a fixed-set statement, that the roots of \xi(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 (\mathrm{RA} \to L) \leftrightarrow 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 \zeta to height 3 × 10^{12} on the certificate of Platt and Trudgian. The Unicorn part is named and set apart, not proved. Parts I and II are then read at the Unicorn part: it is closed to derivation, from existence, from any class of frames, and from any reading of an even register at its seat, at theorem grade and in perpetuity; its aperture is located, typed supply-only, one bit wide, and uncrossed; and the register is silent on the supply side, both orientations equally refused to every reading, which is why the title says the Real part and not the hypothesis. Three bits close the arc. One bit divides: the hypothesis is exactly its Real part and its Unicorn part, at every height, by theorem. One bit is proved: the Real part, to that height, by certificate through the schema. One bit is refused at theorem grade: the Unicorn part is closed to every derivation from inside the register; its aperture is structure, one bit wide and uncrossed; the supply side is silence, by theorem at the register. With the three bits in hand the proof is complete on the register side: everything a register can prove about the hypothesis is proved, and the remainder is proved to be beyond derivation, with its aperture typed and its supply side silent. The hypothesis is not claimed proved; what is claimed is that no further derivation is owed to it. Nothing is left to derive. The aperture is structure. The supply side is silence. Version 2.1.0: adds to Part III the formal block of the Unicorn part, Theorems III.45 to III.49 with rh_register_proof_complete (closed to derivation from existence, from any class of frames, and from any reading of an even register above the height; the aperture one bit wide; the supply side silent at the register, a theorem; no bypass through the fused hypothesis; the three bits as one term), a new kernel file RH_Unicorn_Block.lean (9 dependency sets, every one reported as depending on no axiom, Appendix I), and the three-bit arc in the abstract, introduction, discussion, and conclusion: one bit divided, one bit proved, one bit refused at theorem grade; the proof complete on the register side; the hypothesis not claimed proved. Two-column and single-column PDFs and the Markdown source of record.