A Formal Proof of the Real Part of the Riemann Hypothesis: The Seat, the Address, and the Division of One Bit, Proved in Core Lean 4 from Existence Alone, Carried into ZFC Along the RA–RAM Bridge, and Axiom-Free Under a Bare Computation Arrow

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.