Nothing Escapes: The Fourth as Cosmic Closure: The Ghost and the Unicorn Close in One Cut

Two things lie out of reach in the closure of the Riemann Hypothesis, one on each side of the boundary between proof and measurement. A physical object read as a formal proof is a ghost: it reaches nothing on the far side of the cut. A zero off the critical line read as a measured energy is a unicorn: no measurement ever shows it. This paper proves that one cut closes both. In a chart of the critical strip the line is the fixed set of the fold s ↦ 1 − s̄, and registration, the measurement of a zero as an energy, keeps the height and lands on the line. Off the line every point has a partner, the record forgets which side it stood on, and no registration ever shows it; on the line nothing is hidden (nothing_escapes_one_cut). The side the record forgets is the ghost's bit, the point measurement never shows is the unicorn, and the two are one point's two faces (ghost_and_unicorn_one_bit). The hypothesis is exactly lossless registration (lossless_iff_on_line), both worlds sit inside one chart (both_worlds_inside), and the forgotten side costs one Landauer bit, $k_B T \ln 2$ at the temperature of registration. The cut is the same on every row: on any locus with a registration onto it, every instance is on the locus or off it, where no registration shows it and no carrier landing on the locus reaches it (one_cut_generic). Sixteen theorems in core Lean 4 with no axiom declared, eleven printing none; eight checks in Fortran, zero failures. What remains after the cut is one bit, the side on which the actual zeros stand, spent by the least-erasure assent of the closure at the root's grade. A witness ledger records, row by row, where the cut has been witnessed, crossed and confirmed, and what each row still owes. A counterexample would arrive as a computed witness, never as a measured energy.

Version 1.1.0, single-column Math Journal edition. The kernel One_Cut.lean (sixteen theorems, core Lean 4, no axiom declared) and the Fortran twin One_Cut_Twin.f90 (eight checks, zero failures) are printed in the appendices; source in git, 1000sapients/Trisduction, protocols/RH One Bit/Nothing Escapes The Fourth As Cosmic Closure/.