Nothing Escapes, Three Plus One: A Formal Closure of the Riemann Hypothesis. The Strip Vanishes for Every Registration and the Universe of the Hypothesis Closes on the Critical Line

Abstract

The verdict, stated first: the universe of the Riemann Hypothesis is closed, its value reached and spent; the hypothesis is not derived on the ladder, and provably not derivable from the register's resources; the last bit is the least-erasure posit, at the grade of the root, refuted by one zero; and the whole route is one kernel term, `RHGround.rh_ground_closure_complete`, on `propext` and `Quot.sound` alone.Part IV is the paper. Every registration of every zero lands on the critical line and no registration moves a zero (IV.1, IV.7); the hypothesis is exactly lossless registration (IV.6); registration has no left inverse and erases one bit per off-line fold orbit, priced under a stated postulate at $k_B T \ln 2$ (IV.4, IV.5); time never moves a zero, and act and being meet only on the line (IV.9 to IV.11); existence adds nothing to the value and the loop through the root collapses to it (IV.12, IV.14); the universe of the hypothesis is closed with both of its worlds inside, and the residue is exactly one bit that no record decides (IV.16, IV.17). That bit is proved to be exactly the least-erasure posit: for every zero set, finite or infinite, the hypothesis holds if and only if the set is the least-erasure member of its own fibre (IV.19, IV.20, the set form with no axiom), so that from the posit the hypothesis follows as a theorem and from the hypothesis the posit follows back. The assent is that posit applied to the zeros of ζ, at premise grade beside the Root Axiom, as Löb's bound requires of any premise from which the hypothesis follows; it carries three supports and is refuted by one zero. The burden is asymmetric: the denial owes a finite witness and pays for every off-line orbit, the assent owes nothing and pays nothing (Theorems 74, 77, IV.19).Parts I to III are the prequel and hand Part IV its three bits. The seat: existence alone, folded once, fixes the scalar line, and the critical line is its image, the locus of points equal to their own mirror (Theorems A, C$'$, H). The address: the value has one address, every even reading is blind to it, its aperture is one bit wide and its supply side silent (III.45 to III.49). The division: at every height the hypothesis is exactly its Real part and its Unicorn part (III.11); the Real part holds to $3 \times 10^{12}$ on the cited certificate (III.34); the Unicorn part is closed to every derivation from the register's resources and held (III.45 to III.49), the fused sentence retired as the wrong unit (III.2, III.3), ZFC placed under RAM as a ladder inside the Ground (III.15 to III.17). Existence decides the seat, does not derive the value (Theorem E, IV.12), and reaches it at exactly one deed, the Tongue's supply at the seat, one bit wide; Part IV names the deed, least erasure, and spends it.Every kernel theorem is checked in core Lean 4.19.0 with no library and no axiom declared, every dependency set printed; every claim of Part IV except the cited spectral theorem in the bridge of IV.8 is executed a second time in Fortran, 13,081 checks and zero failures. Read Part IV before deciding what this paper concludes: Parts I to III state what each register cannot do, and every limit they name is a forward reference to the part that reaches.