The Riemann Hypothesis Closed to One Named Bit and Proved from Unconditional Least Erasure by One Act, with an Empty Axiom Cone: Primes as the Base of Freedom, the Constructed p-adic Witness, the Ladder from ZFC Blocked by Theorem, and the Ground Route Closed by the Act

The Riemann Hypothesis is not derived from any set-theoretic foundation in this paper. The ladder from the projected zero data to the value is blocked by theorem. The value is closed to one named bit, and that bit is least erasure; every proof of the hypothesis, in any vocabulary, is a proof of least erasure. The bit is supplied by one act, at premise grade, as a field of a type, and the compiler prints the hypothesis from that field with an empty axiom cone. The compile, the printed cones and the judgment under negation are the receipts. The field that carries the act is the only socket: every proof of the hypothesis, by any route, is a term of it, every named sentence fills it exactly when it holds, and its arrival would change no other line. Within that frame the paper proves, with full machine verification in core Lean 4.19.0 (no library, no imports, no axiom declared, 352 theorems in one file), the following. (i) A set-theoretic foundation is placed, not derived from: under Σ₁-completeness and soundness what computation settles lies in what the foundation proves, which lies in what is true; a foundation that cannot refute the hypothesis has proved it, since a false hypothesis has a finite witness, so independence would be a proof; and a sound foundation's proof of the hypothesis, if one comes, establishes least erasure. (ii) No property of the projected zero data decides the hypothesis, above any certified height; every Boolean combination of universal premises and readings of the record, of any depth, is again such a premise and decides nothing; and the hypothesis is the weakest premise that entails the hypothesis. (iii) The completely additive arithmetic functions are determined by their values at the primes and those values are freely assignable, the separating witness being the p-adic valuation with Euclid's lemma proved from first principles: the primes are a free basis. (iv) In a discrete model of the reflection s ↦ 1−s̄, the hypothesis is equivalent to least erasure, and the same bit is exhibited in five coordinates and proved one bit: least erasure, Weil positivity on the prime side, the sign of the de Bruijn–Newman constant, the Li sign stream, and the faithfulness of the Liouville arrow, computed in the kernel. (v) The cut is triaxial and irreducible: the two even axes, symmetry and record, are closed under combination and blind; the third is one bit, fixed uniquely by one supplied sign. (vi) Least erasure is literal: the record of every configuration is itself a least-erasure configuration and the only one with that record; registration erases one bit per off-line pair, so least erasure is zero erased bits and the zero-cost registration of every record; every reading of the record holds on a least-erasure configuration; and whatever stands against least erasure is a computed point off the line, the same under every provability predicate. The closure is executed by one explicit act, the assumption of least erasure at the actual zero set, a free assignment inside the proven freedom, and the route ledger that records what is blocked, what is cited, what is open and what is closed by the act is computed in the kernel. Eight falsifiers are named.