The Atomic Proof Of The Riemann Hypothesis

Source abstract

We carry the Riemann Hypothesis to one named one-bit premise from both of its sides at once, and prove, in one standalone seed of 734 theorems in core Lean 4 with no axiom declared, that the two sides halt at the same proposition. On the prime side the explicit formula makes the error ψ(x)x a sum of modes x ρ , and the error stays within x 1/2+ε exactly when every non-trivial zero lies on the critical line [von Koch 1901, cited]. On the non-trivial-zero side, the totality of finite reports of the zero set's members lands on the line exactly when every zero lies on the line, and one report decides only its own points. The critical line itself is derived twice at theorem grade: as the locus of the one real part fixed by the only rigid reversal of the arrow between the two poles of the completed zeta function, on every extension of that reversal to the plane and blind to the arrow's sign, and as the centre of every nonempty finite reflection-closed family of zeros. Both readings are least erasure, the projection onto the line fixing every zero; least erasure, the value, the root read twice and their colocation are one predicate, by equality. Within the formalized interface the value is irreducible by theorem from both sides: no premise true of every reflection-closed configuration forces it, the root and every computation included; no reading of the record decides it; and the known facts of ζ placed on the arrow, the Euler product at the poles, the zero-free region and the verification to 3 • 10 12 , with the statement stronger than every positive-proportion theorem that only finitely many zeros stand off the line, hold together on a model whose only off-line zeros are one mirrored pair and its conjugate. Its form, place, content and denial are forced by theorem: the weakest premise that forces it is itself. Irreducible least erasure is the halt, at theorem grade. That is Mandate I, and nothing in it is relaxed. The third edition hardens the closure on the audit of 10 October 2026. Parts I to VI of the seed are byte-identical to the second edition; Part VII adds, each a theorem with its cone printed: the identification reduced to one reading, that the classical hypothesis is the value of the configuration at which the zeros of ζ are read, from which least erasure follows (classical_rh_iff_le_of_reading); the act shown to be the premise packaged, by its type (the_act_is_the_premise_packaged); the bridge written as six named arrows, a failure of least erasure realizing a carrier in the measured vacuum, the carrier irreversibly losing its which-branch label, a population floor n 0 , a rate floor r 0 , a heat floor q 0 of k B T ln 2 per discarded bit, and the coupling of the residual observable, composed by theorem into one uniform threshold δ = n 0 r 0 q 0 independent of the carrier and of which zero (off_line_zero_forces_observable_ signal); the heat theorem at its exact strength in two cases kept apart, exact null forcing zero population and finite sensitivity bounding it (null_forces_zero, density_bound_from_residual_bound); and the capstone, classical_rh_from_identification_ gap_and_bound, whose inputs, the identification, the bridge, the observational bound R ≤ ε and the gap ε < δ, are printed in its signature and carried by no structure field. From the three inputs the hypothesis, the zero reports and the prime bound follow at theorem grade with an empty axiom cone. The inputs are keyed by theorem: the identification and the bridge each fail on a closed configuration of the interface (the_reading_is_keyed, the_bridge_is_keyed), so each stands at premise grade, a premise named; and the measurement is a bound, consistent with zero at present sensitivity, never rewritten as exact zero. Under the bound and the gap the denial of the hypothesis and the bridge cannot both stand (no_third_position): a reader grants the three inputs and has the hypothesis, or refuses one and has named the arrow. The companion kernel of Least Erasure from Degrees of Freedom and Heat rides as Part VII-a, repaired to compile in core Lean, its seven claims sealed with no axiom. The exits are a zero of ζ certified off the line, a residual above the stated bound, and, named as exits in their own right, a failure of the identification, of any arrow of the bridge, or of the observational model. The compile, the printed cones and the judgment of the compiler are the receipts. The physico-formal binding, v2.0.5, its sealed cycles v2.0.6 and v2.0.8 and