The Riemann Hypothesis — Complete Proof

Riemann Hypothesis — Complete Proof (Maximus Shlygin, 2026) presents a proof-by-certificates “corridor” for the Riemann Hypothesis (RH) built around a single collision door in the de Bruijn–Newman heat-flow deformation HtHₜHt. The proof reduces RH to excluding multiple real zeros (collisions) of HtHₜHt on a fixed time slab t∈[0,t∗]t ∈ [0, t*]t∈[0,t∗]. In the corridor language, define the vector detector

so that a collision on the real axis is detected exactly by V(t,x)=0 (equivalently G(t,x)=0). The condition NZ1 (“no collisions on the slab”) is therefore equivalent to V(t,x)≠0 for all (t,x) ∈ [0,t*]×ℝ. Because the slab is non-compact in x, the proof splits the domain into a compact core and a non-compact tail, and reduces NZ1 on the full slab to:

1. a finite cover of the core by rectangles, each excluded either by a canonical Lipschitz propagation rule (C4) or by a direct box gap enclosure (C3), formalized as slab-guard (C5.1) and initialized by an anchor gap (D1) at t=t*; and

2. a uniform tail gap (E1) valid for all t ∈ [0,t*] and all |x| ≥ R.

Once NZ1 is established, a standard open+closed propagation argument transports the “all-real-zeros” property from the anchor regime down to t=0. The final bridge identifies the t=0 statement in the canonical normalization with the Clay Mathematics Institute formulation of RH.

External certificates and acceptance condition

All quantitative work is confined to three external certificates plus reproducible audit logs:

• D1: uniform anchor gap G(t*,x) ≥ η* on x ∈ [−R,R]

• E1: uniform tail gap G(t,x) ≥ η∞ on t ∈ [0,t*], |x| ≥ R

• L1: certified Lipschitz constants Lᵥ(B) for V on each core rectangle B (or certified derivative bounds from which Lᵥ(B) is computed)

• C5.1 audit logs: the finite rectangle cover and per-rectangle closure evidence (C3 or C4), with deterministic partition parameters and hashes

Acceptance criterion: the proof is accepted conditional on independent reproduction of D1/E1/L1 and the slab-guard audit logs as outward-rounded enclosures under the stated deterministic environment.

Document structure

The proof is presented as a sequence of short node-documents Node 00–20, progressing from the Clay RH target, through canonical normalization and an Integrability Pack, into the certified C-pipeline (C1–C5.1), then propagation, bridge to RH, and final TCB/reproducibility criteria.