15 August 2026 · this programme's own result · v2.1, revised 4 September 2026 — certified twice

More than 67.311% of the zeros of zeta are simple and lie on the critical line

The third and fourth links in a chain five days old. Anthropic proved 67.2501% unconditionally; within hours, kaizero_ainta's AI-generated draft refined it to 67.3009% with a certified seven-point inequality; this page carried the next refinement to 67.3054% — a nine-zero window with a retuned pressure, certified in 27,955,544 nodes — and now carries a second, independent refinement on top of it: 67.3118%, reached by tilting the kernel inside the class the analytic estimates actually permit, adding a deliberately asymmetric one-step correction to the window, and sharpening the defect credit to its exact square-root form. Both certificates are published beside this page.

Ondřej Dvořák · Riemann zeta geometric decomposition programme. Every claim carries its evidence tier; both computer-assisted claims are certified and their certificates are published beside this page.

Part I — the result

The theorem this page is built to bank

Let N(T,2T) count zeta zeros with ordinates in (T,2T], with multiplicity, and let N₀ˢ(T,2T) count the simple zeros on the critical line among them. With H₀ = 3/2 − (1/√2)cot(1/√2) = 0.672500703679…:

liminf N₀ˢ(T,2T)/N(T,2T) ≥ (660,000·H₀ − 1,315)/657,504 = 0.6730536459526…  — and, with the tilted kernel, ≥ 0.6731181924… Verified

The steps are +4.51×10⁻⁵ and then +6.45×10⁻⁵ over the previous link, and +6.17×10⁻⁴ over Theorem D in total. Two calibrations, and the sharper one is new. The route's own roof: the defect this argument harvests can never exceed the pair energy of the simple zeros, and a configuration of density H₀ can always arrange itself into low-energy "detuned fences" — so the whole Ψ-defect route is capped at 0.67351275, a number derived by two independent consultants of this programme and verified by a third route in its parallel session. The two theorems realise 54.6% and now 61.0% of everything that route can ever give.

The wider class ceiling, stated exactly, because v2.0 stated it loosely. Remark 1.1 of Anthropic's paper caps every certificate that reads band-width-one data configuration by configuration at 0.6818287. That number does have a formal derivation, and v2.1 names it: the theorems lawN256_rows, ceiling_law256 and ceiling_law256_decimal in Zeta23/PairCeiling/ of their Lean development. Two things v2.0 did not say. First, it is a ceiling per regularity budget — the theorem reads 0.6818287 + 2.55×10⁻⁶(|r′(1)| + ∫|r″|), and the flat 0.68185 is what that gives for a test function of budget about 8.4. Second, every one of those theorems carries the hypothesis EnclOK, which the development does not discharge: it is checked by interval arithmetic outside Lean against a certificate file that is not published. So the ceiling is kernel-checked conditional on EnclOK — not reproduced by us and not refuted by us, and we use it here only as a sanity gate. Detail, and the parts of it we did recompute, on our feedback page. Reaching that class at all needs certificates that couple in the total pair energy of all zeros, which no argument in this chain, ours included, has yet touched.

Credit, stated once and exactly

Theorem D and its machinery are Anthropic's — the compressed Weil form, the optimised window, the rank–trace argument, the analytic estimates; all imported here as proved there, and independently audited, reproduced, and kernel-checked by this programme's first feedback. The stability refinement is kaizero_ainta's — the defect term Δ(M) = tr Ψ(M) that the two-trace argument discards, the seven-point certified inequality, and the block-averaging scheme; audited, measured, and reproduced to the hash by our second feedback. What is ours: the window design — nine zeros, pressure 1/4000, target 39/10,000, block 264 — selected from a measured design curve; the exact-rational deduction at those constants; the independent verifier; and both certificates. And, in the second refinement: the kernel tilt with the admissibility argument that licenses it, the reversal-asymmetric sub-action, the sharp square-root form of the defect credit with its proof and its sharpness example, the 99-point design sweep that chose the parameters, and the ergodic cross-check run before certification. The mechanisms behind the second refinement were identified in a four-lens consultation commissioned by this programme — the kernel angle as a first-order lever, the parity theorem that kills symmetric corrections and the asymmetric repair that survives it, and the sharp credit lemma — and are credited as such.

Part II — the construction

One paragraph of mechanism

For the optimised test family, the vectors attached to simple zeros overlap by the Montgomery–Taylor kernel k evaluated at the gap between their ordinates — and k cannot vanish at u, v and u+v simultaneously, so consecutive zeros always leave a quantitative trace in the Gram matrix. The stability refinement converts that trace into extra count. The price of reading the trace through a local window of p consecutive zeros is a linear "pressure" term on the gaps; the wider the window, the more of the trace is read but the more pressure is charged. The previous link read seven zeros at pressure 1/3000. The design curve — minimum of the window functional against the bound it supports, measured across window sizes and pressure coefficients — peaks at nine zeros and pressure 1/4000, and that is the whole of what this page changes. Measured

The certified inequality it needs

For nonnegative gaps g₁,…,g₈ define, with w = k² and coefficients 2/(9−s) on a pair spanning s gaps,

F₈(g₁,…,g₈) = (1/4000)Σgᵢ + Σ_s=1..8 (2/(9−s)) Σᵢ w(gᵢ+⋯+gᵢ₊ₛ₋₁)   — all 36 pair separations of nine ordered points.

Proposition F₈: F₈ ≥ 39/10,000 for all nonnegative gaps. Verified — certified 12 August 2026, certificate published. Measured first: the true minimum is 3.92793×10⁻³, located by a systematic multi-start sweep, so the target keeps a 0.7% safety margin — the same relative margin at which the seven-point certificate proved out. Certification is by our own interval verifier: Arb enclosures at 128 bits rebuilt from formulas on every run, outward-rounded binary64 combination, one-body and pressure reduction, range-minimum pair bounds, a convex tangent prune re-proved in ball arithmetic before use, loud failure on any unresolved cell.

From the window to the theorem, in exact rationals

Everything after the Proposition is bookkeeping, and all of it is checked in exact arithmetic. Summing F₈ over the m−8 nine-point windows of m ordered points: a pair spanning s gaps appears in at most 9−s windows with coefficient 2/(9−s), total at most 2; each gap in at most eight windows, total pressure 8/4000 = 1/500 per unit span — the same 1/500 the previous link paid, which is what lets the wider window win:

E_m + span/500 ≥ (39/10,000)(m−8)

Take m = 264, the largest block with A₀ = (39/10,000)·256 = 624/625 < 1 — strictly below one, exactly what the defect lemma's min{1,·} needs. Convex pinching over the 264 shifted block partitions, the span-escape for sparse blocks, and the offset averaging — all verbatim from the audited seven-point chain — give

Δ(M°) ≥ (624/165,000)·N₀ˢ − (263/132,000)·N − o(N)

and substituting into the stability master inequality N₀ˢ ≥ H₀N + Δ(M°) − o(N) and clearing denominators:

liminf N₀ˢ/N ≥ (660,000·H₀ − 1,315)/657,504 = 0.6730536459526…

Every rational above recomputes exactly; the deduction differs from the audited seven-point chain only in the constants (9, 1/4000, 39/10,000, 264, 624/625) and was checked step by step at those constants. Verified — deduction and Proposition both; the same bookkeeping at the second refinement's constants (147/100, 1/3300, 4733/1,000,000, 234) carries its exact square-root credit instead of the clipped one.

Part III — what is closed, and what is left

Both propositions are closed

Each rests on one 8-dimensional branch-and-bound over the surviving gap cells, the same procedure that proved the seven-point inequality in 707,901 nodes: the nine-point inequality in 27,955,544 nodes across two external machine stops it was checkpointed through, and the tilted-kernel inequality in 78,344,600 nodes over 16.3 hours, searching all 256 root boxes because its added term is deliberately not reversal-symmetric. Both certificates are published beside this page with their table hashes and full prune counters, and both were produced by an instrument that first reproduced the previous link's committed certificate bit for bit.

What is left, stated as a boundary rather than a plan

Two of the three levers behind the second refinement are not spent: the sub-action's coefficients sat against their own bound during the design solve, so that lever still has roughly a part in 10⁴ to give, and the pressure-window family has an accessible ceiling of about 4.7×10⁻⁴ above Theorem D that this chain has now taken 61% of. What no amount of that work reaches is the 0.6818287 class ceiling: crossing from 0.6735 toward it needs a certificate that couples the simple-zero count to the total pair energy of all zeros, which nothing in this chain, ours included, does. That is the honest boundary of the method, and it is where the interesting work now is. Measured