RIEMANN PROJECT · 23 SEPTEMBER 2026 · v1.61

Geometry gives a useful rule. Arithmetic remains the proof problem.

The simpler curvature rule captures 97.63% of the available improvement at size 256. That means an 8.13% reduction in squared error, compared with the full optimum’s 8.33%. RH and G2 remain open.

What the geometric work delivered

The corrected model kernel has an exact positive energy written in terms of derivatives. Its selected numerator is comparable to the bending energy of a cubic spline, with constants independent of size. This produces a local coefficient rule requiring a linear number of operations once the actual residual correlations are known. Finding those correlations remains a separate arithmetic calculation.

The three arithmetic templates use the Möbius function with constant, linear-logarithmic and quadratic-logarithmic weights. Even their best combination falls short of the curvature rule on every checked block. This rules out that particular explanation of these finite results; it does not exclude other formulas or their eventual behavior.

A control that prevents overinterpretation

We constructed an altered family with exactly the same complete Gram matrices but a provable positive error floor. At size 256 it also captures 97.63% of the available block improvement, although at least 69.34% of its current squared error can never be removed. A second, smaller deformation gives still closer finite statistics.

The altered family inserts a known zero through a unitary factor. This is not a zero of the original zeta function, and it establishes no error floor for our original problem. It proves that finite efficiency and shared geometry alone cannot settle convergence.

Where the next attempt must go

Use the actual signed arithmetic residual to prove a lower bound for its curvature or model numerator, together with an upper bound for the energy of the chosen correction. The resulting improvements must accumulate without a finite total as the block sizes grow. No such estimate has been proved. Larger finite tables cannot substitute for it.

A concrete next test is to form correction coefficients from the divisor sums of the existing optimized coefficients. Unlike the three fixed Möbius templates, this uses the arithmetic defect of the current residual itself.

What was checked

All finite calculations were repeated with 256- and 384-bit ball arithmetic and a doubled tail cutoff at the largest size. Full tails, old-space projections and cross terms were retained. Maxima passed 67 exact algebra checks. The manuscript was built and its new pages inspected; the archived validation file records the actual counts.

Complete new proofs · Validation record · Local self-review

Exact identities, proved implications and certified finite computations are distinguished from the still-open arithmetic input. Local self-review only; no independent review or publication-readiness claim.