RIEMANN PROJECT · SEPTEMBER 23, 2026 · VERSION 1
A better finite correction direction
The corrected kernel captures 97.06% of the full optimal block improvement at size 256. The unmodified correlation direction captures 38.41%. Both precision replays and the doubled-cutoff check passed.
RH and G2 remain open. We have a certified finite improvement and a sharper description of the arithmetic estimate still needed. We have not proved that sufficient improvement continues indefinitely.
What the experiment actually measures
Starting from the best approximation using the first N functions, we compare possible directions for the next block of corrections. Every candidate is measured with the complete arithmetic Gram matrix, including the old-space projection and the infinite analytic tail.
| Block size | Unmodified direction | Corrected direction |
|---|---|---|
| 16 | 23.32% | 83.90% |
| 32 | 32.58% | 82.53% |
| 64 | 27.23% | 95.77% |
| 128 | 29.47% | 97.81% |
| 256 | 38.41% | 97.06% |
Percentages are shares of the full optimal improvement for that block. “Corrected” uses the projected repaired kernel to choose a direction; its energy is evaluated with the actual kernel. Values are rounded from certified ball enclosures.
At size 256, this removes 8.08% of the squared residual error, close to the full optimum’s 8.33%. Efficiency is not monotone across sizes. The archive gives the replay results and the exact conditional statement.
Why the correction matters
The elementary part of the Gram formula assigns the wrong leading energy to two very close dilations. Subtracting an explicit term, d/max(u,v), fixes that mismatch. In adjacent-difference coordinates, the correction is exactly diagonal.
The corrected kernel still cannot bound the actual kernel uniformly over every coefficient vector: known zeta zeros and large values prevent that. Our experiment uses it only to choose coefficients, then retains the true arithmetic energy. This is why the finite improvement is compatible with the comparison obstruction.
Other routes tested
- Canonical cancellation: exact signed cells and complete tail bounds retain the open growth criterion. After an explicit centering, cancellation between the two complete Abel vectors cannot improve their power exponent. Useful cancellation must occur inside the arithmetic integral or cell sums.
- Positive renewal: the geometric averaging operator has a controlled inverse, but it transfers the same growth exponent. The signed forcing still needs an RH-equivalent estimate. Its critical weighted total energy diverges; this does not say the canonical norm itself diverges.
- Endpoint approximation: the actual signed remainder exceeds the leading model at sizes 64 and 128. At size 128, both endpoint directions together capture only about 3.3% of the available gain. Treating that remainder as a small correction is unsupported on those blocks.
The next mathematical target
The selected-direction formula separates two quantities: the residual’s model-inverted numerator relative to its error, and the true energy cost of the chosen direction. At size 256 they are approximately 0.76685 and 9.48848, giving the observed 0.080819 relative improvement.
We need independent bounds on those quantities along arbitrarily large blocks, strong enough that the relative improvements have a divergent sum. That would force convergence in one fixed norm and hence RH. Neither the finite table nor the elementary kernel supplies these bounds.
Verification and Maxima
The five tasks add 137 exact Maxima checks, complete Arb calculations at 256 and 384 bits, separate cutoff replays, and independent overlapping computations. The new manuscript builds to 330 pages, with zero unresolved/duplicate references or overflow warnings. All 13 new-section pages were visually checked. Review was local self-review only.
For Dad, the most focused starting point is the 14-check kernel calculation: Fourier density, cusp removal, and the exact diagonal correction. The renewal checks show precisely what positivity does and does not provide.
The complete Gram formula builds on Werner Ehm’s work. The canonical criterion uses classical results recorded in Báez-Duarte’s strengthened criterion. The large-value comparison uses Soundararajan, Theorem 1. No priority claim is made for the general reformulations. Local review and limitations.