RH RESEARCH · NS89 · SEPTEMBER 23, 2026 · VERSION 1
Back to the upper bound
The target remains EN ≤ C / log N. We have not proved this decay. This update supplies an explicit normalized approximation and a complete upper-error certificate.
Alex redirected the work away from another lower-bound obstruction test. That proposed test was stopped. The final work concerns construction and upper bounds.
A restriction we can impose without losing the rate
We can require the approximating function to vanish exactly for x > 1 by imposing the single coefficient constraint ∑cn/n = 0. This changes the approximation itself; it does not discard part of its error.
For every N ≥ 256, the best normalized squared error satisfies
EN ≤ EN0 ≤ 1.144579 EN.
This preserves any possible logarithmic decay exponent. It compares two errors whose limiting behavior is unknown; it does not show that either tends to zero.
The exact normalization cost is ηN2/qN, where ηN = ∑cn/n and qN is the squared length of the exterior-tail function’s projection onto the first N atoms. Projection gives EN0 ≤ EN/qN. The qN increase with N, so the certified size-256 comparison holds at every later size.
An explicit complete upper certificate
The repository now contains exact rational coefficients for sizes 16, 64 and 256. Each vector satisfies the exterior constraint exactly. Rounding only proposes the coefficients; their complete original quadratic error is then verified afresh.
| Size | Normalized squared error, approx. | Normalization cost |
|---|---|---|
| 16 | 0.00003195291 | 0.01636% |
| 64 | 0.00001976097 | 0.01248% |
| 256 | 0.00001414367 | 0.0088342% |
The percentages are relative to the unrestricted optimum at the same size. At size 256, independent complete Gram and physical calculations certify squared error below 0.000014144. These normalized errors are slightly above the previously certified unrestricted optima; they are not improved best finite bounds.
The remaining arithmetic estimate
The complete upper certificate consists of the exact signed divisor-cell energy up to a cutoff, followed by the logarithmic main tail and an explicit bound for the entire remainder, including its cross term. The remote region near x = 0 remains included.
To prove the candidate upper rate, we still need an explicit family of coefficients and a uniform bound on that complete quantity by C / log N. The finite table and the normalization comparison do not provide this missing estimate. The present optimized coefficients still require complete Gram solves; no fast algorithm or closed-form family is supplied.
Checks and scope
Arb checks use 256 and 384 bits, a doubled analytic Gram cutoff, and a doubled physical cutoff. Twelve exact Maxima checks cover the normalization algebra and integration formulas. The physical calculation shares the exact rational coefficients with the Gram calculation but uses a different full-norm representation. Review is by the author only.
All 96 prior registered conclusions were reviewed for scope and dependencies at main commit ba02640. This is not a new proof audit of all historical work. Both missing-original evidence groups remain open. The manuscript remains v1.66; RH and G2 remain open.
The classical setting is described in Báez-Duarte’s integer-dilation criterion and his generalized criterion and Mellin identity. The rate comparison here is an elementary projection calculation, not a claim to a new RH criterion.
Reproduction guide · Complete argument · Exact rational coefficients · Maxima worksheet