RIEMANN PROJECT · 23 SEPTEMBER 2026 · NS-86

One arithmetic correction fails the finite descent test

The first concrete attempt to turn the optimized coefficients into an improvement across several scales does not give a reliable full step. It increases the squared error at two of the three sizes checked. We retain an exact arithmetic identity and a complete tail estimate, but have no new asymptotic gain bound. The manuscript remains v1.66; RH and G2 remain open.

What was tried

Starting with the actual best coefficients at size N, we formed a nonlinear correction through N². Its formula combines coefficients according to integer factorizations. The resulting divisor defect is exactly the Dirichlet convolution square of the old defect throughout that range. This means summing products over divisors; it does not mean squaring the error point by point.

The calculation measures the complete squared error, including the exterior and the entire remaining tail. It also measures the best step length in this direction, and then allows all old coefficients to be refitted. These modifications generally sacrifice the exact convolution-square identity.

The last column is a fraction of the improvement available from a completely unrestricted fit at the larger size. It is not a fraction of the entire remaining error. At size 16, that unrestricted fit reduces the error by about 55.73%, while our damped, refitted direction reduces it by about 15.22%.

Independent integration explains the failure of the full step at size 16: its new energy beyond the reciprocal coordinate t=256 alone exceeds 21 times the entire old error. Its interior energy also increases. The exact local convolution identity does not control either contribution. This finite result does not exclude eventual success at larger sizes or other arithmetic correction rules.

What remains useful

The optimized residual now has an exact integral-versus-lattice-sum description. Define a potential P by integrating the residual twice in the appropriate logarithmic coordinates. Its value P(1) is exactly the optimal squared error. At every old integer dilation, the complete difference between the integral of P and its lattice sum vanishes.

We proved a complete bound for the part of each observation beyond a finite cutoff. We also found a shared tail term that cancels exactly before numerical evaluation; retaining that cancellation improves the certificate. The signed finite sum still needs an arithmetic estimate from below. Neither identity supplies it.

This operator belongs to the classical Müntz / Nyman–Beurling framework. A hypothetical off-critical zero would remain invisible to all the corresponding complete observations. Rewriting the equations does not exclude that obstruction.

Controls and verification

The altered family from NS-74 was tested using its exact loads and complete shared Gram matrix. It retains its positive error floor. Its coefficient convolution identity is algebraically valid, but the original physical divisor formula is not silently transferred to altered atoms.

Checks pass at 256 and 384 bits and after doubling the Gram and physical cutoffs: 156 scalar comparisons, 728 coefficient comparisons, 144 potential comparisons and 58 cutoff comparisons. Fourteen independent physical/Gram norm checks and 56 potential/Gram pairing checks pass. All 82 exact Maxima checks pass; Python diagnostics are clean. The unchanged manuscript builds to 358 pages with zero reference, LaTeX, PDF or box warnings. Review is local author review.

Before this continuation, all 91 existing conclusion entries were reviewed at main commit 253df4f, together with the current checkpoints and evidence ledger. This was a scope and dependency review, not a new proof audit of all history. The remaining README “only positivity result” wording is corrected: lambda=4 is the largest certified positive window, and lambda=3 is also certified. Both historical missing-original evidence groups remain open.

Where this leaves the direction

I would stop tuning this convolution correction. The next advance needs an estimate that controls the actual signed arithmetic observations together with their full energy cost. The precise normalization sums and tail terms are now explicit, so a proposed argument can be tested without hiding them.

The self-contained Maxima worksheet is suitable for checking the algebra with Dad. It covers the convolution identity, all three signed normalization sums, and the potential’s tail cancellation. Finite symbolic checks support those identities; they do not establish the missing limit.

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