RH PROJECT · NS93 · 23 SEPTEMBER 2026 · VERSION 1

Averaging does not rescue the unsigned cost bound

The new attempt isolates exactly where the scalar cost needs arithmetic cancellation. Its diagonal contribution is bounded uniformly. The remaining signed pair contribution has no proved upper average. Replacing the pairs by their absolute values creates a cost bound that grows at least linearly in N, even before averaging.

What was proved: for every dyadic N ≥ 2048, the fully separated absolute-product majorant is at least N log²(4/3) / 9216. This rejects that particular estimate. It does not prove growth of the actual signed cost or exclude the smaller projected cost.

The precise split

Write w_n = log(2N/n)/n. The signed suffix-square cost from NS92 is

S_N = Σ_m,n μ(m)μ(n)w_mw_n[min(m,n) − N] = D_N + C_N, with N < m,n ≤ 2N.

The diagonal D_N is at most (log 2)²/4. The off-diagonal C_N contains every distinct pair, with its sign. Bounding the sum of those C terms by a constant times the number of scales would supply the cost half of an averaged argument. That estimate remains unproved.

Why absolute values fail, uniformly

The interval (5N/4, 3N/2] contains at least N/32 squarefree integers for dyadic N ≥ 2048. This follows from a union bound over square divisors and Σ_k≥2 1/k² ≤ 3/4. Each such integer has |μ(n)| = 1 and w_n ≥ 2 log(4/3)/(3N). Their pair overlaps are at least N/4. Keeping just those positive contributions proves the linear lower bound above.

Across doubling indices J through 2J − 1, even the largest term makes the mean absolute-product majorant grow at least proportionally to 2^2J−1/J. A growing group of scales cannot make that bound uniformly small.

What the actual finite arithmetic says

These are rounded displays of interval certificates, not asymptotic estimates. At N = 65536, the signed cost is about 0.00223001 while the absolute majorant is about 218.643. This comparison concerns S, not the actual projected direction cost K or a new residual improvement.

The lower numerator is still essential

Let r_j be the positive part of the certified numerator margin divided by the actual error at N = 2^j. If its mean stays above ε > 0 and the mean signed cost stays below C, weighted Cauchy–Schwarz and the existing error lower bound yield persistent improvement across growing groups of doublings.

Both averaged arithmetic hypotheses remain open. Stating their sufficiency is an application of the existing descent framework, not a new RH criterion. This task does not compute a new lower numerator, improve the best finite error, or prove convergence.

Next mathematical target: estimate the signed off-diagonal Möbius correlations and the actual residual’s numerator together. Preserve the full PR42 observation route if the single-direction scalar budget loses too much. The useful new restriction is precise: taking absolute values of all Möbius products cannot prove the proposed bounded-average cost.

Validation: 256/384-bit arithmetic, 112 precision overlaps, 112 doubled-sieve overlaps, seven agreements with NS92, direct quadratic checks at N = 16, 32, 64, and eight exact Maxima checks. The finite sums have no omitted physical tail. Author review only; no independent review. Both historical evidence-recovery gaps and RH remain open. Manuscript v1.66 unchanged.