RIEMANN PROJECT · 23 SEPTEMBER 2026 · NS-84 / NS-85

Review first, then test the arithmetic

The full conclusion review found summary errors worth correcting before more research. All 91 entries in the project’s conclusion register were read for their scope, assumptions, evidence status and implications. The repository now requires this review at the start of every turn. This pass did not independently reprove all historical theorems or rerun their large numerical calculations.

What the review corrected

The comparison page still asserted that the matrix was a standard single-symbol Toeplitz–Hankel pencil and that the relevant Fourier mass had no high-frequency tail. The prior audit had already rejected those claims. The page now distinguishes the actual structured split from the unproved applicability of those asymptotic theorems, and it retains nonzero Fourier tails.

The map also drew three arrows from zero-distribution or quantization-growth assumptions, combined with special packets, toward the signed-floor target. Their known implication only obstructs a specified class of low-complexity certificates; it does not establish positivity. Those arrows are removed and the missing signed estimate is named separately.

Smaller corrections identify lambda=4 as the largest certified positive window in this record, separate the no-positive-gap result from the bounded-floor criterion, and state the cutoff inequality as a necessary lower barrier rather than an optimal asymptotic.

These are interpretation and summary corrections. No new failure of the fixed-window certificates or the v1.66 critical-zero argument was identified. The manuscript remains v1.66.

Evidence status

All registered proposition labels and declared evidence paths resolve. Of 34 version manifests checked against the current tree, 30 pass. The four legacy manifests, v1.34–v1.37, refer to moved or subsequently edited publication files. All 18 affected publication artifacts match their original expected hashes at the corresponding immutable version tags. The raw current-tree failures are retained in the review record.

This does not recover the two known missing groups of original scientific evidence. Their status remains open, and four map entries still explicitly mark missing originals. Publication text and later replacement certificates are not substitutes for the missing original computations.

The research continuation

I tested the full classical logarithmic Möbius taper, with every coefficient prescribed. This differs from the earlier test of new-block directions. The calculation includes the exterior, every reciprocal cell through the cutoff, and a bound for the entire remaining physical tail.

The table gives complete squared errors, rounded for readability. “One coefficient fitted” minimizes the full norm by changing only the first coefficient; “Exterior balanced” instead chooses that coefficient to cancel the exterior exactly.

For comparison, the earlier unrestricted N=256 optimum is about 0.00001414. Even fitting the taper’s first coefficient leaves more than 3,200 times that squared error. This family decreases on the finite list but is a weak replacement for the optimized coefficients at these sizes. The comparison neither proves a rate nor excludes eventual asymptotic convergence.

What remains to estimate

The taper has an exact interior description. Set L=log N, S_N=Σ_n≤N μ(n) log(N/n)/n, and Ψ₁(t)=Σ_m≤t Λ(m) log(t/m), where Λ is the prime-power weight. For 1≤t≤N, its residual obeys

L R_N(1/t) = S_Nt − Ψ₁(t).

Thus we need control of the integrated squared discrepancy over a growing interval, as well as the remaining tail. Pointwise prime-counting asymptotics alone do not supply that estimate. This identifies the arithmetic issue but does not solve it.

The logarithmic taper is established prior art. Bettin–Conrey–Farmer’s theorem gives an optimal unsmoothed asymptotic under RH and an additional inverse-zeta-derivative moment hypothesis. We have used neither hypothesis as an unconditional input.

Validation and next direction

The new calculation passes 64 comparisons between 256-bit and 384-bit enclosures, six comparisons after doubling both cutoffs, and six independent full-Gram checks at N=16. All 135 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 only.

The next useful work is an arithmetic estimate for the adaptive optimized residual, or a coefficient construction that controls this complete growing-interval discrepancy. Another finite decrease or a rearrangement of the same unknown sum would not establish that estimate. The original Weil floor, cofinal convergence, G2 and RH remain open.

All 91 conclusion entries and findings · Exact arithmetic argument · Finite replay record

Reviewed main commit 7db0b91. Earlier reports and evidence are preserved. This review creates no new manuscript version or RH claim.