RIEMANN PROJECT · SEPTEMBER 23, 2026 · NS91 · RESEARCH UPDATE V2

A bound that preserves the cancellation

The failed estimate in PR45 can be repaired at the tested finite sizes. Combining the terms first, and removing an irrelevant old-basis contribution, gives a positive lower bound with the entire tail included. The uniform arithmetic estimate needed for RH remains unproved.

What changed

The earlier estimate took absolute values of separate divisor terms. That destroyed their cancellation and produced a lower bound below −186 times the current error at size 256.

I combined those terms into an integral involving a bounded sum of fractional parts. Then I used the existing orthogonality equations to subtract one old basis function. This leaves both the gain numerator and its true projected cost unchanged, while making the remainder exactly zero throughout 0 < t < N.

The remaining integral is grouped over the fixed intervals [N, 2N], [2N, 4N], and so on. Signs are retained inside each interval before its absolute value is taken. The grouping is fixed by powers of two; it is not fitted to the signs of the data.

What the checks establish

At size 256, the repaired bound guarantees removal of more than 2.821% of the current squared error using the already known Möbius/logarithmic direction. Before the old-basis correction, the grouped bound guarantees more than 0.792%.

This is a stronger certificate for an existing direction. It is not a new best finite approximation, a new discovery of that direction’s gain, or a claim that the same percentage persists at larger sizes. The bound is also positive at sizes 16 and 64. Some grouped terms are negative at size 16, so positivity of every interval is not assumed.

The infinite tail is included

A bounded primitive built from the second Bernoulli polynomial controls the sum of the absolute values of all omitted dyadic interval contributions. Shared boundary terms and the complete residual remainder are included.

Combining this inequality with the existing coefficient and critical-zero bounds proves that, at T = N⁶, the regrouped absolute tail divided by the current error is O(N^−5/2 log^5/2N). It is therefore smaller than the target 1/log N scale eventually. No numerical starting threshold is claimed.

The attempted uniform estimate

I also tested whether Möbius inversion turns the gain numerator into a positive quadratic form in the residual. The inversion itself is already in NS78. After the old normal equations are imposed, the new-looking expression is exactly a scalar multiple of the original numerator. It supplies no independent sign estimate. This does not rule out proving a bound for its arithmetic terms.

Where to push next

The remaining task is to bound the finite grouped arithmetic integrals uniformly relative to the actual optimized error, while controlling the true projected direction cost. That would turn these finite certificates into a persistent improvement estimate. Neither uniform estimate is proved here.

The whole joint observation route in PR42 remains available even if this selected direction does not admit the needed bounds. RH, G2, and the central arithmetic lower-gain problem remain open.

Verification and publication status

Arb at 256 and 384 bits; doubled physical and inherited Gram cutoffs; 108 scalar precision overlaps, 114 grouped-integral precision overlaps, 38 common integrals checked across cutoffs, 14 complete cutoff overlaps, and 32 exact Maxima sanity checks. Python diagnostics report zero errors and warnings. Original optimized coefficient enclosures and full projected costs are inherited; no fresh Gram solves were performed. Mathematical review was by the author only.

All 98 previous registered conclusions were reviewed for scope and dependencies at main 8e03860. This was not a fresh proof audit of the entire archive. Both historical missing-original evidence groups remain open.

The previous approval-service block has resolved. The evidence is now integrated in the claimed NS91 research branch, with fresh replays. The earlier standalone draft is preserved unchanged. The manuscript stays at v1.66; no new manuscript version or uniform gain theorem is claimed.

Companions: complete mathematical argument · Maxima worksheet · validation record · reproduction notes.