RIEMANN PROJECT · 23 SEPTEMBER 2026 · NS88 · VERSION 1

An upper bound, with the arithmetic gap exposed

The candidate remains a whole-error bound such as E_N ≤ C/log N. We have now derived an exact upper envelope in terms of one cumulative arithmetic quantity. We have not proved that this quantity grows without bound.

What the optimized equations actually say

Take the old optimized approximation and dilate it by two. Its overlap with the remaining error is a signed number d_N. It is also the difference between the complete potential P_N(2) and the residual’s complete dilation autocorrelation.

The normal equations cancel every part inherited from the preceding half-size approximation. What remains is exactly the overlap between two successive improvement increments. This yields the complete inequality

2|d_N|² ≤ (E_N/2 − E_N)(E_N − E_2N).

When the previous improvement is positive, this already gives an explicit upper certificate:

E_2N ≤ E_N − 2|d_N|²/(E_N/2 − E_N).

The certificate uses previous and current optimized fits plus cross-correlations. It does not need the next full matrix inverse. Applying the identity again at the next size still requires the actual optimized fit.

The cumulative upper bound

On sizes N = 2^j, add the nonnegative quantities |d_2ʲ|/E_2ʲ⁻¹. Call their sum through step J S_J, starting at a fixed step. The complete error satisfies

E_2ᴶ⁺¹ ≤ K exp(−√2 S_J).

Here K is the geometric mean of the two initial errors. If we could prove S_J ≥ a log J − b for any a > 0, this would give E_N ≤ C/(log N)^√2a, sufficient for convergence. Even divergence of S_J without a rate would suffice.

That lower growth estimate is unproved. The envelope follows from Hilbert-space geometry; growth must come from the original arithmetic. The earlier critical-zero lower bound also forces S_J ≤ (log J)/√2 + O(1). Neither the exact envelope nor the finite table proves a vanishing whole-error upper bound.

One tempting shortcut is now ruled out

A uniform bound |d_2ʲ| ≥ aE_2ʲ looked like a possible way to force continued progress. The exact increment identity shows it is impossible. More generally, an eventual lower bound aE_2ʲ/j^p is impossible for 0 ≤ p < 1. It would force decay faster than the known critical-zero lower bound allows.

The scale aE_2ʲ/j remains possible only with a ≤ 1/√2 under these bounds. No positive constant at that scale is established. This excludes one proposed numerator estimate, not all multiplicative constructions or the RH project.

What the complete finite checks show

The sign change and sharp dip provide no basis for an eventual lower bound. These percentages concern available gain, not the whole error. The cumulative envelope through these four steps gives E₁₂₈ < 0.000336728, much looser than the directly optimized value about 0.0000164943. Its value is the explicit arithmetic growth target, not a better finite numerical bound.

All identities passed Arb checks at 256 and 384 bits, with a doubled Gram cutoff and a doubled physical cutoff. Independent complete integrations checked the potential and autocorrelation, including their infinite tails. There are 80 scalar and 120 coefficient precision overlaps, 20 scalar and 64 coefficient Gram-cutoff overlaps, 27 physical/Gram checks, and 13 exact Maxima checks. Python diagnostics are clear. Review is by the author; no independent human or agent proof review is claimed.

Where the work stops today

We have a concrete cumulative quantity whose divergence would prove the desired error decay. We do not yet have an arithmetic argument forcing that divergence. Further progress must establish it, find a different sufficient lower estimate, or show that this particular observation misses too much cancellation. Another finite table would not resolve that question.

RH and G2 remain open; manuscript v1.66 is unchanged. All 94 prior conclusions were reviewed at main 99640fe. Both historical missing-original groups remain open. The critical-zero lower bound is used only asymptotically, with no effective onset claimed.

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