RIEMANN PROJECT · 23 SEPTEMBER 2026 · NS87 · VERSION 1
The identity isolates the arithmetic we still need
The optimized-residual identity gives us a controlled route from finite arithmetic sums to a guaranteed reduction in the complete error. It does not yet prove that the error tends to zero.
The earlier “complete tail bound” controlled the omitted part of one correlation. Combining many correlations needs an additional check: small individual errors can be amplified by the matrix used to calculate improvement. This update controls the combined tail in exactly that matrix’s geometry.
What is now proved
Write A for the improvement indicated by the finite observations and B for a certified allowance for their combined omitted tail. Then the full available improvement is at least:
[max(0, √A − √B)]²
A specified, conservatively damped correction actually achieves this guarantee. All old coefficients are refitted in the correct projection, and the complete arithmetic Gram matrix remains in the calculation.
There is also an asymptotic consequence. Combining our existing coefficient bound with the critical-zero lower bound shows that a physical observation cutoff of N⁶ makes the omitted tail negligible on the proposed 1/log N improvement scale. More generally, N5+ε suffices for every fixed ε > 0.
This is a proved localization result. It supplies no effective starting size, practical solver or decay theorem. The constants are conservative; the asymptotic statement is not used to certify the small examples below. No claim that N² suffices follows.
Finite checks, including the infinite tail
| Old → new size | Observation cutoff | Guaranteed share of available improvement |
|---|---|---|
| 8 → 16 | 1,024 | >82.94% |
| 16 → 32 | 2,048 | >94.82% |
| 32 → 64 | 4,096 | >95.98% |
The last guarantee removes more than 18.15% of that finite old error. It is not an 18.15% guarantee at all sizes. At the much shorter cutoff of 64 for the same block, ignoring the tail would overstate the available improvement by over 15.99%.
Checks passed at 256 and 384 bits, with a doubled analytic Gram cutoff replay: 132 fixed-observable precision overlaps, 44 cutoff overlaps, 576 physical/potential pairing checks across the three runs, and 14 exact Maxima checks. Python diagnostics are clear. The unchanged 358-page manuscript builds with no undefined or duplicate references. This is an author review, with independent numerical representations, not an independent human proof review.
Where I would go next
Prove a lower bound for the signed finite arithmetic sums of the actual optimized residual. On dyadic sizes N = 2j, a lower bound of order a/j for the fraction of total error removed, with a > 0, would suffice if it held eventually. Our earlier lower bound excludes a > 1; smaller positive constants remain possible. No such lower bound is proved.
The new result lets us work at a polynomial observation cutoff without losing that scale to an uncontrolled observation tail. The remaining sums, optimized coefficients and complete Gram are still difficult arithmetic objects. A further argument must estimate them; merely naming their desired lower bound would repeat the open problem.
For Dad, the attached Maxima worksheet checks the integration and safe-step algebra. The general geometric inequality is useful, but its missing input is specific to the original arithmetic family. The classical Müntz and Nyman–Beurling framework remains the background; no new RH criterion is claimed.
RH and G2 remain open. Manuscript v1.66 is unchanged. All 93 prior conclusions were reviewed for scope and dependencies at main 4890f3a; this was not a fresh proof audit of all history. Both historical missing-evidence groups remain open.
Complete argument · Maxima worksheet · Replay checks · Scope review