RIEMANN PROJECT · 23 SEPTEMBER 2026 · v1.63

Local review of the complete sampling comparison

Scope: NS-78–80 proofs, complete finite replays, coefficient and tail conventions, manuscript integration and literature scope. This is author self-review, not an independent audit.

Complete geometry and signed arithmetic

The interpolation uses logarithmic coordinates on every reciprocal cell, with its actual varying measure. The leftover inverse-coordinate term is retained as a square-integrable bubble. The exterior tail contributes exactly the square of its coefficient. The infinite sample sum is proved convergent by finite-cell comparisons and monotone passage, not assumed convergent.

The exact inverse recovers each coefficient from finitely many sample differences and a divisor sum. Fixed-index coefficient convergence does not justify exchanging that limit with the increasing number of coefficients. The manuscript keeps the earlier sharp Möbius-truncation obstruction intact.

Independent norm replay

The physical-cell integral retains all six square and cross terms. Its remaining tail uses an explicit periodic Bernoulli remainder with the correct Stirling constant. The infinite sample tail has a separate positive bound. Every reconstructed full physical enclosure contains the published Gram enclosure at the tested sizes.

Combined printed ball intervals can be wider because of decimal serialization. The finite part, complete main tail and analytic uncertainty are separately archived, and the verifier reconstructs the tighter bound from those components.

Tail balance and projected gain

The repair preserves the exterior coefficient only before old-space reoptimization. It cancels every new jump except the final one. The four-direction span equality follows because each original and balanced vector differs by the same endpoint direction; all mixed terms remain in the positive Gram calculation.

The finite results include two raw improvements at size 128 and thirteen increases elsewhere. Reporting all fifteen as failures would be incorrect. Repeated raw tail preservation with a fixed nonzero initial exterior coefficient has that fixed error floor; initialization at zero or old-space reoptimization is not excluded.

Uniform efficiency does not supply convergence

The arithmetic Gram comparison survives the different old-space minimizers by taking the infimum of both complete quadratic forms. The actual residual correlations are retained. The Kantorovich argument and twenty-step bound concern full available gain within one fixed finite block.

The comparator includes infinitely many actual arithmetic samples. Neither a finite-prefix replacement nor a fast inverse is assumed. The earlier no-go theorem for the smooth model is unaffected. A uniform fraction of the full gain does not remove a possible fixed component of the residual.

The related general strong Nyman–Beurling theorem has smooth-kernel hypotheses that our logarithmic target does not satisfy directly. The manuscript relies on the previously justified fixed-order integration argument instead. No new priority claim is made for these classical mechanisms.

Source hashes, precision and cutoff replays, exact Maxima counts, Python diagnostics, rendered pages and the actual build are recorded in validation.json. No unresolved issue was identified in this pass. Cofinal arithmetic control, the original Weil floor, G2 and RH remain open.

Exact identities, proved implications and certified finite computations are distinguished from open arithmetic inputs. Local self-review only; no independent review or publication-readiness claim.