RIEMANN PROJECT · 23 SEPTEMBER 2026 · v1.63

The remaining arithmetic is now an explicit weighted sum.

We proved a uniform comparison between the complete residual error and its values at every reciprocal integer. The result keeps the exterior tail and the infinite remaining cells. It gives a concrete arithmetic target, but does not prove that target tends to zero. RH and G2 remain open.

What geometry has delivered

Write S for the exterior-tail energy plus a weighted sum of the squared residual values at 1, 1/2, 1/3, and so on. The complete squared error lies between S/16 and 21S/16, for every finite choice of coefficients. These constants do not grow with the problem size.

The samples also recover the coefficients by an exact divisor inversion. That exposes the remaining constraint: small sample values must satisfy infinitely many arithmetic compatibility equations. Picking arbitrary small values at a finite collection of points does not solve the problem.

As an independent check, direct integration over the physical cells, with the entire remaining tail bounded analytically, reproduces the Gram-matrix norms at sizes 16, 64 and 256. At the largest size, the refined analytic tail uncertainty is less than 0.0042% of the complete squared error.

A uniform guarantee, with its computational assumption

The full arithmetic sampling sum defines a comparator whose condition ratio is at most 21, including after the old space is projected out. With its exact inverse, one ideal correction captures at least 21/121 of any finite block’s available improvement. Twenty such steps capture more than 97.79%, independently of size.

This comparator retains every arithmetic sample. We have not supplied a fast way to apply its inverse. The guarantee belongs to this complete comparator, not to the simpler curvature rule. Capturing a fixed fraction of available improvement also does not prove that the available improvement is large enough to remove the entire error.

Testing a concrete tail repair

The earlier divisor correction changed the exterior tail badly. We adjusted its final coefficient to preserve that tail, explicitly giving up cancellation of the final derivative jump.

The raw balanced correction still increases squared error by about 53 times at size 256. Of fifteen raw steps checked, thirteen increase error; two tapered steps decrease it at size 128. Preserving the tail is useful structure, but it does not provide uniform descent.

The next mathematical target

Prove decay of the complete signed arithmetic sample sum, or a lower bound on available block improvement relative to the entire current error. The geometric norm comparison and uniform block-efficiency theorem do not establish either statement. This is where further work must retain arithmetic information.

The update includes 56 exact Maxima checks, certified calculations at 256 and 384 bits, complete cutoff replays, a 350-page manuscript build, and visual review of the new pages.

Complete new proofs · Validation record · Local self-review

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.