RIEMANN RESEARCH RECORD · NS-55 · 22 SEPTEMBER 2026 · VERSION 1
Local geometry does not yet give arithmetic uniqueness
We proved that two tempting shortcuts fail for the actual arithmetic operator: vanishing on a local patch does not force global vanishing, and its semigroup does not preserve pointwise positivity. The whole-window kernel question remains open.
These are exact analytical results. They do not exhibit a full-window nullvector, a negative Weil direction, or a uniform lower bound. G2 and RH remain open.
The construction that changes the local argument
Inside any interval with half-width a > (log 2)/2, choose three small, separated patches. Put an arbitrary smooth function on the middle patch W. Choose the values on V so that the prime-2 shift exactly cancels the integral contribution on U. The resulting nonzero function satisfies both f = 0 and Aaf = 0 on U.
values chosen
by an inverse
arbitrary smooth
nonzero input
f = 0
Aaf = 0
V is U shifted left by log 2. All other prime shifts miss these patches. A convergent Neumann inverse makes the cancellation exact. The equation is imposed on U, not on the whole interval.
The witnesses belong to the complete operator domain. A symmetric construction works in each parity for a > log 2. Its infinite-dimensional family allows finitely many linear constraints, including the actual source orthogonality, without eliminating every witness.
This disproves a proposed local property; it is not merely a failed numerical bound. It also explains why a theorem for the logarithmic Laplacian plus a local potential cannot be transferred by treating the prime shifts as a harmless perturbation.
What still works, and its exact limitation
| Statement | What it gives |
|---|---|
| Prime-free exterior observation cell | Uniqueness follows if the observation interval lies beyond the support, inside the window, and avoids every prime-shifted support. An exponential-moment argument retains the pole cancellation exactly. |
| A hypothetical first zero eigenvalue | Every nullvector must reach both window endpoints. It therefore supplies no exterior-of-support interval inside that window. Reaching the endpoints is compatible with zero boundary trace. |
| Inward dilation | An exact finite difference retains all prime autocorrelation changes and both poles. The archimedean contribution is positive and at most 5 log(1/r)‖f‖². The required signed arithmetic estimate is still missing. |
| Full semigroup | A positive off-diagonal pairing of disjoint nonnegative tests prevents ordinary pointwise positivity preservation. This says nothing against nonnegative diagonal Weil energy. Positive bounded multiplication cannot repair this cone property. |
The supporting notes also show why the raw bounded prime perturbation is not analytic at a newly entering prime shift. A valid compact Fredholm reduction exists at a fixed window, but a discrete exceptional set does not exclude the physical coupling from that set.
What the geometry represents
The logarithmic principal operator has a known harmonic extension in three dimensions. Its boundary formula contains a renormalized trace. The arithmetic equation leaves a nonlocal prime-and-pole term in that trace; drawing or lifting the operator into a higher-dimensional space does not determine its sign. The needed theorem remains global arithmetic injectivity, not local vanishing alone.
The relevant primary sources are Chen–Hauer–Weth, extension and unique continuation, Harrach–Lin–Weth, the logarithmic Schrödinger equation, and Suzuki, the complete Weil operator. The new scoped statements above are proved directly in the accompanying evidence; no worldwide novelty is asserted.
Validation and remaining input
Three assisting agents developed separate arguments and cross-reviewed them. The full manuscript builds to 285 pages, with 0 undefined references, 0 duplicate references, and 0 overfull boxes. The eight affected pages were visually inspected. No new numerical window or tail metric was computed.
What had to change: local PDE uniqueness was replaced by an exact prime-shift analysis; a differentiated boundary argument was replaced by finite dilation on the complete form domain. The obstruction: local cancellation and signed prime/pole overlap terms remain. The scope: the local shortcuts fail; whole-window injectivity and the physical Weil positivity question are unresolved.
The next useful step must deliver an arithmetic estimate or a construction with control, rather than another equivalent name for injectivity. The separate Nyman–Beurling residual-correlation route remains an available next test; this checkpoint supplies no convergence claim there.
Integrated proof · Independent integration review · Build and validation record · Detailed evidence