SEPTEMBER 22, 2026 · VERSION 1 · NS-58

The boundary test reaches its stop condition

The exact boundary pairing can be derived, but its vanishing for every translated radical is equivalent to the original nullvector equation. No independent sign or uniqueness estimate emerged. RH remains open.

What was established

Cut a translated arithmetic radical at the two window endpoints. Its inside part belongs to the complete operator domain, even though the sharp cut may have jumps. Its outside part creates a square-integrable response inside the window. The response includes the singular continuous kernel, both pole terms, and every exterior prime shift, including primes beyond the finite-window cutoff.

The exact identity is boundary pairing = −q(f, inside radical). For a nullvector the right side vanishes. Conversely, vanishing for every translate forces the original nullvector equation. The converse uses a compactly supported distribution and Fourier uniqueness; it does not assume that ordinary norm density is also density in a global form norm.

Both even and odd sectors are included. The odd pole retains its negative sign. The argument does not assume an ordinary square-integrable derivative or introduce an unsupported endpoint flux.

What the altered-weight controls reveal

Changing finitely many prime coefficients by the operator B adds the explicit forcing term Bφ_t. The altered nullvector equation therefore requires the boundary pairing to equal that forcing pairing.

For every nonzero test, the forcing profile is nonzero for some translate. This follows because the finite cosine multiplier of B vanishes only on a set of measure zero, and the translated radicals are dense in ordinary L². This makes the loss of the original radical identity precise.

However, the forcing map is compact. There is no positive lower bound for its norm over the entire unit sphere of either fixed-window parity space. This statement does not exclude a lower bound restricted to a particular altered operator’s finite-dimensional kernel.

Why this does not exclude a first zero

The original arithmetic family has a homogeneous boundary equation and the altered controls have a forced one. That distinction is exact. But the original homogeneous equation is already the condition we are trying to exclude. Its derivation supplies no sign that contradicts a hypothetical nullvector.

A successful continuation would need an independently proved estimate on the complete signed expression, possibly using the already known first-zero support and variational constraints. Renaming its required nonvanishing would add no information.

Disposition

NS-58 is complete at its agreed stop condition. This particular boundary reformulation supplied no additional exclusion estimate. Arithmetic affine-potential injectivity, the uniform finite lower bound, G2 and RH remain open. Geometry remains available as a way to discover an estimate; the identities alone do not establish one.

The analytic proof, finite exact Maxima checks, local integration audit, and actual validation record are archived together. The algebra checks verify signs and forcing bookkeeping, not the functional-analysis arguments. No independent review or new numerical certificate is claimed.