RIEMANN RESEARCH RECORD · NS-51 · 22 SEPTEMBER 2026 · VERSION 1

An exact nullvector equation, with the domain repaired

A hypothetical zero-energy eigenfunction can be characterized by a continuous screw potential that is affine inside the window. This avoids assuming a square-integrable derivative. Injectivity of the arithmetic potential remains open.

What had to change

The proposed reduction through the ordinary derivative Df required f to belong to H₀¹. The complete Weil form has a logarithmic form domain, and that additional regularity was not established. We instead convolve the zero extension of f with the exact screw function, retaining the derivative in the sense of distributions.

For f in the complete form domain Vₐ, the proved equivalence is:

Aₐf = 0 ⇔ (g ∗ Eₐf)(x) = α + βx throughout (−a,a).

The potential is continuously differentiable. For even f, it must be constant; for odd f, it may be βx. Dropping the odd constant derivative would impose an unjustified extra condition. The relevant integral operators are compact, but compactness does not supply injectivity.

A precise failure of the regularity shortcut

A rank-one bounded self-adjoint perturbation of the same restricted logarithmic Laplacian has an exact zero eigenfunction with logarithmic boundary decay, yet that eigenfunction is not in H₀¹. Its construction uses the positive torsion function and a rank-one operator that cancels its constant right-hand side.

This proves that the principal operator, bounded perturbation and zero eigenvalue alone do not justify the proposed bootstrap. It does not disprove H₀¹ regularity for a hypothetical nullvector of the actual arithmetic operator, and it does not produce a negative Weil direction.

The remaining arithmetic input

Affine-potential injectivity (API): for every finite window, the only f in Vₐ whose exact arithmetic screw potential is affine on that window is f = 0.

With the known continuity of the lowest eigenvalue, positive small-window starting point and compact resolvent, API prevents a first zero crossing. This is a conditional use of the established nondegeneracy criterion. No uniform positive gap is needed; growing blocks of near-zero eigenvectors remain compatible with injectivity. API is not proved here.

Prior art and scope

Suzuki’s derivative-completion framework already distinguishes generalized derivatives from ordinary square-integrable ones. The present result gives the zero case as an explicit continuous-potential equation; it does not claim a new RH equivalence. The complete even and odd operators are retained, with no source projection substituted.

Suzuki: screw-function realization and derivative completion
Hernández-Santamaría, López Ríos and Saldaña: sharp logarithmic torsion boundary behavior
Complete propositions and proofs

Claim categories: exact identities, proved implications, one explicit countermodel, and a named open input. No numerical experiment, new window or new tail metric. G2 and RH remain open.