RIEMANN RESEARCH RECORD · INDEPENDENT REVIEW

The complete nullvector equation without an H¹ assumption

NS-51 · 22 September 2026 UTC · review v1

Findings: 0 MAJOR, 0 MINOR, 2 NOTE. The proof supplies a valid weak integral reformulation on the complete physical form domain. It does not establish arithmetic injectivity. G2 and RH remain open.

Reviewed artifact: evidence/ns51_kernel/proof.tex, SHA-256 ee27a3cd7a43df40605eccb468588854bafe05bb2a91b6e47c7af1e3f3c20cba. Review performed independently by the NS-51 domain reviewer, with access to the manuscript’s endpoint theorem and primary sources. This review is analytical; it is not a numerical certificate or a manuscript build.

What was checked

The form-domain converse is sound. For f in the closed physical form domain, the distributional identity q(f,φ)=−∫F_fφ̄″ holds against the smooth form core. An affine potential annihilates these tests. Continuity in a shifted form norm extends the equality to the entire form domain, and the representation theorem gives f in the complete operator domain with Af=0. The argument uses the sesquilinear equation against every test, not merely q[f]=0.

The convolution and compactness claims are sound. The screw kernel is continuous; its derivative has a logarithmic singularity at zero and finitely many finite jumps on compact intervals. Hence its derivative is locally square integrable. Local translation continuity and Cauchy–Schwarz give F_f∈C¹ for compactly supported L² data. Both finite-interval kernels are Hilbert–Schmidt; the derivative’s squared kernel norm is ∫_−2a^2a(2a−|t|)|g′(t)|²dt.

All derivative coefficients and parity constants check. Differentiating the pole term gives −4 sinh(t/2). Differentiating the Lerch term gives −½e^−t/2Φ(e^−2t,1,¼). The strict prime-power endpoint changes no almost-everywhere formula. The even potential must be constant; the odd potential may be bx. Thus the odd first derivative may be a nonzero constant and must be removed only by the stated mean-zero projection.

The countermodel is valid in its stated scope. The cited torsion theorem applies in one dimension when r<e^−γ: its general condition reduces to 2r<2e^−γ. For its positive torsion solution τ, put M=∫τ and Kf=−(∫f)1/(2M). Then K is bounded self-adjoint rank one, with every L^p norm bounded by |I|/(2M), and (½L_Δ^Dir+K)τ=0. The lower bound τ≳1/√|log d| contradicts the H¹₀ estimate |τ|≤√d‖τ′‖₂. This is not a counterexample to the actual arithmetic kernel.

Notes to preserve through integration

NOTE 1 — retain the complete form-domain intersection. The proof establishes ker A=V∩ker C=V∩ker T. It does not show that every L² solution of the compact integral equation belongs to V. Compactness supplies no injectivity by itself. The reviewed text already respects both limits.

NOTE 2 — the derivative completion is prior art. Suzuki’s sections 8.3–8.5 already distinguish the derivative-side form completion from ordinary L². The manuscript correctly attributes this and calls its contribution a derived continuous integral formulation. Ordinary L² injectivity of G cannot simply be substituted for injectivity on the relevant completion. The reviewed text makes no such substitution.

Result and limitation

The H¹ shortcut has been replaced by a weak equation that is valid on the existing complete domain. A bounded perturbation of the logarithmic principal operator can have a zero eigenfunction outside H¹₀, so general endpoint theory cannot provide that shortcut. What remains is uniqueness for the specific signed arithmetic convolution potential. The conditional sign-continuation proposition is valid and needs no uniform positive gap.

Primary sources: Suzuki, Weil’s quadratic form via the screw function, Theorems 1.1, 1.3–1.4 and sections 8.3–8.5; Hernández-Santamaría, López Ríos and Saldaña, Theorems 1.1–1.2, published in Discrete and Continuous Dynamical Systems 45 (2025), 1–36, DOI 10.3934/dcds.2024084.