RIEMANN RESEARCH RECORD · v1.54 · 22 SEPTEMBER 2026
Three first lemmas, with the arithmetic questions exposed
The complete nullvector problem now has a valid integral formulation. Paired heat-zero transport has an exact collision identity. Arithmetic approximation has an unconditional block-gain bound. None yet proves the missing estimate needed for RH.
1. Kernel exclusion: the primary route
For a function in the complete logarithmic form domain, being a nullvector is exactly equivalent to its screw potential being affine inside the window. This uses a continuous integral equation and keeps the boundary contribution of zero extension. It does not assume a square-integrable derivative.
A rank-one perturbation of the same logarithmic principal operator supplies a rigorous countermodel to the generic derivative-regularity shortcut. This is a failure of that inference, not a counterexample to the arithmetic operator.
Open input: affine-potential injectivity, API. The actual arithmetic equation must have only the zero solution at every finite window. With the established continuity theorem, this would prevent the first sign crossing even while positive spectral gaps approach zero.
Kernel report · Independent domain review
2. Heat deformation: cancellation survives, a proposed comparison may not
Counting a finite zero cluster with multiplicities removes the inverse-gap singularities of individual zero velocities. The correct heat coefficient in Xi coordinates is 1/4. The exact derivative still contains interaction with zeros outside the selected cluster.
The elementary estimate grows with support. There is also a precise conditional obstruction: if the real-zero deformed function has one zero absent from the original Xi zero set, a uniform comparison using the same tests is impossible. This checkpoint does not prove or certify that unmatched-zero hypothesis at a chosen time.
Open input: a suitable complete signed transport estimate, potentially with a justified change of tests. The general heat approach is not closed, and the original Weil form has not been shown negative.
3. Arithmetic approximation: a surviving gain bound
Adding integer dilations gives the exact residualized block improvement hᵀS⁻¹h, where S is the full Schur complement. An unconditional trace estimate bounds this below by ||h||²/[κ log(M/N)], with κ = log(2π) − γ. The old/new interactions remain in h and S.
Open input: a sufficient lower estimate for the corrected arithmetic correlations, named RBC. No such asymptotic estimate is proved here. Any rigorous approximation rate tending to zero would suffice; the optimal rate is not required. The exact finite formula alone does not establish convergence.
Review and verification
The kernel argument received a separate domain review; the heat and approximation agents independently cross-reviewed each other's proofs. No mathematical MAJOR finding remains. Two mechanical heat typesetting corrections were applied and the final source was rechecked. The integration review distinguishes its scope from the mathematical reviews.
Actual manuscript build: 278 pages; 0 undefined references; 0 duplicate references; 0 overfull boxes. These are analytic results. No new numerical window or tail metric was used.
Integrated propositions and proofs · Integration review · Validation record
Next priority: arithmetic uniqueness for the affine screw-potential equation. The approximation correlation bound is the parallel alternative. G2 and RH remain open; no uniform floor has been proved.