P51: proof, priority and significance assessment

Henry Zweiman. September 16, 2026.

Internal decision: provisionally admit the complete Q60 contribution as one preprint, P51, subject to verified GitHub publication. This is a revisable research judgment by the originating assistant, not independent expert certification, a journal acceptance decision or a prediction of citations. Earlier zero-flux and flux-transition checkpoints were held uncounted; the final package includes the geometric theorem in Section 11.

The proposed contribution

Theorem 1.1 gives an explicit rank-one expansion of the planar exterior magnetic Dirichlet-to-Neumann operator, in the norm of bounded differences on boundary L2, uniform over every nonnegative scalar/magnetic ratio. Theorem 10.3 resolves the transition through integral Aharonov–Bohm flux by an analytic effective operator with a uniform algebraic error. It controls every ordered eigenvalue. The lowest branch has a universal leading crossover and a third inverse-logarithmic signed-flux difference that detects the regularization ratio.

The geometric conclusion is substantial additional content. Theorems 1.3 and 11.2 show that every fixed smooth simply connected nondisk has strictly smaller lowest eigenvalue than the equal-perimeter disk throughout an entire sufficiently small field, mass and flux neighborhood. There is no symmetry or convexity hypothesis and no prescribed relation between the flux and the logarithm of the field. Theorem 11.1 identifies the capacity and harmonic-measure contributions to the second-order transition. Formula (G10) gives the precise positive limiting gap. The neighborhood and lower-bound constant depend on the obstacle.

The paper is one contribution: the radial estimates, boundary matching, spectral consequences, asymmetry and geometric comparison are not separately counted.

Open status and closest antecedents

Helffer–Nicoleau's inspected v2 raises the general-domain weak-field problem; the June 2026 Bundrock–Girouard–Grebenkov–Levitin–Polterovich preprint still describes the magnetic equivalence as expected and suggests a qualitative route. Consequently qualitative convergence alone is not the significance claim here. The proposed result is a quantitative operator expansion and uniform multi-parameter transition, with a geometric theorem beyond those inspected statements.

Christiansen–Datchev's final planar scalar expansion is prior work. Its shifted logarithm and capacity mechanism are credited. Grebenkov–Chaigneau's scalar asymptotics are likewise credited. Helffer–Nicoleau's fixed-flux disk estimate does not yield the transition by direct substitution, since its error loses smallness when flux has inverse-logarithmic size. The Kummer connection identity, compact-cylinder coth formulas, classical elliptic boundary calculus and conformal covariance are not new results of this paper.

Kachmar–Lotoreichik's current v1 gives a specified magnetic field interval under symmetry and geometric assumptions. The full exterior theorem proof was inspected. The present theorem removes those shape assumptions at an obstacle-dependent threshold, and also includes flux and scalar mass. It does not settle their entire field range or give a shape-uniform threshold. Colbois–Provenzano–Savo's final Aharonov–Bohm Weinstock theorem and conformal argument were inspected and credited; the auxiliary flat-flux coefficient here is used to identify the two-parameter Hessian, not promoted as a separate open problem.

Primary arXiv histories for Helffer–Nicoleau, Kachmar–Lotoreichik and Colbois–Provenzano–Savo were rechecked on September 16, 2026. The source audit records other current versions and exact inspected passages. Targeted searches found no equivalent combined theorem in the inspected primary texts. This is bounded priority evidence, not proof that no unpublished or unindexed work exists.

The complete final Helffer–Nicoleau body was not accessible; its current author preprint and final preview were compared. The final Provenzano–Savo annulus body was not compared; its current arXiv cylinder and small-flux proofs were read. These access limits remain explicit. The final Christiansen–Datchev, Cekic–Siffert and Colbois–Provenzano–Savo passages were available. Neither access failure is represented as evidence of novelty.

Why the strengthened package supports admission

The contribution handles a singular interaction of weak confinement, topological flux and geometry. The cancellation at integral flux and the unbounded scalar/magnetic ratio require uniform estimates that are absent from fixed-parameter substitutions. Keeping the analytic obstacle-dependent operator allows spectral and geometric information to survive that transition. The positive gap extends the range of admissible shapes in a current magnetic isoperimetric problem. The explicit normal form is reusable for spectral clusters, perturbation of obstacles and more general flux configurations, although those later problems are not claimed solved.

These features support the internal judgment that the complete theorem package is substantial enough to pursue at the user's Transactions-of-the-AMS-or-comparable target. That judgment is less certain than the displayed algebraic identities. No particular journal's acceptance or future citation count can be established here. The proof uses classical tools, and the importance of this particular combination remains a matter for external mathematical assessment.

Rigorous scope and review limits

The hand proof is complete as written, with the exterior realization, uniform radial estimates, all-mode variational bounds, bounded-region perturbation, matching and second-order coefficients included. The review records checks of the singular gauge and the flat limit, one-sided affine perturbation, cancellation of the enclosing radius and uniform remainder domination. No fatal gap was identified in the originating-assistant audit. Floating-point radial and Fourier checks are supplementary diagnostics, not proof or interval certification.

The work does not cover arbitrary nonconstant fields, higher dimensions, independent flux vectors on multiply connected obstacles, a geometric-class-uniform field threshold, or all field strengths. Expert review of the uniform boundary-operator topology and the singular limiting realization would be especially valuable. No expert has been contacted by this workflow.