P45: Spectral localization and uniform reactive capacitance of thin planar patches

Henry Zweiman. September 15, 2026.

AI-assisted research preprint. Not peer reviewed or independently verified.

The manuscript establishes a strong operator limit for thin planar patches around fixed embedded curves. Every fixed eigenvalue is controlled by maximum width, while the input spectral measure converges to the full area-weighted width distribution. This yields vanishing fixed spectral weights for thin ellipses and rhombi, and a capacitance formula with relative error tending to zero uniformly over all positive reactivities.

The same paper gives an exact criterion for asymptotic accuracy of the standard one-pole approximation, bounded remainders for the two Grebenkov-Maurette tube conjectures on regular center curves, and a connected smooth example with F1>F0. The source's operator, spectral representation, variational principles and approximation are credited prior work. No separate count is assigned to these related consequences.

Sharp spectral corrections, fixed-index weight limits on positive-length maximum sets, and singular or thickness-dependent center curves remain outside the result. The connected weight-reversal example is not asserted to be convex. See the assessment for current-source comparison and its limits.

To rebuild, install Pandoc and Tectonic on PATH and run python3 build.py. The authoritative source is manuscript.md. Formula parsing and visual PDF checks establish artifact integrity, not proof correctness or historical priority.