Covariogram Uniqueness with One Regular Body in R³: A Candidate Proof via Area Measures and Fourier–Laplace Zeros

This preprint presents a candidate proof for the following extension of the covariogram problem, identified as open in the cited work of Gabriele Bianchi: a convex body K in R³ with C⁹ boundary and positive Gaussian curvature is determined, among arbitrary convex bodies L, by its covariogram up to translation and central reflection. The proposed argument combines area measures for a possibly nonsmooth competitor with an orientation argument based on Fourier–Laplace zeros. Two central propositions remain subject to independent mathematical scrutiny. This is a candidate proof, not a validated solution. The accompanying Lean project verifies 33 explicitly stated supporting and conditional theorems; it does not formalize or certify the complete uniqueness theorem or its two central propositions. Andrea Comparini developed the candidate through AI-assisted mathematical exploration as a nonprofessional author outside convex geometry research. The technical manuscript concentrates on the mathematics. Supplement S1 describes the exact scope of the Lean checks; Supplement S2 describes computational diagnostics. The source archive includes the manuscript, supplements, Lean and Python sources, verification records, bibliography, research notes, and timestamp evidence for the first GitHub publication. No exported PDF is included. Version: 1.0-candidate. Author: Andrea Comparini. This record was deposited using the author’s Zenodo account linked to GitHub andreacomparini. The DOI identifies this deposited version; it does not establish correctness or independently certify identity or priority. Exact source snapshot: Git commit 7fb853f751ab8238c033c5f3f3d2783f83681431, tag v1.0-candidate. Browse this version at https://github.com/andreacomparini/covariogram-uniqueness-candidate/tree/7fb853f751ab8238c033c5f3f3d2783f83681431 . The manifest lists SHA-256 digests of all 55 source files and identifies the deposit files. Licensing: the original paper and documentation are CC BY 4.0; original Lean, Python and shell code is MIT. These licenses apply to their respective components, not as alternative licenses for the entire deposit. Third-party materials retain their own terms. See LICENSE.md in the source archive for attribution and scope.