Universality of the inverse-matrix sign-overlap limit
Year: 2026
Type: preprint
Abstract: Let M and M′ be k×k random matrices with independent entries that coincide on an r×m block and are independent elsewhere, and let σ denote the conditional sign overlap, given the shared block, of two coordinates of M⁻¹𝟏, that is, the conditional expectation of the product of their signs. We show that k · E σ² → (4/π²) · a / (1 − ab), as r/k → a ∈ (0,1), m/k → b ∈ [0,1), for every fixed entry law that is centered, has unit variance and is subgaussian. No symmetry or density is assumed, so atomic and skewed laws are included; a corollary removes the centering. The Gaussian case with a square block was obtained earlier by a different method. The proof reduces the signs to a normal vector of a random kernel, transports them by a second-order Edgeworth expansion, expresses the even moments of the resulting overlap as sums with nonnegative weights, and identifies each weight by a moment computation on random frames. The contributions of the third and fourth cumulants are removed by a parity argument, by cancellation between coupled laws together with a second-moment bound on mixed tensors, and by a radial cancellation. Files in this record: universal_theorem.pdf — the manuscript (20 pages). universal_supplement_2026-11-04.zip — the supplementary package: the long-form proof with a map to the sections of the paper, the term-by-term ledger of equation (16), the record of how the proof was checked, the code and aggregated data behind every number in the paper, and the Lean 4 files the paper cites. MANIFEST_SHA256.md gives the SHA-256 hash of every file; ./validate_all.sh quick re-checks the package with one command (see REPRODUCE.md). Status: preprint; it has not undergone human peer review. The mathematical arguments were produced and checked by AI systems under the author's direction; the AI Declaration of the manuscript and AI_CONTRIBUTIONS.md in the package give models, roles and what entered the paper.
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225346733