The Gaussian rectangular-block sign-overlap limit by rotation invariance

For real Gaussian k×k matrices M and M′ that coincide on an r×m block and are independent elsewhere, let σ be 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. This note proves k · E σ² → (4/π²) · a / (1 − ab), as r/k → a ∈ (0,1), m/k → b ∈ [0,1), m ≥ 2, by a route independent of the universality proof of the companion paper (doi:10.5281/zenodo.23967834). For b > 0 it uses only rotation invariance, the exact even conditional law of the solution direction, a spherical-harmonic projection of the sign observable, Gaussian integration by parts and the Marchenko–Pastur law: no Edgeworth expansion, no cumulant and no frame-weight series. It extends the square-block proof of SSRN abstract 7083460 (version 2), and the constant arises from a Marchenko–Pastur fixed point. For b = 0 a shorter argument applies Sheppard's formula to the exact even law and reduces the sign correlation to the angle between two projected columns. Files in this record: rectangular_gaussian.pdf — the note. rectangular_gaussian_supplement.zip — the LaTeX source, the Lean 4 file for the elementary algebra, the numerical controls (code, printed outputs and JSON) and the AI review records cited in the note; SHA256SUMS.txt gives the hash of every file. Status: preprint; it has not undergone human peer review. The note was written by an AI system (Claude, Anthropic) under the author's direction and reviewed by AI systems of the same and of another family; the note states the models, their roles and what each review found.