A complete Lean 4 formalization of the simplicial Kirchhoff index theory for Linial--Meshulam random complexes, verifying the results of "Exact mean Laplacian polynomials and concentration for the simplicial Kirchhoff index of Linial--Meshulam complexes" (Bell, 2025). The formalization covers the coefficient-ratio identity relating the pseudoinverse trace to consecutive coefficients of det(L + xI), the exact factorization of the mean Laplacian determinant polynomial, the full-simplex Cartan identity and its spectral consequences, the Sherman--Morrison pseudoinverse update with diminishing-returns convexity, and the concentration bounds via matrix Bernstein and Weyl perturbation.

The project builds against Mathlib (v4.28.0) with zero sorry statements and zero custom axioms. All definitions (pseudoinverse via eigendecomposition, Kirchhoff index as effective-resistance sum, determinant polynomial coefficients) and all theorems (trace representation K_f = n tr(L^+), coefficient-ratio identity, binomial expansion of the mean polynomial, eigenvalue localization, K_f sandwich bounds, and explicit concentration with constants 100/71 and 10d) are machine-verified.