Cell averages of hitting times in harmonic graph condensation

Abstract

Harmonic extension with its induced mass gives the classical Guyan approximation of graph Laplacian eigenvalues, and its leading error coefficient is a killed-Green quadratic form. We quantify how first hitting times govern this coefficient. For graphs assembled from cells with one retained vertex each, the Green form is compared with a retained-trace-weighted cell mean of the hitting potential to the retained set. The error is explicitly O(sqrt(mu)), where mu is the harmonic Rayleigh quotient, with constants determined by Dirichlet pinning, conductance bounds, crossing multiplicities, path lengths and congestion, uniformly in the number and sizes of the cells. A two-vertex-cell family shows that this order is sharp for general traces. Together with the condensed spectrum and one Poisson solve, the comparison gives computable same-index eigenvalue enclosures; a principal variant replaces modal Green solves by an inverse gap and a reverse Schwarz bound. On connected weighted finite-range periodic graphs with one retained type, Schur elimination and Bloch analyticity identify the coefficient of all ordered low eigenvalues on finite tori with the uniform mean hitting time, up to O(lambda). A six-vertex cell has mean one and an exact anisotropic correction. For its self-similar assembly, an interface tail proves a mean limit, which exact integer residual checks enclose in [1.50318,1.50837]; identifying the spectral coefficient with this limit still requires inverse-gap and mode-weighting estimates. Status and files: Version 1.0 of the focused manuscript Cell averages of hitting times in harmonic graph condensation, 20 pages with 41 references, prepared for submission to Linear Algebra and its Applications. This record includes the public manuscript PDF, its LaTeX source, and the complete verification archive with scripts, numerical records and exact integer mean witnesses. Detailed postal contact information is omitted from the public files. The original broader FDCL Part II working paper is at https://doi.org/10.5281/zenodo.23227296; proof and certification methodology is documented in FDCL Part IX v1.1 at https://doi.org/10.5281/zenodo.23340664. AI use: OpenAI ChatGPT and Codex and Anthropic Claude assisted mathematical arguments, code, numerical analysis, literature comparison, and drafting and revision. Bin Seol directed the experiments and stress tests, co-developed verification methods, managed the data and evolving definitions, and selected, reviewed and revised the final claims. Bin Seol is the sole human author and accepts responsibility for the content. Finite floating-point spectral observations are distinguished from exact rational and stored integer residual checks; they are not formal machine verification.