Discrete approximants, spectrum, and a variational bridge toward the canonical Laplacian on the Menger sponge

Abstract

This paper develops a reproducible discrete approximation program for the normalized local-symmetric Laplacian on the standard Menger sponge established in Paper A 1.1. It constructs both nodal and cell-based graph approximants, fixes their energy scales through geometrically defined effective resistances, and derives the corresponding Neumann and exterior Dirichlet operators. At finite graph levels, the work computes effective resistances, low-lying spectra, exact and stochastic heat traces, finite estimators of walk and spectral dimensions, and tests for log-periodic behavior. The numerical results are audited across refinement levels, boundary realizations, solvers, stochastic estimators, and two distinct discretization schemes. No robust nontrivial log-periodic signal or universal spectral scaling factor is asserted from the available finite data. A variational bridge between the nodal and cellular representations is established through explicit transfer operators and bilateral energy comparisons. Harmonic elimination produces an intrinsic Dirichlet-to-Neumann boundary form. The paper then develops an auxiliary geometric Besov form, proves its projective Mosco convergence, constructs an exact orthogonal decomposition by birth scales, and identifies a Thompson–Schur tower of effective forms. By enlarging the state to include both the effective forms and their minimum-energy recovery operators, intrinsic Mosco convergence is reduced to three explicit and falsifiable stability conditions, denoted Z1–Z3. The manuscript carefully distinguishes finite theorems, reproducible numerical results, conditional propositions, conjectures, and open continuum problems. In particular, it does not yet claim unconditional convergence of the discrete spectra, multiplicities, or heat traces to the continuum operator, nor does it derive a physical mass spectrum. Instead, it provides the discrete operators, computational evidence, correction record, and precise mathematical conditions required for the next stage of the TCA-F5D research program. This publication is synchronized with Paper A 1.1:https://doi.org/10.5281/zenodo.21918417