Functional and Spectral Foundations of the Scalar Laplacian on the Menger Sponge

Paper A 1.1 establishes a rigorous functional and spectral framework for the scalar Laplacian on the standard Menger sponge and clarifies its possible role as an internal sector of a free scalar field within the TCA-F5D research program. The construction begins with the metric measure space ((M,ρ,μ)) and the locally symmetric canonical class of non-zero, local, regular, conservative Dirichlet forms. Within this class, the energy is unique up to a positive multiplicative constant. Paper A fixes that remaining freedom through an explicit unit-resistance convention between two opposite exterior faces. The normalized form determines two nonnegative self-adjoint operators: a conservative Neumann realization and an exterior Dirichlet realization defined using the full six-face exterior boundary. The paper develops their compact resolvents, discrete spectra, positive spectral measures, heat semigroups, heat traces, and spectral zeta functions. It also incorporates the internal operator into a free scalar model on (R1,3× M), while clearly separating the mathematical construction from any future phenomenological interpretation. Version 1.1 substantially strengthens the precision of the original publication. It clarifies the scope of the locally symmetric class; makes explicit the capacity argument and the use of the trace theorem on exterior faces; states the comparability between the path metric and the restricted Euclidean metric; distinguishes the six-face exterior Dirichlet boundary of Paper A from the two-face condenser studied in Paper B; corrects the notation for the Neumann determinant and spectral truncation; and updates the status of the discrete and variational program developed in Paper B. This paper does not calculate a numerical continuum spectrum, determine exact values of the walk or spectral dimensions, identify modal thresholds with observed particles, or constitute a general validation of TCA-F5D. The discrete approximations and numerical investigations belong to Paper B. Their full variational identification with the continuum operator, spectral compactness, and uniform convergence of heat traces remain open problems. Paper A 1.1 supersedes version 1.0 as the current mathematical foundation of the functional and spectral sector, while preserving the previous publication as part of the documented development of the program. Current version: https://doi.org/10.5281/zenodo.21918417Previous Paper A version: https://doi.org/10.5281/zenodo.21705566Historical TCA-F5D corpus: https://doi.org/10.5281/zenodo.20652631