THFR-E2: Continuum Limit of the Kronecker Macrograph: Spectral Convergence, Coarse-Graining, and Conditions for Emergent Effective Geometry — Part I: A Controlled Theorem Program for the Discrete-to-Continuum Transition

Abstract

The Theory of Harmonic Field Resonance (THFR) employs a discrete architecture for generating a Kronecker macrograph. However, the transition from this discrete construction to smooth effective geometry, Lorentzian signature, and the infrared sector of General Relativity has not previously possessed a completed mathematical derivation. In particular, the growth of the finite graph representation through the recursive Kronecker construction $A_G^{(k)} = \mathcal{M}(\mathcal{I}^{AB}, \chi)^{\otimes k}$, with $D_G(k) = 11^k$, does not by itself establish the existence of a continuum limit. Exact hierarchical self-similarity of the microscopic construction requires a separate demonstration that its discrete and potentially fractal-spectral features become irrelevant to long-wavelength infrared-scale quantities. The purpose of this work is not to prove the existence of the continuum limit, but to formally decompose the previously underderived transition into a system of distinct, dependency-ordered, independently falsifiable mathematical obligations. At a schematic ontological level, the research program proceeds from pre-geometric relations through infrared equivalence, continuum organization, effective dimensionality, causal structure, Lorentzian geometry, and effective gravitational dynamics. The corresponding proof architecture introduces graph coarse-graining, measured Gromov–Hausdorff and Mosco convergence, Riemannian regularity, infrared spectral dimension, causal-orientation structure, Lorentzian metric emergence, and effective gravitational dynamics as separate theorem targets. The main result of this work is the construction of a controlled architecture of proof obligations, an explicit dependency graph, and a matrix of local falsifiability. Neither the existence of the continuum limit, nor four-dimensionality of the infrared sector, nor emergence of Lorentzian causality, nor Einstein dynamics is claimed here as proven.