The LMC Framework: Structural Geometry of Physical Fields - From Measurement Operators to Topological Certification

We present a rigorous mathematical framework unifying the structural analysis of heterogeneous physical fields through spectral geometry. The Leue Modulation Coefficient (LMC) framework establishes a three-layer architecture connecting classical field theory to compact geometric representation. The central construction is an explicit measurement operator ℒ_phys: H^s(M) → Q³ mapping electromagnetic, acoustic, and thermal fields on spacetime M = 𝕋³×[0,T] to points in the hypercube Q³ = [-1,1]³. We prove continuity and injectivity modulo gauge transformations, establishing Q³ as the quotient of the Sobolev space by the kernel of the spectral energy functional. The framework introduces structural equivalence classes via discrete invariants (sign signature, effective support, dominance ordering, normalized ratio profiles) and defines Rechenäquivalenz (computational equivalence) through a categorical formalism. The universal property (Theorem 2.2) guarantees that all computations on physical systems factor uniquely through the LMC quotient, making media computationally transparent while preserving physical distinctions. We establish the Hodge decomposition as the natural coordinate system: the third LMC component η(v) = ‖δω‖L²/‖ω‖H¹ measures field transversality via the codifferential norm, justified by the Gaffney-Gårding inequality. Electromagnetic waves map to (1,0,1), thermal diffusion to (0,1,0), and acoustic modes to intermediate positions, providing geometric signatures for field classification. The S³ certification layer provides finite-resolution topological guarantees via simplicial homology (explicit boundary operators given), while Prime Time Quantization emerges as a logarithmic sampling lattice with constant relative resolution. Numerical validation confirms six-nines precision for the spectral decomposition and homological sphere conditions. This work provides rigorous foundations for universal signal processing across heterogeneous media without reductionist collapse.