0.8 Spectral Action Decomposition Theorem: Seeley-deWitt Expansion and Three-Sector Numerical Verification

Abstract

This article carries out the explicit computation planned in 0.7 §2: starting from the Spectral Triple constructed from Cl(8), we perform an 8-dimensional heat kernel expansion of the spectral action , compute the Seeley-deWitt coefficients through term by term, and map each coefficient to CSG's established Sector parameters. **The global constraint running through the entire article is that Total Action is Zero** (0.1 §2.3): , i.e., the sum of the three-Sector contributions in the Seeley-deWitt expansion must cancel exactly. Core result: the three-Sector splitting is not an artificial division, but a natural grouping of by curvature source, and this grouping is rigidly locked by the zero-sum constraint; locks the Sector gravitational coupling, locks the Sector gauge coupling (), locks the Sector topology (). The form of the cutoff function is uniquely determined by the Coding Orbit layer structure—— is not a free function, but a smoothing of , where corresponds to the eigenvalue of Seven-Layer Truncation E₇. Explicit expressions are given for all 8 coefficients –, and cross-validation with existing CSG values is consistent to within precision, satisfying .