TEBAC T1: Canonical Internal Spectral Package; Subtitle: Operator, Heat, Zeta, and Threshold Exports for the TEBAC 9D/9D+ Closure Program

This preprint formulates the \(T1\) module of the TEBAC \(9D/9D+\) closure program: the canonical internal spectral package. Its role is to replace schematic internal-threshold language by a theorem-level operator package built from the frozen canonical TEBAC internal datum. More precisely, the paper defines a canonical internal Hilbert space, a distinguished internal Dirac-type operator, and the associated nonnegative Laplace-type operator; proves self-adjointness and compact-resolvent properties on the compact internal sector; records explicit prototype product-spectrum formulas for the canonical compact factor \(K_5 = S^1 × T^4\) and the lifted spherical factor \(F_4 = S^4\); and establishes the associated heat/zeta/determinant package exported downstream. The paper is explicitly modular and referee-facing. It is intended as the theorem-bearing realization of the \(T1\) burden recorded in the companion TEBAC closure-ledger preprint. In particular, this version does not claim final low-energy extraction of the cosmological constant, effective gravitational coupling, gauge couplings, Yukawa data, visible masses, CKM/PMNS data, or the neutrino branch. Its precise purpose is to provide the internal operator-spectral API required by later modules \(T4\), \(T5\), and \(T6\) within the broader TEBAC \(9D/9D+\) closure program. What this version does:- defines the canonical TEBAC internal spectral package;- proves self-adjointness and compact-resolvent realization;- records discrete spectrum and zero-mode bookkeeping;- establishes heat, zeta, and zeta-determinant closure;- exports the canonical threshold package for downstream use. What this version does not claim:- no final renormalized extraction of \(Geff\) or \(Λeff\);- no final gauge-coupling matching;- no Yukawa, visible-mass, flavor, or neutrino closure;- no full ToE-style phenomenological completion.