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 \ (K₅ = S¹ T⁴\) and the lifted spherical factor \ (F₄ = S⁴\) ; 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 \ (G₄₅₅\) or \ (₄₅₅\) ; - no final gauge-coupling matching; - no Yukawa, visible-mass, flavor, or neutrino closure; - no full ToE-style phenomenological completion.