One-Loop Spectrum and Quadrupole Suppression from the Pre-Geometric Instanton

The companion paper 'Cosmological Structure Without Inflation' (Weller, 2026) derived the primordial perturbation spectrum from the combinatorics of tetrahedral edge-sharing, with the spectral index, tensor-to-scalar ratio, and perturbation amplitude determined by the topology of viable three-dimensional tessellations and an information-theoretic principle that fixes the per-edge failure probability with no free parameters. It identified three open problems: the one-loop calculation, the transfer function, and the EPRL vertex amplitude. This paper closes the first and reframes the second. The present paper presents seven exact results on the S⁴ instanton, derived without free parameters: • Eigenvalues: λₙ = (n−1) (n+1) (n+3) /n (n+2) — the complete one-loop scalar spectrum • Quadrupole ratio: λ₃/λ₂ = 128/75 — regularization-independent • Determinant: log det H = ζ (3) / (4π²) + ln (27π³/256) — closed-form • Curvature identity: Tr (D) = 4π per Pachner defect — shape- and generation-independent • Action suppression: ΔS (g) = 3 × S₀/4ᵍ — quadrupole suppression from the path integral • Spectral identity: Σ (−1) ⁿ/λₙ = ln 2 − 11/32 — the instanton's encoding of binary information • Freed-vertex mechanism: p ~ exp (−ΔS) × (det ratio) ^1/2 / S₀ — the path-integral origin of algebraic suppression The action suppression result connects a long-standing CMB anomaly to the fraction of tetrahedra destroyed by a single edge failure. The quadrupole is suppressed in every daughter universe, not as a statistical fluctuation, but as a necessary property of the action-weighted path integral over the instanton. The spectral identity connects the per-edge failure probability p = ln 2/ (2π²) to the eigenvalue spectrum, reformulating the information principle of the companion paper as an exact sum over instanton modes. The freed-vertex mechanism identifies why the failure probability scales as 1/S₀ rather than as exp (−S₀): when an edge fails, the disconnected vertex drops out of the Regge action entirely, contributing a phase-space volume Vol (S³) = S₀ to the path integral. Section 7, new in this version, takes the amplitude from leading order to a one-loop closure, derived end to end up to one structural postulate that is stated explicitly and flagged as such. The construction measure is 4/3 bits per mode in every shell (the shell theorem), so the remaining factor is a property of the root alone. The root carries six frozen construction degrees of freedom against five excluded continuum modes – an exact ledger, with the construction-to-continuum correspondence verified at the first countable rung (9 = 9) – so the measure question involves a single degree of freedom. Among the counting-native candidates for the resulting factor, only the root's index extent n₀ = 3^1/3 survives the observational window, and the postulate (P-ROOT) that selects the index measure is stated with its provenance. The result, Aₛ = 3^1/3 e^−2π² (det H) ^−1/2 = 2. 1014 × 10⁻⁹, agrees with the observed (2. 101 ± 0. 034) × 10⁻⁹ at 0. 01σ with no free parameters. The conformal and ghost sectors net to unity within a structural obstruction menu, and the conditions that would falsify the closure are registered, including the precision at which Aₛ would distinguish the derived value from nearby numerical rivals that current precision cannot exclude.