One-Loop Identities on the S4 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 six 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 - Defect action cost: ΔS (g) = 3 × S₀/4ᵍ — the exact, parameter-free action weight of a coarse defect - Spectral identity: Σ (−1) ⁿ/λₙ = ln 2 − 11/32 — a consistency condition on the information principle of the companion paper The defect-action-cost result connects a long-standing CMB anomaly to the fraction of tetrahedra destroyed by a single edge failure. It is the exact, parameter-free expression of an amplitude deficit at the largest scales -- the construction is anchored and most constrained at its coarsest scales — and a one-sided pressure: every daughter universe is deficient at the largest scales, never enhanced, unlike the inflationary account in which the low quadrupole is a statistical fluctuation drawn from the same distribution as every other multipole. Its classical weight e^ (−ΔS) is offset by the one-loop determinant ratio of the defective instanton, so the observable depth of the deficit is the net of the two and is left open here. A second low-multipole prediction accompanies it: under the tetrahedral symmetry Td the ℓ = 2 representation restricts to E + T₂ exactly, so the root's framing organizes the quadrupole's angular content along preferred axes -- a tetrahedral alignment bearing on the observed quadrupole–octupole alignment, contingent on the framing remaining coherent to the largest scales. 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 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. New in this version, the root mode map identifies that degree of freedom in closed form as the root's uniform scale mode, and a non-realizability theorem shows it cannot be expressed by first-order saddle geometry – a necessity result: the postulate is required and first-order-irreducible, its content validated structurally and empirically rather than derived. 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. Eliminating S₀ between the tilt and the amplitude leaves a scale-free relation between the two observables; the Planck amplitude, run backward through the closure, implies S₀ = 19. 739 ± 0. 015, with the construction's 2π² = 19. 7392 at its center.