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 T_d 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 = 31/3 survives the observational window, and the postulate (P-ROOT) that selects the index measure is stated with its provenance. The result, A_s = 31/3 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 A_s 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.