Cosmological Structure Without Inflation: The Perturbation Spectrum from Pre-Geometric Construction

This paper demonstrates that all four problems the inflationary paradigm was introduced to solve are addressed by the pre-geometric seed proposed in 'The Last Evaporation: Planck Remnants as Cosmological Seeds in Empty Spacetime' (Weller, 2026). The primordial perturbation spectrum (the decisive test for any alternative to inflation) is produced by the correlation structure of tetrahedral edge-sharing during the construction of geometry from the seed. The per-edge failure probability is determined by an information-theoretic principle: each edge is a binary degree of freedom carrying ln 2 nats of information, and the saddle-point action S₀ = 2π² is the only available scale, giving p = ln 2/(2π²) with no free parameters. The spectral tilt n_s = 1 − ln 2/(2π²) = 0.96489 matches observation at 0.004σ in central value (intrinsic theory spread ≈ 0.001). The tilt is constant across CMB scales: the adjacency structure of iterated Pachner moves produces a bulk spectrum that is exactly Harrison–Zel'dovich at leading order, and the logarithmic running of S₀ through an operator trace identity gives n_s − 1 = −p at all scales, with predicted running α_s ≈ +p² ≈ +0.0012 (Planck: −0.0045 ± 0.0067; the positive sign is the framework's sharpest discriminator against slow-roll, which generically predicts negative running). This is the combinatorial tilt: the observed spectrum is the combinatorial content after forward evolution, P_obs = P_comb · T_early² · T_standard². T_standard is the ordinary, tilt-preserving radiation transfer common to every cosmology; T_early, through the brief de Sitter→radiation conversion, is the one construction-specific factor and the central open computation -- diagonal at linear order but otherwise uncomputed. Across the measured band the combinatorial tilt passes through as the observed tilt, the agreement itself indicating a non-distorting early transfer on these scales; any departure is confined to the lowest harmonics, the low-ℓ deficit below. The EPRL vertex amplitude |A_v| ≈ 0.035, computed independently in the spin foam program, agrees with this value – a prediction of the framework, not an input. Phase coherence (the adiabatic initial condition that produces the observed acoustic peaks) is derived from the regularity of ζ at the origin, without an inflationary phase. The tensor-to-scalar ratio vanishes at leading order from the T_d symmetry of the regular tetrahedron, providing a dynamical mechanism for Penrose's Weyl curvature hypothesis; the first correction gives r = 4(1+3p)/[135(1−p)] = 0.033943 – at the BICEP/Keck bound r < 0.036, and decisively testable by operating experiments within years. The tensor power further splits exactly 5:1 between the magnetic and electric Weyl sectors (r_B = 0.0283, r_E = 0.0057), a zero-parameter sub-prediction. Non-Gaussianity is derived from the per-block third cumulant of the binary defect distribution: f_NL is positive, equilateral-type (f_NL^local = 0 at leading order, since different Pachner generations are statistically independent), and falls as ℓ⁻³, giving f_NL ~ 7.5 at the quadrupole but ~10⁻⁴ at Planck scales – consistent with all current bounds. The generation-to-harmonic mapping n_g = (3 · 4^g)1/3 connects the discrete Pachner generations to the continuum S³ harmonic decomposition; the lowest multipoles are suppressed because the construction is most constrained at its coarsest scales – a structural prediction inflation does not make, in the direction of the observed low quadrupole. Its mechanism and magnitude are an open computation, developed in the companion one-loop paper alongside a companion prediction: a tetrahedral alignment of the lowest multipoles, bearing on the observed quadrupole–octupole alignment. The perturbation amplitude is exp(−2π²) at leading order from the combinatorial budget; the companion one-loop paper closes the prefactor, giving A_s = 31/3 e−2π² (det H)−1/2 = 2.1014 × 10⁻⁹ against the observed (2.101 ± 0.034) × 10⁻⁹ – agreement at 0.01σ with zero free parameters, resting on one stated measure postulate. The spectral observables are rigid functions of this one derived constant, bound by parameter-free relations – α_s = (1 − n_s)², r = (4/135)(4 − 3n_s)/n_s, and the amplitude relation – and the paper states explicitly which number each future tilt measurement tests: the point, the curve of relations, or the generation-to-harmonic identification. Flatness, homogeneity, and the absence of magnetic monopoles are default properties of the initial state.