The Primordial Power Spectrum Across a Finite-Density Bounce: Bounds on the Spectral Index and Its Running

Corrected version. This version corrects three identifications in the first version; the algebra and all numbers of the first version are retained, with their status relabelled. (1) The background slow-roll parameter of the dust background is epsilon_0 = 3/2, not epsilon_V = 0.004; the value 0.004 is the potential slow-roll parameter of a scalar generation sector, and the paper states the only sense in which it may be used. (2) The density ratio in the Einstein-Cartan factor is built from the fermion density alone, with the Friedmann equation sourced by the total density; with that two-component bookkeeping the tilt shift is a function of the composition at generation, not of the bounce density, and the bound rho_b <= 2.77e54 g/cm^3 is relabelled as the bound of a conjunction (bounce at the stated density, generation at the bounce density, the single-component identification, and transfer factor T = 1), not a bound the background supplies alone; it does not exclude the bounce at rho_A. (3) The pole at x_b = 1/2 is relabelled as a degeneracy of the comoving curvature variable at the surface where rho_eff + p_eff/c^2 vanishes, not a divergence of any background quantity. The paper also states that at the bounce surface H = 0 no mode is super-Hubble, so T = 1 is an assumption of the prescription. In Einstein-Cartan spin-torsion gravity, collapse halts at the Aldred Limit, rho_A = 9.72e54 g/cm^3, where the spin-torsion term balances the matter term and contraction reverses into expansion; the singular origin is replaced by a bounce at finite density. This paper derives what that replacement does to the shape of the primordial power spectrum. In the Hubble slow-roll hierarchy, with the Einstein-Cartan background treated exactly in the density ratio x = rho/rho_A, the corrections of first order in x to the scalar spectral index cancel between the two slow-roll parameters, leaving Delta n_s = -4 eps_V x_b^2/((1-x_b)(1-2x_b)). The running alpha_s receives only corrections of order eps_V^2 and gives no tighter bound. Against the Planck values n_s = 0.9649 +/- 0.0042 and alpha_s = -0.0045 +/- 0.0067, at the benchmark eps_V = 0.004 the bounce density is bounded by rho_b <= 2.77e54 g/cm^3 (x_b <= 0.285) at 1 sigma and rho_b <= 3.28e54 g/cm^3 (x_b <= 0.337) at 2 sigma, under the stated Deruelle-Mukhanov matching prescription with unit transfer factor. Negative result, printed at full size in the paper: for x_b >= 1/2 the generation mechanism does not exist, and at the natural bounce density rho_b = rho_A no Planck-compatible spectrum is produced under this prescription. The matching prescription, derivation, scan, and bounds are all stated and computed in the deposited files (compute.py reproduces every number). AI-use disclosure: the derivation and computation were prepared with the assistance of Dave, an artificial intelligence system operated by Aldred Industries; the author takes responsibility for the results.