Second-Order Running of the Primordial Spectrum Through the Bounce: A Prediction Below the Planck Central Value

This paper computes the second-order running of the scalar spectrum, beta, for a bounce in which the scale dependence of the shear fraction itself changes across the observable window. Allowing the scaling exponent to vary linearly in the logarithm of wavenumber, with slope delta = -0.60, gives alpha = -0.002109 and beta = 0.01042 at the pivot, the beta value confirmed by an independent finite-difference evaluation in the deposited script. The Planck 2018 analysis reports beta = 0.025 plus or minus 0.013, so the prediction sits -1.1 standard deviations from that central value: a mild tension, stated plainly, in the direction of the bounce predicting far less second-order running than the Planck central value suggests, though the two remain compatible at the edge of the two-standard-deviation region. The corresponding variation of the index across one decade in wavenumber is 0.003985. If the Planck preference for a positive beta hardens with future data, this class of smooth bounce spectra will be in genuine difficulty. The study is computed from stated assumptions, not measured.