The Planck-Scale Engine: Analytical Derivations of the Holographic Bounce

This technical supplement provides the explicit mathematical derivations supporting the findings of the research article “The Holographic Big Bounce and the Resolution of Time Zero”. We rigorously construct the string‑theoretic effective action at the Planck scale, detailing the T‑dual regularization of the potential that enforces the geometric bounce at Rₘᵢₙ ∼ √α′. We perform the complex‑time WKB analysis to resolve the initial singularity into a unitary quantum turning point. Crucially, we derive the topological origin of matter by calculating the spectral flow of the Dirac operator, proving that the bounce induces an integer Chern–Simons charge shift of ΔN꜀ₛ = 3, which serves as the seed for baryogenesis. Finally, we demonstrate that the post‑bounce relaxation of the radion field is governed by the “walking dynamics” parameter ε ≈ 0.0318 (fixed by F‑theory fluxes), naturally generating Nₑ ≈ 1000 e‑folds of inflation and connecting the Planckian origin to the 55.3 TeV confinement scale.