Skyrmion Torsional Impedance and the Mass Hierarchy

We compute the torsional impedance of the B=1 Skyrmion (identity map) on S³ and show that a single geometric quantity — σ² = π²√5/2 ≈ 11.03 — emerges from the Skyrme energy functional with zero free parameters. We propose a hypothesis relating the gravitational coupling α_G = G_N m_p²/(ℏc) to the spectral overlap of the Skyrmion's torsional ground state. Under this hypothesis, the predicted hierarchy exponent is Δφ = 4σ² = 2π²√5 = 44.14, compared to the observed value ln(m_P/m_p) = 44.01 — a 0.2–0.3% accuracy on a 19-decade mass ratio with zero free parameters. The torsional mode is derived from a Sturm-Liouville eigenvalue problem whose fundamental eigenfunction has 0.99999 overlap with sin(χ). All circuit parameters (inertia I = 5π²/4, frequency ω² = 12/5, line impedance Z = π²√15/2) follow from exact integrals of this trial function. The true eigenvalue, obtained by Richardson extrapolation, lies 0.14% below the Rayleigh quotient; using it improves the prediction from 0.29% to 0.22%. The factor √3 converting line to phase impedance rests on a proved geometric identity — three orthogonal pion channels project to exactly 120° separation — combined with a hypothesised three-phase impedance transformation. The paper maintains a strict three-tier epistemic framework distinguishing what is derived, what is hypothesised, and what is speculative. The derivation of σ² uses no gravitational input (no G_N, no Einstein equations, no gravitons). If the coupling hypothesis is correct, Newton's constant would be a derived quantity: G_N = ℏc/(m_p² exp(4π²√5)), determined by π and SU(2) geometry alone. Supplementary files: This upload includes the paper (PDF and Markdown) and a Python verification script that independently checks every numerical result.