Hyperbrane Relativity V20: One-Parameter Theory from Helical Mode Geometry

We demonstrate that all dynamical parameters of Hyperbrane Relativity (HBR) - gravitational coupling A, precession correction C, singularity avoidance B, vortex interaction κ, and W-axis tension α - derive from a single geometric quantity: the brane thickness Δw. Starting from V18's polynomial potential as a near-field expansion of Yukawa modes in a volumetric brane, we show that quantized helical modes with wave numbers κ_n = nπ/Δw produce overlap integrals whose ratios fix all parameter relationships. The vortex coupling constant emerges analytically as 4Δw/ℏ² from the mutual inductance of helical filaments in 4D space, while the W-axis tension α(r) is identified as the W-axis curvature of the same unified potential, requiring no independent calibration. The theory reduces HBR from three phenomenological parameters to one geometric parameter Δw, with the constraint Δw/w_0 ≈ 0.54 from Mercury's perihelion precession. New testable predictions include:(i) a force-law crossover from 1/r² to 1/r³ at separations d ∼ Δw;(ii) mode-dependent vortex coupling with selection rules;(iii) the asymptotic equality ω_W → ω_orbital at large distances.