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 κₙ = 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/w₀ ≈ 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 ωW → ωₒrbital at large distances.