Spiral Dynamics and the Geometric Invariant of the 3D Logarithmic Spiral: Critical Scale, Helix Transition, and G = omega/(2z_0)

We present a complete geometric analysis of the 3D logarithmic spiral r(s) = (R cos(omega s) exp(-delta s), R sin(omega s) exp(-delta s), z_0 s) parametrized by arc length. Three results are established. (1) There exists a critical arc length s* = (1/delta) log(R sqrt(omega^2+delta^2) / z_0) at which the spiral undergoes a geometric phase transition: the xy and z velocity components are equal, and the Frenet phase is aligned to an integer multiple of pi (proven analytically for s*, verified numerically for the phase alignment). (2) Every damped logarithmic spiral (delta > 0, z_0 > 0) converges exponentially to a limiting helix with curvature kappa* = omega^2/(omega^2+z_0^2) and torsion tau* = omega z_0/(omega^2+z_0^2). The helix is identified as the FRD fixed-point attractor (Tatai, 2026a; DOI: 10.5281/zenodo.19728746). (3) The geometric invariant G = max_s |kappa(s) cos(Omega(s))| satisfies G = omega/(2z_0) + O(delta/omega) in the regime omega >> z_0. G is independent of R (proven), achieved at s ≈ s*, and verified numerically to < 0.3% across 175 parameter combinations. The analytical proof of this formula is the primary open problem. This paper is the third in a series: FRD (DOI: 10.5281/zenodo.19728746) provides the mathematical framework; TMF (DOI: 10.5281/zenodo.19733996) provides the physical realization; Spiral Dynamics provides the geometric foundation for both.