Smooth-Path / Minimum-Curvature Principle

This record presents TA16 (Smooth-Path / Minimum-Curvature Principle), part of the Q5 Transport Architecture Series developed under the Zero-Point Hypothesis framework. TA15 derived the continuum depth evolution equation governing transport along the normalized Mobius-Hamiltonian depth coordinate d. TA16 identifies what that equation selects for: stable transport trajectories are those with low depth roughness, while high-roughness modes are dynamically suppressed. The central object is the depth roughness/curvature functional, defined as the integral of the squared second derivative of psi along d. This measures total second-derivative energy, a surrogate for local depth roughness. Large values indicate rapid variation, jolts, or sharp bends along the depth path; small values indicate smooth evolution. The mechanism is spectral damping. The symmetric spreading term in the TA15 equation acts as a diffusion operator in depth, exponentially damping Fourier modes at a rate proportional to Dd times the squared frequency nu². High-frequency modes (high depth roughness) are damped faster than low-frequency modes. The oriented transport term translates modes without changing their roughness profile. The rotational phase term mixes orientation sheets without affecting the depth-roughness spectrum. Therefore low-roughness modes persist preferentially under the full TA15 evolution. Three lemmas support the main theorem: diffusion suppresses high-roughness modes (Lemma 1, via explicit spectral decay rate) ; smooth modes are stable (Lemma 2, by identifying the surviving low-roughness sector) ; and curvature generates leakage corrections (Lemma 3, connecting high depth roughness to amplified mismatch with the admissible projection structure and enhanced higher-order leakage terms). A critical epistemic distinction is maintained throughout: the theorem states suppression, not minimization. The system does not minimize the roughness functional in the sense of a global variational or least-action principle. It suppresses high-roughness modes dynamically through diffusive damping. Claiming minimization would require deriving an Euler-Lagrange condition from the Q5 geometry, which has not been performed. This distinction is stated explicitly as a corollary to prevent overclaiming. Together, TA14, TA15, and TA16 form a coherent emergence chain: discrete adjacency transport (TA14), continuum approximation of adjacency flow (TA15), and spectral selection of smooth transport sectors (TA16). The depth coordinate moves from a combinatorial indexing label to an emergent coarse coordinate with local generator flow, continuum dynamics, and a geometric selection mechanism favoring smooth evolution. The theorem chain progressively derives the structure of the effective transport generator \ Gₑff = PiY G PiY + K†BK \, from which observable phase, leakage, decoherence, and residual correction emerge as structural consequences of projected transport closure on Q5.