Continuum Limit of Depth Evolution

This record presents TA15 (Continuum Limit of Depth Evolution), part of the Q5 Transport Architecture Series developed under the Zero-Point Hypothesis framework. TA14 established that the effective transport generator G acts as a local adjacency operator on the 320-slot depth graph, inducing weighted averaging over neighbouring depth values rather than deterministic stepping, making depth a continuum-inducing parameter. TA15 performs the continuum limit explicitly. Taking the elementary depth spacing \ (epsilon = 1/320 \) as a small parameter, and noting that after choosing any ordering of the 320 slots compatible with local transport adjacency, the induced transport law reduces locally to nearest-neighbour form up to higher-order corrections, a standard Taylor expansion yields a differential evolution equation in the continuous depth coordinate \ (d = n*epsilon \). The result is: \ partialₜ (psi) = cd * partiald (psi) + Dd * partiald² (psi) + omega * A * psi + O (epsilon²) \ where \ cd * partiald (psi) \ is the oriented transport term arising from the Mobius/Gray directional bias cd = vepsilon, Dd * partiald² (psi) is the symmetric spreading term from symmetric adjacency averaging (Dd = kappaepsilon²), omega * A * psi is the local rotational phase term from the reduced generator A = i*sigmaᵧ, and leakage-return corrections from the K†BK block enter at higher order. Three lemmas support the derivation: the symmetric continuum limit (equal forward and backward rates produce a diffusion equation in depth), the oriented continuum limit (Mobius/Gray directional bias produces a first-order transport equation), and the phase rotation term (the A-block contributes independently of the depth derivative structure). Two regime corollaries identify the transport-dominated case (strong directional bias gives directed propagation along depth) and the diffusion-dominated case (symmetric transport gives depth diffusion). A third corollary identifies the continuum equation as the starting point for TA16 (Smooth-Path / Minimum-Curvature Principle). The theorem does not claim a physical wave equation, Schrodinger equation, gravitational field equation, or any observable identification of the depth coordinate. The continuum limit is a structural result about the 320-slot transport lattice. Explicit coefficient values for cd, Dd, and omega remain conditional on the full Q5 construction of the leakage generator (open from T126). Together, TA14 and TA15 mark a conceptual transition point in the architecture: the depth coordinate moves from a combinatorial indexing label to an emergent coarse coordinate with local generator flow, continuum approximation, and transport/diffusion decomposition. This transition occurs without abandoning the discrete substrate. 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.