Generator Compatibility of Depth

This record presents TA14 (Generator Compatibility of Depth), part of the Q5 Transport Architecture Series developed under the Zero-Point Hypothesis framework. TA13 introduced the Hamiltonian depth coordinate \ (d = n/320 \) as a normalized label on the 320 oriented transport slots (x, i, σ) of the Q5 hypercube. TA14 establishes how the effective transport generator G acts on this coordinate. The central result is that G does not act as a strict stepwise \ (n → n ± 1 \) shift on the depth label. Instead, G acts as a local adjacency operator on the 320-node transport graph, sending each basis slot to a superposition of neighbouring slots. The mechanism decomposes into two contributions: an orientation flip via the A-block \ (A = iσᵧ \), mixing the two transverse sheets while preserving vertex and direction, and a leakage excursion via the \ (K†BK-block \) (moving through the complement sector and returning to a generally different admissible slot). The resulting evolution of the depth coordinate is a weighted average over neighbouring depth values, discrete diffusion rather than deterministic stepping. This establishes depth as a continuum-inducing parameter: the evolution operator acts as a graph-local averaging process over the discrete depth structure, preparing the natural ground for the continuum limit performed in TA15. The depth coordinate \ (d = n/320 \) remains discrete at this stage; what is continuum-like is the generator's action, not the coordinate itself. Three lemmas support the main theorem: the A-block produces a local orientation flip (Lemma 14. 1) ; the K-block produces an inter-fibre leakage move dominantly supported on the local Q5 neighborhood of the source vertex, grounded in the locality of the leakage generator \ AL = Anti (KYQ5 K) \ from T123 (Lemma 14. 2) ; and the full generator produces a local superposition over neighboring slots (Lemma 14. 3). Two corollaries establish depth as a continuum-inducing parameter and identify the discrete-to-continuous bridge leading to TA15. The theorem is careful to assert no physical identification of depth with any observable quantity. Explicit edge weights remain conditional on the full Q5 construction of AL (open from T126). 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.