Hamiltonian Depth Coordinate from MÅNobius Transport Resolution

This paper derives the elementary Hamiltonian transport-depth increment ε = 1/320 from a pure counting argument: no dynamics required. The Möbius transport architecture on Q₅ admits a normalized internal depth coordinate with elementary resolution \[ Δd = 1/320 \]. The elementary transport-slot space is: \[ 𝒯 = V(Q₅) × {1,...,5} × {+,−} \] with cardinality \[ |𝒯| = 32 · 5 · 2 = 320 \], where the three factors are vertex localization (32 vertices of Q₅), directional transport (5 coordinate directions), and Möbius orientation sheet (2 sheets from the double cover). Normalizing total traversal length to 1, each elementary slot has depth increment \[ ε = 1/320 \]. For any ordering \[ n: 𝒯 → {0,...,319} \]compatible with the oriented Hamiltonian traversal structure of Q₅ (any admissibility-preserving ordering consistent with the Gray–Möbius cycle), the normalized depth coordinate is \[ d(x,i,σ) = n(x,i,σ)/320 \]. The coordinate d depends on the chosen admissible ordering, but the elementary increment \[ Δd = 1/320 \] and slot count 320 are ordering-independent. The 32 vertices are coarse floors; the 320 elementary substeps are the true depth resolution: 32 floors × 5 directions × 2 orientation sheets = 320 elementary substeps. The Hamiltonian traversal has 32 levels but 320 transport-resolution slots. This corollary immediately recovers the TA5 doubling result: \[ µ = 2ε = 1/160 \], and the TA12 transport denominator \[ Δ_M = 320 = 1/ε \]. The depth coordinate d is an internal transport-depth coordinate, not an emergent physical radial coordinate. No claim is made regarding gravity, inverse-square law, or spacetime curvature. The 1/320 scale is the elementary resolution of Hamiltonian transport through the Möbius-doubled Q₅ incidence structure.