Motion, Locality, and Cross-Tile Interaction in Foam Cosmologies: How matter propagates when spacetime is discrete — the compact-dimension foam, the photon-only cross-tile relay, and the cosmic-time evolution of inter-cell coupling from a (3+3) spacetime framework

Every theory that posits a discrete spacetime structure must answer the question of how matter moves through it. We present a systematic resolution of this question within the (3+3) spacetime framework, demonstrating that the (3+3) framework's compact-dimension foam structure (the third time dimension t₃ as a discrete S² with 2¹⁵² Planck-area cells, attached identically to every point of continuous 3D space) sidesteps the Lorentz-invariance problem at the structural level. Motion is c-budget redistribution across the six dimensions of the manifold; cross-cell information transfer is photon-mediated, with the W and Z bosons borrowing the photon's face-centre access (the structural origin of the weak force's suppression and short range), gluons unable to fit through any single face-gate (the structural origin of confinement), and gravity emerging as the elastic response of the foam itself. We further show that the inter-cell coupling evolves with cosmic time through the T₂cosmic threshold sequence, with empirical predictions confirmed by Planck (τ ≈ 0. 054, nₛ ≈ 0. 968) and DESI 2024 (deceleration-acceleration crossing at z ≈ 0. 22). Comparison with causal sets, loop quantum gravity, causal dynamical triangulations, group field theory, Wolfram hypergraph rewriting, and quantum graphity shows the (3+3) approach is uniquely operationally specific, committing to a definite geometric structure that produces sharp falsifiable predictions across multiple physics sectors.