Abstract We present a mathematical result connecting two independently developed frameworks: Tick Compression Cosmology (TCC), a discrete causal graph theory in which spacetime emerges from events carrying tick-demand and remainder operators, and Scale-Recursive Non-Uniform Domain Tiling (SRNUDT), a proposed organizational principle in which sovereign domains with connected borders recur at every scale of complex systems, each boundary generating unresolved remainder content that seeds organizational structure at the next scale. Under a conjectured two-component (Dirac-like) embedding of the TCC Hamiltonian, we apply the Foldy-Wouthuysen transformation to a coupled two-node TCC system and prove the Boundary Tick Theorem: the leading-order center-of-remainder correction term takes the form TN+1ᵉff = - (hbarN² / 4mN² vN²) · (∂VN/∂ρ₊) · ∂/∂ρ₊ This operator is first-order, vanishes in domain interiors where the potential is flat, and is maximal at domain boundaries where the potential gradient is large. Scale- (N+1) tick events are dynamically concentrated at scale-N domain boundaries — a mathematical mechanism consistent with, and partially derivable from, the SRNUDT organizing principle. All results are derived in TCC remainder space (ρ̅ coordinates) ; translation to physical position space requires the ρ̅→x map, which is Open Problem OP4 of TGT and is the primary obstacle to direct empirical relevance. We state clearly what this result proves and what it does not. The theorem is conditional on a conjectured two-component embedding of the TCC Hamiltonian, which is stated explicitly as a postulate rather than derived from first principles. The theorem is proved for two Foldy-Wouthuysen reduction steps (Theorems 1 and 2) ; the general N-step recursion requires additional work. We identify six open problems and situate the result relative to prior work in causal set coarse-graining, multiscale entanglement renormalization, Born-Oppenheimer effective field theory, and Energetic Causal Sets.
Bradley Ploof (Sat,) studied this question.