CF03: Complete Formalism — A8 Scaling Theorem (Hierarchical Tachyonic Interfaces)

We formalize the VDM candidate axiom A8 (“hierarchical partition with logarithmic depth”) for tachyonic interface-forming scalar-field systems by connecting three ingredients: (i) a phase-field (Ginzburg–Landau) energy with a tachyonic/double-well potential and a smallinterface width ε, (ii) its sharp-interface Γ-limit as a perimeter functional (Modica–Mortola), and (iii) a definition of active hierarchy depth N(L) as the number of resolved interface scales between a macroscopic domain size L and a microscopic cutoff ℓ0 (lattice spacing / minimum interface width). The Γ-limit yields boundary-energy concentration Eexc ∝ Per(A), implying the familiar interface-area scaling in d dimensions for boundedshape partitions. We then show why N(L) is at most logarithmic in L/ℓ0 and provide a falsifiable interpretation of “hierarchy” in terms of a scale-by-scale active interface census. Finally, we define decisive gates that separate true log-depth organization from combinatorial blow-ups (e.g., 2^k interfaces at scale k) on lattice simulations.