T-V5b-T — Pre-Objective Goldstone on Translation-Invariant Graphs (W4-Extended Capstone)

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1. #1

   Crossover scale — refined at CV-1.5.1. The transition scale between sub-lattice and super-lattice regimes is graph-class dependent and -dependent; theoretical derivation as an analytic function of dimensionality and lattice topology remains open (NQ-174).

   $${\zeta }_{\ast }(2D\text{ torus},c)\approx 0.40+(c-0.10)\cdot 0.30,c\in [0.10,0.30],$$
2. #2

   (V5b-T-zero, Cat A definitional, CV-1.5.1.) On the sub-spinodal interior of the regime, the Goldstone mass is exact zero by translation invariance:

   $${\mu }_{\mathrm{Gold}}=0\text{exactly on the sub-spinodal interior.}$$
3. #3

   (V5b-F-empirical, Cat B target, CV-1.5.1.) On boundary-modified / partially-lifted graphs, the Goldstone mass scales empirically with the boundary measure:

   $${\mu }_{\mathrm{Gold}}^{V5b-\mathrm{F}}\approx C(\beta )\cdot \frac{\Vert \partial S\Vert }{n},$$