{Paper 134: Large-$L$ Hierarchy Depth Scaling

Paper~116 derived the wave-coherence crossover size $L^* = 2004 /5 317$: below $L^*$ a single VCML lattice operates as one coherent level; above $L^*$ it spontaneously stratifies into multiple levels. We now characterise how deep the hierarchy grows as a function of $L$. Simulating $L \{192, 256, 320, 512, 768, 1024, 1536\}$ with physics-derived equilibration rules ($N_equil L^2$, lazy-bulk stride $K 1/ (L)$), we find that the spatial coherence length $ $ increments by exactly one block ($64$ cells) at each golden-ratio rung $L^* ^d$ ($d = 0,1,2,3, $). The result is $ (L) = 64\,(d+1)$ where $d = _ (L/L^*) $ for $L > L^*$. The inter-domain gap (a proxy for distance to criticality) closes monotonically from $1.23$ toward zero. The external boundary fractal dimension grows from $d_H^{ ext} 1.49$ toward $ 1.71$, while the internal fractal dimension grows from $d_H^{ int} 0.26$ toward $ 1.12$. Together these confirm that each $ $-rung adds one new level of spatial self-organisation, and that the hierarchy approaches its own incompleteness boundary ($gap 0$) as $L $.