H: The Hexahedral Hologram — On the Crystalline Imperative: Why Space Must Be Three-Dimensional and Cubic

The causal graph, under the B=1 bandwidth constraint, crystallizesinto the C8 cubic scaffold as the unique zero-rent minimum of agraph-theoretic rent functional. Uniqueness is proved by exhaustiveenumeration at N=8 (five isomorphism classes, Q3 unique; Lean 4zero-sorry) and by a 1D Lemma on Hamiltonian cycles with ±1 steps(the only Hamiltonian cycles are forward and backward; Theorem-grade).The general uniqueness theorem for all L follows: locality andconnectivity conditions uniquely select the L×L×L cubic lattice,with global uniformity of step size emerging as a deductiveconsequence of the 1D Lemma rather than a separate assumption.Convergence of the full edge-rewiring Metropolis dynamics isguaranteed by Hájek's theorem (1988), with measured barrier depthc ≈ 1.13. Spatial dimension d=3 is uniquely selected by a two-sided bound:d≥3 from the force-fold correspondence (three gauge forces requirethree topological folds), and d≤3 from Bertrand's theorem (stableclosed gravitational orbits exist only in d=3), with independentsupport from Mermin-Wagner and the topological triviality of gravityin d=2. A direct corollary of C8 uniqueness and d=3 uniqueness is thePythagorean bandwidth structure: the internal and external bandwidthcomponents of a node are orthogonal vectors in emergent 3D space,so they compose as f_int^2 + f_ext^2 = B^2. The normalization B=1is a unit choice; the Pythagorean structure itself is a theorem,not an axiom. The C8 scaffold structurally anchors the four-equation unified forceframework, the topological mass chain, and the topological scalerelations of Z. The scale relations are identified asnumerical candidates; the C8 crystallisation, the d=3 uniquenesstheorem, and the Pythagorean bandwidth theorem are Theorem-grade. All Lean 4 formalizations are zero-sorry.Companion code: https://github.com/hzhangy/Hex