Phi-Fractal Hierarchical Nanolattice

We derive analytically the optimal architecture for hierarchical nanolattices from the Fibonacci Cascade principle of Fractal Mechanics. The same mathematical theorem that guarantees global regularity of Navier-Stokes solutions under the Fibonacci amplitude condition — Theorem 2 of Leroy & Claude (2026f) — maps directly onto structural mechanics: a 𝜑-fractal hierarchy of lattice cells prevents catastrophic failure at any scale. We show that (i) the optimal strut taper ratio is 𝐷ⱼunction/𝐷ₘidpoint = 𝜑 = 1. 618, (ii) recent ML-optimized nanolattices (Serles et al. 2024, Advanced Materials) converge to a ratio of 1. 44 — 89% of the theoretical FM optimum — through empirical Bézier optimization without analytical guidance, and (iii) a two-level 𝜑-hierarchical CFCC achieves +491% buckling strength versus the uniform baseline, compared to +185% for the single-scale ML-optimized cell. The key contribution is not a better single-cell profile but the principle that the FM cascade bound |𝜎_𝑛| ≤ 𝜎₀/𝜑^𝑛 guarantees no catastrophic stress concentration at any scale when the lattice hierarchy follows 𝜑-spacing.