Nature's Latent Catalog: A Formally Verified Tour of Biological Shape Optimization
Abstract
Why does a honeycomb look regular while trabecular bone looks chaotic? We propose that the visual complexity of a biological structure is governed by the Latent dimension \(d_L\): the number of independent parameters generating the optimal geometry from a constrained optimization problem. We survey eight structures spanning \(d_L = 0\) (honeycomb) to \(d_L = 6\) (bone) and prove, for each, the mathematical core of its optimality. The 170 theorems are formally verified in the proof kernel with zero axiom debt and zero unresolved goals, and exported to Lean 4 for independent checking. The \(d_L\) ordering recovers the intuitive complexity ranking: low-\(d_L\) structures are periodic and visually simple; high-\(d_L\) structures are aperiodic and visually complex. We argue that \(d_L\) is not merely a classification convenience but a fundamental invariant of the optimization landscape — it determines whether closed-form solutions exist (\(d_L \leq 2\)) and predicts the computational cost of numerical design (\(O(N^{d_L})\) scaling).
Keywords: biological optimization, Latent dimension, formal verification, isoperimetric inequality, phyllotaxis, Murray's law, Wolff's law, gyroid, myomere, proof kernel
Novelty
The Latent dimension d_L as a unified complexity measure across biological optimization problems is a genuine organizing idea, though the individual results (isoperimetric inequality, Murray's law, phyllotaxis, etc.) are classical and the formal verification adds rigor rather than new mathematics.