Geometric Efficiency Bounds for Mycorrhizal Networks: A Freiman–Villani Framework

We introduce a new mathematical framework for the thermodynamics of mycorrhizal networks, connecting Tao's quantitative Freiman–Ruzsa theorem in additive combinatorics with Villani's HWI inequality in optimal transport theory. To each hyphal branching network Γ we associate a local Freiman index σ_r(Γ), a computable geometric observable measuring how closely the spatial arrangement of junction nodes approximates the hexagonal lattice. The main result (Theorem 6.1): any mycorrhizal network satisfies Ψ(Γ) ≥ C* · D · ε⁻² · (σ_r(Γ) − K_hex)² where K_hex = 19/7 ≈ 2.714 is the local doubling constant of the hexagonal lattice (exact, proved by enumeration), C* ≈ 21.8 analytically, and C_sim ≈ 20.9 numerically (4% agreement). The non-perturbative lower bound is proved in full generality using the explicit Tao–Vu covering constant N_TV(K) = (2K)2⌊log₂K⌋+1 (Theorem 5.32 of Tao–Vu 2006). Under simulated drought stress, σ_r rises from 3.1 to 4.6 and metabolic dissipation increases by factor ≈ 24, consistent with quadratic scaling. Numerical validation uses the MycoNet Python package (pip install myconet, DOI: 10.5281/zenodo.20342304). Submitted to the Journal of Mathematical Biology.