Basic principles drive self-organization of brain-like connectivity structure

The paper defines the three rewiring principles precisely (P1 by distance, P2 by external field, P3 by heat-kernel diffusion with H(τ) = −e^{−τL}) and implements a probabilistic algorithm S0–S2; it then demonstrates via simulations that combining the principles produces modular small-world networks and that peaks of small-worldness occur along lines q + p ≈ 0.23 (for the tested τ), i.e., r ≈ 0.77, with lateral and radial fields alike. These are presented empirically, not as theorems, and the authors themselves note the dependence on τ, not universality. Consequently, a general mathematical guarantee (e.g., that the dynamics must reach S > 1 for broad classes of initial graphs and parameters, or that maximizers always satisfy p + q = c(τ)) is not proved in the paper and, per our audit, remained open at the stated cutoff. The candidate solution offers plausible heuristics and partial arguments (e.g., small-τ expansion for P3 and coupling to small-world models) but no complete proof; it also correctly cautions that the constant 0.23 is model/parameter-dependent rather than universal. Therefore, the mathematically general claims are likely open as of the cutoff, while the paper’s empirical findings are consistent with its stated scope.