Emergence of Arithmetic Signatures in Discrete Networks Under Constraint Redistribution. The Case of Fibonacci.

We study the redistribution of a charge injected at a point of a discrete network, with fixed conductances, local leakage, and a conservation rule. The equilibrium solution is computed in exact fractions, without eigenvalues, without continuous functions, without floating-point arithmetic. We show that the arithmetic signature of the solution depends on the valence of the network. A linear network with two neighbours per site produces the Fibonacci sequence. A binary tree with three neighbours per site produces powers of two. A square grid with four neighbours per site produces no known integer sequence. The result suggests that the constants observed in nature are not fundamental, but emerge from local structures. A discussion of the consequences for the notion of a universal constant is proposed.