Quantization causes waves: Smooth finitely computable functions are affine

Every automaton (a letter-to-letter transducer) A whose both input and output alphabets are Fp = {0, 1,..., p - 1} produces a 1-Lipschitz map fA from the space Zp of p-adic integers to Zp. The map fA can naturally be plotted in a unit real square I2 ⊂ R2: To an m-letter non-empty word v = γm-1γm-2... γ0 there corresponds a number 0.v ∈ R with base-p expansion 0.γm-1γm-2... γ0; so to every m-letter input word w = αm-1αm-2 ··· α0 of A and to the respective m-letter output word a(w) = βm-1βm-2 ··· β0 of A there corresponds a point (0.w; 0.a(w)) ∈ R2. Denote P(A) a closure of the point set (0.w; 0.a(w)) where w ranges over all non-empty words.We prove that once some points of P(A) constitute a C2-smooth curve in R2, the curve is a segment of a straight line with a rational slope. Moreover, when identifying P(A) with a subset of a 2-dimensional torus T2 ∈ R3, the smooth curves from P(A) constitute a collection of torus windings which can be ascribed to complex-valued functions ψ(x, t) = ei(Ax-2πBt) (x, t ∈ R), i.e., to matter waves. As automata are causal discrete systems, the main result may serve a mathematical reasoning why wave phenomena are inherent in quantum systems: This is just because of causality principle and discreteness of matter.