Quantization causes waves: Smooth finitely computable functions are affine

Given an automaton (a letter-to-letter transducer) A whose input and output alphabets are Fp = {0, 1, . . . , p − 1}, one visualizes word transformations performed by A by a point set P(A) of real plane R2 as follows: To an m-letter non-empty word v = γm−1γm−2 . . . γ0 over the alphabet A put into the correspondence a rational number 0.v whose basep expansion is 0.γm−1γm−2 . . . γ0; then to every m-letter input word w = αm−1αm−2 · · ·α0 of the automaton A and to the respective m-letter output word a(w) = βm−1βm−2 · · · β0 (rightmost letters are feeded to/outputted from the automaton prior to leftmost ones) there corresponds a point (0.w; 0.a(w)) of the real unit square [0, 1]2; denote P(A) a closure (in the topology of R2) of the point set (0.w; 0.a(w)) where w ranges over the set W of all non-empty words over the alphabet Fp. For a finite-state automaton A, it is shown that once some points of P(A) constitute a smooth (of a class C2) curve in R2, the curve is a segment of a straight line with a rational slope; and there are only finitely many straight lines whose segments are in P(A). Moreover, when identifying P(A) with a subset of a 2-dimensional torus T2 ⊂ R3 (under a natural mapping of the real unit square [0, 1]2 onto T2) the smooth curves from P(A) constitute a collection of torus windings. In cylindrical coordinates either of the windings can be ascribed to a complex-valued function ψ(x) = ei(Ax−2πB(t)) (x ∈ R) for suitable rational A,B(t). Since ψ(x) is a standard expression for a matter wave in quantum theory (where B(t) = tB(t0)), and since transducers can be regarded as a mathematical formalization for causal discrete systems, the main result of the paper might serve as a mathematical reasoning why wave phenomena are inherent in quantum systems: This is because of causality principle and the discreteness of matter.