Recursive Maps as a Foundational Physical Metalanguage: A Metatheoretical Justification

This paper proposes and rigorously demonstrates that discrete recursive maps can serve as a more fundamental physical metalanguage than continuous differential equations. Starting from the minimal empirical premise of “the existence of an orderable sequence of observations, ” this paper establishes a **five-condition framework** for evaluating the fundamentality of physical formulations—logical priority, universality, cross-theory validity, relative minimal sufficiency, and foundational prospect—and proves, condition by condition, that recursive maps are superior to the continuous differential equation framework in terms of conceptual burden and universality. Through the **state-space extension method** (Proposition 1), a broad class of dynamical behaviors with finite memory is shown to be subsumable under a first-order recursive form; the complete formal proof of the proposition appears in Appendix C and Supplementary Material SM-01. Through the **non-embeddability argument** (Proposition 2), this paper strictly demonstrates that a generic recursive map cannot be embedded into any continuous-time flow, thereby establishing the independent status of the recursive framework as a broader foundational syntax. Through the complete proof of the **information monotonicity lemma** (Lemma 1), the intrinsic noise ₙ receives a rigorous information-theoretic grounding: in recursive systems containing non-degenerate noise (and satisfying the applicability conditions of Lemma 1), the non-measure-preserving property of the recursive rule together with noise injection jointly lead to a monotonic increase in information entropy; the equality condition corresponds to the joint case of a measure-preserving bijection and zero noise. Through the complete argument of the **relative minimal sufficiency proposition** (Proposition 4), the recursive triple (, R, ) is strictly established as the minimal dynamical skeleton of the state-update syntax, in which the irreducibility of holds under the constraint of “not introducing additional premise commitments. ” This paper also presents a preliminary comparison of the recursive framework with the continuous-time nonlinear contraction analysis work of Lohmiller–Slotine, and points out that the recursive framework in principle provides candidate solution paths for problems such as the origin of multi-valued trajectories and the essence of ; detailed derivations are deferred to future work.