The Laws as the Necessary Emergence of Change — The Physical Foundation of Recursive Constitutive Dynamics

All known theories of physics take the "existence of laws" as a default premise, yet none explains the dynamical origin of laws themselves. This paper constructs a self-consistent generative framework to fill this gap. We begin from the only fact that requires no justification: **change exists** — two measurements performed on the same physical system yield different results. To give mathematical expression to this fact, one must distinguish that part of change attributable to structure from the remainder not yet so attributed, yielding the Recursive Constitutive Equation (RCE): $$ \Psi_{n+1} = \mathcal{R}[\Psi_n] + \xi_n. \tag{1} $$ where \( \mathcal{R} \) is the **constitutive map** and \( \xi_n \) is **intrinsic noise**; the two are mutually defined within the RCE and constitute two sides of the same dynamical process. The RCE is not a law imposed upon the system from without, but the mathematical transcription of the fact of "change": any simpler form cannot distinguish attributable from non-attributable change, and **under the assumption of a normed state space, the RCE is the minimal equality structure that provides a complete description of structural change**. This paper explicitly acknowledges the three axiomatic presuppositions of the RCE: discrete sequential structure (succession), a normed state space \( ( \mathcal{S}, \| \cdot \| ) \), and composability of mappings. These three constitute the minimal logical grammar required to state "change" and presuppose no specific physical laws, symmetries, or forms of matter. Within this framework, the limiting case in which \( \mathcal{R} \) is absent is defined as the **ground state**: all possible states obey the maximum-entropy measure, and \( \xi \) occupies the entire dynamical space. We rigorously prove: **(P3)** ordered dynamical fragments carrying self-maps emerge with positive probability in the ground state; **(P4)** fragments that do not satisfy the membrane-locking conditions dissipate with probability 1 in finite steps, while those that satisfy them survive forever with probability 1; **(P5)** the self-maps carried by surviving fragments constitute a **self-referential closure** on the surviving subspace — namely, a state subspace invariant under the map, with the recovery mechanism internal to the structure of that subspace. Thus, "whence laws?" receives a foundational dynamical answer: laws are the surviving outcomes of membrane-locking selection, not a priori givens. The surviving structures manifest in the RCE as self-referential closures; their specific algebraic forms (such as spacetime dimension or symmetry groups) lie beyond the scope of derivation in this paper and are deferred to subsequent formal treatments based on this framework.