A No-Go Theorem for Informational Invariance in Unregulated Systems

A No-Go Theorem for Informational Invariance in Unregulated Systems Description:This work establishes a formal no-go theorem demonstrating that informational invariance cannot arise or persist in unregulated systems. Specifically, it proves that in the absence of explicit informational constraints, no system—regardless of scale, substrate, or dynamics—can maintain a stable lower bound on informational entropy. The result is formulated at the level of state-space admissibility, not dynamics. It does not posit mechanisms, forces, or evolutionary pathways. Instead, it identifies a necessary condition for the existence of stable informational plateaus: the presence of regulatory constraints that limit the effective configuration space available to the system. The theorem applies broadly across physical, chemical, biological, and abstract systems and is independent of thermodynamic formalism, microscopic dynamics, or equilibrium assumptions. Unregulated systems are shown to exhibit unavoidable entropy drift under coarse-graining, projection, or stochastic perturbation, precluding invariant informational structure. This no-go result serves three primary purposes: Boundary Definition: It delineates the exact conditions under which informational invariance is impossible, thereby sharpening falsifiability criteria for empirical claims of invariant behavior. Constraint Clarification: It demonstrates that informational stability is not a consequence of entropy minimization or dynamics alone, but of admissibility constraints acting on state space. Theoretical Closure: It rules out an entire class of alternative explanations—chance maintenance, unconstrained dynamics, or statistical coincidence—for the emergence of persistent informational structure. The theorem complements prior empirical findings showing invariant informational plateaus in regulated systems by establishing the converse: where regulation is absent, invariance is forbidden in principle. All claims are stated within explicitly defined operational and mathematical limits. The work is intended as a constraint-setting contribution to information-theoretic foundations rather than a comprehensive physical theory. Keywords: informational invariance, no-go theorem, entropy, state space admissibility, constraint systems, information theory, complex systems, emergence