From Coherence Invariants to Composable Intelligence_ Why Identity, Safety, and Emergence Require a Scalar Governing Law

Abstract This paper establishes a necessary structural condition for identity-governed emergence in systems evolving over time. We show that persistence of identity under perturbation, composition, and substrate change is undecidable unless the system admits a scalar identity invariant that totally orders states and bounds admissible drift. In the absence of such an invariant, claims about safety, continuity, replayability, or composability are not merely unproven but ill-defined. Assuming prior results identifying SO(2)-structured phase coherence as the minimal symmetry supporting scalar identity invariants, we analyze the consequences of invariant governance. We prove that operations such as merge, split, federation, nesting, deterministic replay, safety certification, and substrate transfer are definable if and only if identity is governed by a scalar coherence invariant with bounded drift. The result is a sharp demarcation between architectures that admit identity-preserving autonomy in principle and those for which such claims are structurally impossible, independent of scale, data, or implementation