When is a Physical System the Same System? - Identity, Dynamics, and Representative-Independent Evolution

Physical theories determine how states transform, but a same-system claim asks a different question: whether the transformed description concerns the same physical subject. La Profilée (LP) proposes a universal structural law of nature governing identity under transformation; here “universal” denotes a functional necessity claim about determinate same-subject judgments, not a derivation of domain-specific dynamics, constants, masses, or couplings. In LP, persistence is not an independent primitive alongside identity: it is diachronic identity, the positive consequence of preserving the relevant sameness relation through admissible transformation. This paper gives the physical identity formulation of that law and isolates a necessary dynamical condition: No Coreference Without Congruence. If two admissible representatives are identified as the same physical subject through the transformation under analysis, they must induce the same retained future; otherwise no representative-independent reduced evolution exists on the proposed identity quotient. Q1 certifies whether a determinate integrated bearer continues at all. Q2 is the identity verdict on that continuing bearer and, where the physical realization requires distinct carriers, is typed into Q2a, identity-class continuity, and Q2b, continuation-stable profile/history identity. A constrained bridge π_p:H_ℛ ⇀ Ā_ℛ relates those carriers, and Q2b ⇒ Q2a ⇒ Q1. The congruence requirement is stated for deterministic maps, linear quotients, CPTP channels, and quantum instruments. Greatest-fixed-point, bisimulation-like, and quotient-congruence techniques are established mathematical tools; the contribution is their typed integration into the physical identity law supplied by LP. Persistence is then a regime consequence of successful identity preservation, not a primitive used to define identity.