Interchangeability and Entanglement: A Sector-Neutral Framework for Shared Degrees of Freedom

We propose a sector-neutral kinematics for deciding when a statistical description and a physical description are the same state. The core is a pair of linear “interchangeability” maps that place both channels in a common Hilbert geometry and certify equality exactly on the physical side and modulo a canonical projection on the statistical side. A single quadratic residual (“Rsameness”) then quantifies calibrated mismatch and is shown to be norm-equivalent on both sides. For any admissible operation—one that intertwines the calibration and is contractive in the comparison norm—the residual cannot increase, giving a geometry-level, postulate-free form of entanglement monotonicity. With a rank-one “one-budget” statistical carrier, admissible maps preserve global resource, and a finite-speed relay bound yields a causal ceiling on local growth. The framework comes with practical diagnostics (projection and data-processing tests; principal-angle conditioning) and specializes cleanly in three settings: flux–gradient PDE (Neumann potentials/projections), pointer expectations in operator algebras/quantum Markov semigroups, and Ornstein–Uhlenbeck/free-field covariance flows. When a sector provides a Lyapunov identity and a spectral/elliptic gap, Rsameness becomes a Lyapunov functional with an explicit decay envelope, offering a residual-driven “arrow of time.