Algebraic Chrono–Dynamics: Stratified Covariant Phase Space and Boundary Algebra

We provide a coordinate-free characterisation of phase boundaries in field theory on globally hyperbolic spacetimes with boundary. For a complex scalar field, we prove that a diffeomorphism-invariant local scalar functional $P[\Phi,g]$, partitioning spacetime into strata across a level set $\mathcal{H} = \{P=P_\star\}$, induces a stratified covariant phase space in the sense of Sjamaar--Lerman, in which the admissible variation class jumps discontinuously across $\mathcal{H}$. Concretely, on the dense stratum $M_{dens} = \{P\geq P_\star\}$ a diverging phase-stiffness functional $\kappa(P)$ enforces, by a finite-action selection rule, the vanishing of phase variations $\delta\theta = 0$, restricting the tangent space to amplitude fluctuations alone. The principal result, which we call the Phase Boundary Characterisation Theorem, states that this single energetic condition produces two algebraically equivalent effects on the augmented covariant phase space: it enlarges the presymplectic kernel of the augmented form $\Omega^{\mathrm{aug}}_\Sigma$ in the phase sector, and it suppresses the explicitly represented boundary 2-cocycle of the boundary charge algebra, $K_dens = 0$. The phase boundary $\mathcal{H}$ is identified intrinsically as the unique locus of this stiffness-induced phase-sector degeneracy, with no reference to the trigger functional once the construction is complete. Along the way, we exhibit a concrete mechanism by which the Iyer--Wald--Zoupas freedom in a phase-dependent boundary density is removed on the constrained stratum by explicit mixed boundary conditions; this is a model calculation within the IWZ setting, not a general resolution of the ambiguity. We show that the algebraic structure on each stratum is compatible with reduced phase-space quantization carried out independently on each stratum, and verify that the unstratified limit $P_\star \to \infty$ recovers the standard Lee--Wald / Iyer--Wald formalism identically. Throughout the paper the metric is treated as fixed Lorentzian background data, and the trigger functional $P[\Phi,g]$ is prescribed for purposes of the variational problem; dynamical metric variation and fully dynamical trigger functionals are identified as natural extensions rather than assumptions of the present theorem.