Let ( ℳ , g μ ν ) be a smooth, oriented, globally hyperbolic Lorentzian four-manifold, possibly with smooth boundary ∂ ℳ . The covariant phase-space formalism, originating in the work of Crnković-Witten [1] and Zuckerman [2], developed by Lee-Wald [3], and Iyer-Wald [4], and extended to spacetimes with boundary in the systematic treatment of Harlow-Wu [5] and further refined for fluctuating boundaries by Adami et al. [6], associates with any local Lagrangian field theory a presymplectic current ω μ on the space of solutions, a presymplectic form Ω Σ = ∫ Σ ω μ n μ on each Cauchy hypersurface Σ, and, after quotient by the kernel of Ω Σ , a symplectic phase space

. When ∂ ℳ = ∅ this construction is hypersurface independent: Ω Σ 1 = Ω Σ 2 for any two Cauchy slices, by virtue of the on-shell conservation ∇ μ ω μ = 0 . When ∂ ℳ ≠ ∅ , hypersurface independence generically fails because of symplectic flux through the timelike portion of the boundary. A by-now standard remedy is to augment Ω Σ by a boundary 2-form Ω ˜ ∂ Σ , recovering a hypersurface-independent total form Ω Σ aug at the cost of introducing edge-mode degrees of freedom and, generically, a nontrivial central extension in the algebra of boundary charges [5][7]-[11].

The present paper addresses a natural and physically widespread generalisation of this setup that, to the author’s knowledge, has not been treated in a coordinate-free mathematical form for the scalar case. Suppose that the admissible variation class—the subspace of the tangent bundle to the solution space that is relevant to the canonical structure—changes discontinuously across a codimension-one hypersurface ℋ ⊂ ℳ defined by a diffeomorphism-invariant scalar condition P ( x ) = P ⋆ . Such a discontinuity arises whenever an effective stiffness modulus governing one sector of tangent directions diverges across ℋ , so that any finite-energy variation in the region ℳ dense = { P ≥ P ⋆ } is forced into the complementary sector. The result is what one ought to call a stratified covariant phase space: a single global solution space whose tangent structure—and hence whose presymplectic structure—has different ranks on the two sides of ℋ . The aim of this paper is to develop this framework rigorously and to characterise the phase boundary ℋ algebraically.

The principal theorem is most cleanly stated in advance of the technical setup; the rest of the paper is devoted to proving it.

Main Theorem(Stratified Covariant Phase Structure; Phase Boundary Characterisation). Let ( ℳ , g μ ν ) be a smooth, oriented, globally hyperbolic Lorentzian four-manifold with boundary, carrying a complex scalar field Φ satisfying Assumptions 1-3 of Section 2. Then:

(i) Hypersurface independence on each stratum. The augmented presymplectic form Ω Σ aug is hypersurface independent and descends, under presymplectic reduction, to a non-degenerate symplectic form on the reduced phase space

on each stratum.

(ii) Centrally extended algebra on the regular stratum. On

, the boundary charge algebra is centrally extended,

with K an explicitly represented boundary 2-cocycle on g ∂ .

(iii) Cocycle suppression on the dense stratum. On

the cocycle vanishes identically, K dense ≡ 0 , and the reduced phase-space dimension strictly decreases relative to

.

(iv) Algebraiccharacterisationof ℋ . The phase boundary ℋ is the unique locus at which the stiffness-induced phase-sector kernel of Ω Σ aug enlarges and the cocycle simultaneously vanishes; conditions (ii) and (iii) are activated together at ℋ and absent away from it. (Other, unrelated degeneracies of Ω Σ aug —zeros of Φ, topological sectors, boundary-condition-induced degeneracies—are not excluded by this statement; the uniqueness is uniqueness of the stiffness-induced phase-sector mechanism.)

(v) Bulk dynamics unaffected. Each of (i)-(iv) holds without any modification of the Euler-Lagrange equations: the transition is purely algebraic.

The novelty here is not the augmentation Ω ↦ Ω aug , which is by now classical, nor the appearance of a 2-cocycle K in the boundary algebra, which is generic. The novelty is the assertion (iv): that the geometric locus ℋ = { P = P ⋆ } equals the locus on which two ostensibly unrelated algebraic features—a stiffness-induced phase-sector kernel jump and a phase-sector cocycle collapse—simultaneously occur. Once the construction is complete, ℋ is recoverable from Ω Σ aug alone, with no reference to the trigger functional P that originally produced it. This is what allows us to call the result a characterisation: the phase boundary is an intrinsic feature of the augmented covariant phase space, not of any particular dynamical scalar that happens to be used to define it. (The qualification of (iv) is important: the augmented phase space generically supports several distinct sources of degeneracy—zeros of Φ, topological sectors, boundary-condition-induced degeneracies—and we do not claim ℋ is the only locus where any kernel direction appears. The claim is that the specific combined transition described by (ii)-(iii) is supported at ℋ and nowhere else.)

The proof proceeds in three stages, organised into the body of the paper as follows.

Stage 1: covariant phase space with boundary (Section 2-3). We collect the geometric setup, state the three minimal assumptions on which the entire paper rests, and develop the standard Lee-Wald / Iyer-Wald construction with explicit attention to the boundary. We exhibit the variational origin of the diverging phase-stiffness through a stiffness-coupled Lagrangian (Proposition 3.1), so that the finite-action selection rule of Assumption 3 arises from a concrete action principle rather than being postulated as an external kinematic rule. We prove the boundary flux obstruction to hypersurface independence, introduce the augmentation Ω ˜ ∂ Σ , give an explicit formula for the boundary symplectic density α in polar variables, and exhibit a concrete pair of mixed boundary conditions under which the Iyer-Wald-Zoupas freedom in this α is removed (a model calculation, not a general resolution of the IWZ ambiguity). The presymplectic reduction

and the associated Poisson structure are constructed in this stage.

Stage 2: charges and the cocycle (Section 4). We construct integrable Hamiltonian generators of boundary symmetries, derive the boundary charge algebra under the Poisson bracket, and identify the central extension explicitly. The polar decomposition Φ = t ^ e i θ exposes the cocycle as a bilinear form on phase- and amplitude-sector variations; this representation is the key technical input to Stage 3.

Stage 3: stratification and algebraic transition (Sections 5-6). We make Assumption 3 concrete by introducing the diffeomorphism-invariant trigger functional P and the diverging phase-stiffness coefficient κ ( P ) . We show that the resulting finite-action condition forces δ θ = 0 on ℳ dense , prove the kernel-enlargement theorem (Theorem 6.1) and the cocycle-suppression theorem (Theorem 6.2) using the explicit cocycle formula from Stage 2, and assemble these into the Phase Boundary Characterisation Theorem (Theorem 6.3). The paper concludes (Section 7) with the global structural theorem and a discussion of compatibility with reduced phase-space quantization.

The construction is entirely self-contained. Assumptions 1-3 are the complete logical input; no appeal to any specific physical programme is required, although in Section 5 we discuss canonical choices of P (e.g., curvature scalars) that make contact with strong-gravity and dense-matter regimes.

The covariant phase-space methodology we use is that originating in Crnković-Witten [1] and Zuckerman [2], developed by Lee-Wald [3] and Iyer-Wald [4], and given a systematic treatment of boundary contributions by Harlow-Wu [5] and Adami et al. [6]. The boundary flux issue and its resolution by an edge-mode form Ω ˜ ∂ Σ have been developed extensively by Donnelly-Freidel [8], Speranza [9], Wald-Zoupas [11], and Barnich-Brandt [10], among many others; a pedagogical exposition of the closely related asymptotic-symmetry framework for soft charges in gravity and gauge theory is given by Strominger [12]. The role of surface integrals as charge generators in the Hamiltonian formulation goes back to Regge-Teitelboim [13], and the appearance of nontrivial central charges in canonical realisations of asymptotic symmetries was first established in the seminal three-dimensional analysis of Brown-Henneaux [7]; the present paper exhibits an analogous central extension in a different geometric setting (timelike-boundary scalar field theory), together with its suppression at a phase boundary. Stratified symplectic spaces in the sense relevant here were introduced by Sjamaar-Lerman [14], generalising the classical (smooth) Marsden-Weinstein reduction [15]; the broader framework of stratified spaces with smooth structures and singular symplectic reductions is developed systematically by Pflaum [16] and by Ortega-Ratiu [17]. We use exactly the Sjamaar-Lerman notion of a stratification by smooth manifold pieces, each carrying a compatible presymplectic form, and ordered by closure. What is new in the present work is the combination: a stratification of the covariant phase space induced by an energetic (rather than topological or symmetry-reductive) selection rule on tangent vectors, together with the simultaneous algebraic effects (kernel enlargement and cocycle suppression) that this selection rule produces.

Remark 1.1(Scope of applicability). The coordinate-free characterisation of ℋ proved in Theorem 6.3—as the unique locus at which the symplectic rank of Ω Σ aug drops and the central extension K simultaneously vanishes—is intrinsic to the covariant phase space and makes no reference to the trigger functional P once the construction is complete. The framework therefore applies wherever a diffeomorphism-invariant scalar partitions a field theory into strata with different finite-energy variation classes. Strong-gravity interiors (with P a curvature scalar), dense-matter phases (with P an energy-density scalar), and condensed-matter systems with sharp phase boundaries (with P an order-parameter functional) are concrete settings; the structural mechanism is independent of the particular illustrative choice of trigger functional, within the class of models satisfying the stratification and admissibility assumptions of Section 2.

Status of the metric.Throughout this paper, the Lorentzian metric g μ ν is treated as fixed background data, not as a dynamical field. The action principle of Section 3 contains only the complex scalar Φ as a dynamical degree of freedom, and all variations hold g μ ν fixed ( δ g μ ν ≡ 0 ). The phrase “diffeomorphism-invariant scalar functional P [ Φ , g ] ” (Definition 2.2) is read in the standard equivariance sense: P [ φ * Φ , φ * g ] ( x ) = P [ Φ , g ] ( φ ( x ) ) under simultaneous pull-back of Φ and g by background diffeomorphisms. The extension to a dynamical metric—including gravitational backreaction and the disformal-coupling regime—is a natural direction for future work; here we work in the fixed-background regime throughout.

Geometricassumptions. Let ( ℳ , g μ ν ) be a smooth, oriented, four-dimensional Lorentzian manifold with smooth timelike boundary ∂ ℳ , globally hyperbolic in the sense appropriate to manifolds with timelike boundary [18]-[20]. The boundaryless case follows the classical formulation of Geroch [21], with smooth-Cauchy-hypersurface and metric-splitting refinements due to Bernal-Sánchez [22], both adapted in [19][20] to the timelike-boundary setting used here. Specifically:

(G1) ∂ ℳ is a smooth embedded timelike submanifold of codimension one, with Lorentzian induced metric.

(G2) There exists a smooth Cauchy temporal function τ : ℳ → ℝ whose level sets Σ τ are smooth spacelike Cauchy hypersurfaces with boundary, meeting ∂ ℳ transversally in smooth ( n − 2 ) -submanifolds ∂ Σ τ = Σ τ ∩ ∂ ℳ [19].

(G3) The foliation { Σ τ } is smooth with induced Riemannian metric h i j , uniformly bounded on compact τ -intervals.

(G4) Cobordism regions ℛ τ 1 , τ 2 ⊂ ℳ bounded by two slices and the corresponding boundary portion ℬ τ 1 , τ 2 are piecewise smooth manifolds with corners; Stokes’ theorem applies in the form ∂ ℛ = Σ τ 2 − Σ τ 1 + ℬ , with corner contributions controlled by 2.

We fix one such foliation throughout, denote a generic Cauchy slice by Σ, with future-directed unit normal n μ , induced Riemannian metric h i j , Levi-Civita connection D i , and corner ∂ Σ = Σ ∩ ∂ ℳ . (G1)-(G4) ensure that all bulk and boundary integrals below are well defined and that the Stokes-theorem manipulations in Corollary 3.1 are valid. Traces of Φ and its first derivatives onto ∂ ℳ are well-defined elements of H k ( ∂ ℳ ) for k ≥ 2 (Definition 2.1) by the trace theorem on Lipschitz domains.

The dynamical field is a complex scalar Φ : ℳ → ℂ . We work with the configuration space and admissible variations

Definition 2.1(Configuration space and admissible variations). The configuration space is

A tangent vector at Φ ∈ C —an admissible variation—is a smooth complex function

Throughout the paper we identify the tangent space T Φ C with the space of such variations.

The Sobolev regularity at the boundary ensures that every boundary integral encountered below is well defined: bulk integrals require only C ∞ regularity, while boundary integrals involving derivatives of the field—including the normal derivatives ∇ n t ^ and ∇ n θ that appear in the boundary symplectic density—require traces in H k for k ≥ 2 . The choice k = 2 suffices for the variational analysis below; the construction is unchanged for higher k .

It will be convenient throughout to use the polar decomposition of the field on the open set where it is non-vanishing,

with t ^ = | Φ | 2 ≥ 0 the squared modulus and θ the phase, defined modulo 2π. The polar decomposition is regular wherever Φ ≠ 0 , and we restrict the analysis to that open set; zeros of Φ are excluded from the polar chart and are treated by the regularization procedure in AppendixA; on each positive-amplitude stratum the polar decomposition is used without ambiguity. Variations in this representation decompose as δ   Φ = 1 2 t ^ − 1 / 2 e i θ δ t ^ + i t ^ e i θ δ θ , separating cleanly into amplitude and phase sectors. This separation is essential for the analysis of the boundary cocycle in Section 4 and the suppression mechanism in Section 6.

The entire construction rests on three assumptions, stated here precisely. The first identifies the geometric stratification, the second guarantees that the phase boundary is regular, and the third encodes the energetic selection rule that drives the algebraic transition.

Definition 2.2 (Trigger functional). A trigger functional is a scalar function P [ Φ , g ] : ℳ → ℝ constructed locally from the metric g μ ν , the field Φ, and a finite number of their covariant derivatives at each point. That is, there exists an integer r ≥ 0 such that

for some smooth function F . In particular, P ( x ) depends only on values at the single point x ; it is not an integral or otherwise nonlocal functional. We further require P to be smooth wherever ( g μ ν , Φ ) are smooth, and strictly diffeomorphism invariant: P [ φ * Φ , φ * g ] ( x ) = P [ Φ , g ] ( φ ( x ) ) for all diffeomorphisms φ : ℳ → ℳ .

Assumption1 (Phase Stratification).There exists a trigger functional P as in Definition 2.2 and a fixed threshold P ⋆ > 0 such that P partitions ℳ into two open regions and a separating hypersurface,

where ℳ reg = { P < P ⋆ } is the oscillatory (or regular) stratum, ℳ dense = { P ≥ P ⋆ } is the suppressed (or dense) stratum, and ℋ = { P = P ⋆ } is the phase boundary.

Assumption2 (Smooth Phase Boundary). The gradient ∇ μ P is non-vanishing on ℋ , so that ℋ is a smooth codimension-one embedded submanifold of ℳ by the regular level-set theorem.

Assumption3 (Energetically Stratified Variation Class). There exists a smooth, positive function κ : ℝ + → ℝ + , called the phase-stiffness coefficient, satisfying κ ( P ) → ∞ as P → P ⋆ + . A tangent vector δ   Φ ∈ T Φ C is admissible on the dense stratum if and only if it satisfies the finite-action condition

for every Cauchy slice Σ meeting ℳ dense , where h i j is the induced spatial metric on Σ, h i j its inverse, and D i the associated Levi-Civita connection. The integrand uses the positive-definite spatial metric, not the indefinite spacetime metric g μ ν , since the condition functions as an energy bound on a Cauchy slice. The zero-order term | δ θ | 2 is retained so that constant phase variations are also controlled by the dense-stratum finite-action condition. The constant global U ( 1 ) phase direction is then handled by the phase-redundancy convention adopted in this paper (see the discussion following Definition 3.6). On ℳ reg no such restriction on δ θ is imposed.

Lemma 2.1 (Finite-action phase suppression). Let Ω ⊂ Σ ∩ ℳ dense be open, and suppose t ^ ≥ t 0 > 0 on Ω. Let κ n ( P ) > 0 be a sequence of phase-stiffness coefficients satisfying κ n ( P ) → ∞ uniformly on compact subsets of Ω (representing the limiting regime κ ( P ) → ∞ as P → P ⋆ + ). If δ θ n ∈ H loc 1 ( Ω ) satisfies the uniform finite-action bound

then δ θ n → 0 in L loc 2 ( Ω ) . Consequently, the limiting finite-action admissible tangent space satisfies

Proof. Let K ⋐ Ω be compact. Since t ^ ≥ t 0 > 0 on Ω and κ n ( P ) → ∞ uniformly on K , the quantity

satisfies m n ( K ) → ∞ . By the assumed uniform bound, there is a constant C > 0 such that

for all n . Restricting to K and dropping the non-negative gradient term gives

Since m n ( K ) → ∞ , the right-hand side tends to zero, so δ θ n → 0 in L 2 ( K ) . Since K ⋐ Ω was arbitrary, the convergence is local. The limiting admissible tangent space therefore satisfies δ θ = 0 on ℳ dense .

Remark 2.1 (The selection rule is derived, not postulated). We emphasise that (4) is a theorem, not an assumption: it is the unique consequence of the finite-action condition (3) together with the divergence of κ . We have stated Assumption 3 in this form deliberately, to avoid the appearance that δ θ = 0 is imposed by fiat. The mathematical content is the diverging stiffness; the kinematic restriction follows. This distinction matters in Section 6, where the kernel enlargement and cocycle suppression are derived from (4) and would carry no force if (4) had merely been declared.

Remark 2.2(Two compatible descriptions: finite-energy admissibility and presymplectic quotient). The condition δ θ = 0 characterises the finite-energy sector of the tangent space at Φ ∈ S dense . There are two equivalent ways to use this fact in the symplectic analysis, and we will need both. (a) In the finite-energy admissibility description, the dense-stratum tangent space is restricted to those δ   Φ with δ θ = 0 on ℳ dense ; phase-only variations supported in ℳ dense are not finite-energy admissible and are excluded. (b) In the presymplectic quotient description, one starts with the unconstrained tangent bundle to S , allows phase-only variations as formal tangent vectors, identifies them as null directions of Ω Σ aug on S dense , and quotients them out. The two descriptions agree on the reduced phase space

. Description (a) is the one we use when computing finite-energy actions; description (b) is the one used in Theorem 6.1 and the kernel-enlargement argument, since the statement that a variation lies in ker Ω Σ aug is, by definition, a statement about its pairing under the unconstrained presymplectic form. We will switch between the two as convenient and indicate which is in use when the distinction matters.

We use the term stratified phase space in the precise sense of Sjamaar-Lerman [14]: a decomposition of a presymplectic space into a finite collection of smooth manifold strata, each carrying a compatible presymplectic form, with the strata ordered by closure inclusion. The broader theory of stratified spaces with smooth structures, including their de Rham theory and the analytic machinery used in singular symplectic reductions, is developed by Pflaum [16]; the systematic treatment of momentum maps and Hamiltonian reduction on such spaces, including singular reductions in the finite-dimensional setting, is given by Ortega-Ratiu [17]. In the present work the stratification is two-piece,

(5)

where each stratum is a smooth (infinite-dimensional) manifold equipped with the restriction of the augmented presymplectic form Ω Σ aug constructed in Section 3. The corresponding admissible tangent spaces satisfy the strict inclusion

(6)

proved as a consequence of Lemma 2.1 together with the kernel-enlargement theorem (Theorem 6.1). No Whitney regularity beyond smoothness of each stratum is required; in particular, the stratification does not depend on any embedding into a larger smooth manifold. This is the precise sense in which the covariant phase space admits a two-stratum decomposition as a piecewise-smooth presymplectic space whose presymplectic structure degenerates along the constrained stratum.

The dynamics is specified, in polar variables Φ = t ^ e i θ = ρ e i θ , by the stiffness-coupled Lagrangian

with κ ( P ) → ∞ as P → P ⋆ + as in Assumption 3, and with P [ Φ , g ] prescribed (Definition 2.2); the variation of P does not contribute to the field variation in this paper. The standard complex-scalar Lagrangian

is the case κ ≡ 1 (up to the conventional overall factor of 1 2 , which we absorb into V ). For finite κ the Euler-Lagrange equations from (7) remain smooth across ℋ ; the divergence κ → ∞ on ℳ dense is approached through a sequence κ n → κ . Variation of (7) gives, after integration by parts,

with bulk Euler-Lagrange operator (in polar variables) coming from (7). The presymplectic potential current produced by varying the trigger-coupled phase-sector Lagrangian is

and the corresponding phase-sector presymplectic current is

On the regular stratum, after the normalisation κ = 1 , expressions (10)-(11) reduce to the standard complex-scalar presymplectic potential,

in non-polar variables, with bulk Euler-Lagrange operator

On the dense stratum, finite-action admissibility forces δ θ = 0 in the stiffness limit, so the divergent phase-sector contribution in (11) has zero pairing against admissible dense-stratum variations. The displayed standard form (12) should therefore be understood as the effective admissible-sector presymplectic potential after imposing the fixed-background finite-action condition: it is correct on ℳ reg exactly, and on ℳ dense up to terms that vanish on admissible variations. We use (12) throughout Sections 3-4 for the boundary-flux and augmentation analysis on this understanding.

Proposition 3.1(Action-level origin of the finite-action selection rule). The finite-action condition (3) of Assumption 3 arises as the natural energetic restriction associated with (7) in the prescribed- P setting. The phase-sector contribution of (7) to the second variation on Σ ∩ ℳ dense , on a pure phase variation δ   Φ with δ ρ = 0 , is

the phase-sector kinetic energy in δ θ . As κ → ∞ on ℳ dense , finiteness of S κ ( 2 ) forces δ θ → 0 in L loc 2 on ℳ dense , recovering the conclusion of Lemma 2.1. Equivalently, in the κ → ∞ limit, the term 1 2 κ ( P ) ρ 2 ∇ θ ⋅ ∇ θ acts as a Lagrange multiplier enforcing ∇ θ = 0 on ℳ dense ; configurations with ∇ θ ≠ 0 there are removed from the finite-action space.

Proof. The Lagrangian (7) is bilinear in ∇ θ in its phase-sector contribution. The second variation at fixed ρ and prescribed P is S κ ( 2 ) [ δ θ ] = − 1 2 ∫ ℳ κ ( P ) ρ 2 g μ ν ∇ μ δ θ   ∇ ν δ θ − g d 4 x . Decomposing ∇ μ on a Cauchy slice and reducing to the slice volume form gives the spatial-gradient integrand κ ( P ) ρ 2 h i j D i δ θ D j δ θ in (14); time-derivative and boundary contributions are non-negative on a spacelike slice and do not weaken the divergence-driven suppression. The zero-order | δ θ | 2 term in (3) ensures that constant phase variations are also controlled on the dense stratum (cf. Assumption 3); the constant global U ( 1 ) phase direction is handled by the phase-redundancy convention adopted in this paper (see the discussion following Definition 3.6).

Remark 3.1 (Prescribed vs.dynamical trigger). When P [ Φ , g ] is allowed to vary dynamically—so that δ P contributes through δ κ ( P ) = κ ′ ( P ) δ P —additional terms enter the presymplectic potential and must be controlled by a subleading-correction hypothesis. The dynamical case, including a persistence theorem for phase suppression under admissible dynamical triggers, is a natural extension of the framework and is left for future work. For the prescribed case treated here, no such terms arise.

The solution space is

Definition 3.1 (Solution space).

We will view S as a (formal) submanifold of C , with tangent space at Φ ∈ S given by the linearised solutions δ   Φ of δ E ( Φ ) = 0 .

The presymplectic current is the antisymmetrisation of θ μ over two independent variations:

Definition 3.2 (Presymplectic current). For two tangent vectors δ 1 , δ 2 ∈ T Φ S ,

Substituting (12) and using δ 1 δ 2 = δ 2 δ 1 on configuration space, one obtains the explicit expression

The presymplectic form on Σ is then

Definition 3.3 (Presymplectic form).

Theorem 3.1 (On-shell conservation). For Φ ∈ S and δ 1 , δ 2 ∈ T Φ S , one has ∇ μ ω μ ( δ 1 , δ 2 ) = 0 .

Proof. Take the divergence of (15). Using the Leibniz rule,

The four cross terms cancel pairwise after exchanging 1 ↔ 2 in the antisymmetrised expression. The remaining terms are ( δ 1   □   Φ † ) δ 2 Φ − ( δ 2   □   Φ † ) δ 1 Φ + c . c . Linearising E ( Φ ) = 0 gives δ   □   Φ † = δ ( V ′ Φ † ) = V ′ δ   Φ † + V ″ ( Φ δ   Φ † + Φ † δ   Φ ) Φ † , and substitution shows that the remaining terms are symmetric in 1 ↔ 2 , hence vanish under antisymmetrisation. Hence ∇ μ ω μ = 0 on shell.

Corollary 3.1(Hypersurface dependence in the presence of boundary). Let Σ 1 , Σ 2 bound a spacetime region ℛ ⊂ ℳ with ∂ ℛ = Σ 2 − Σ 1 + ℬ , where ℬ = ℛ ∩ ∂ ℳ is the timelike boundary segment. Then by Theorem 3.1 and Stokes’ theorem,

where n ˜ μ is the outward normal to ℬ .

Proposition 3.2 (Boundary flux obstruction). If ω μ n ˜ μ | ℬ ≡ 0 for some pair of tangent vectors, then Ω Σ is not independent of the choice of Cauchy hypersurface.

Proof. Direct from (16): a non-vanishing right-hand side forces the left-hand side to differ.

This is the obstruction we now resolve.

To restore hypersurface independence, we augment Ω Σ by a 2-form supported on the boundary ∂ Σ = Σ ∩ ∂ ℳ .

Definition 3.4 (Edge symplectic form). The edge symplectic form is

where α is a bilinear, antisymmetric form on T Φ S × T Φ S taking values in 2-forms on ∂ Σ , satisfying the matching condition

The condition (17) is exactly what is needed for Ω ˜ ∂ Σ to absorb the flux on the right-hand side of (16). The existence of such an α is not automatic; we exhibit one explicitly below. The construction proceeds in two stages, the first specifying a boundary polarisation in the polar variables ( θ , t ^ ) , and the second exhibiting the matching density.

Definition 3.5(Polar-amplitude boundary conditions). At ∂ Σ , with n the outward unit normal, the polar-amplitude boundary conditions are

Condition (18) is a Neumann condition on the amplitude: the normal gradient of t ^ is fixed at ∂ Σ . Condition (19) is a mixed Dirichlet-Neumann condition on the phase, weighted by t ^ 2 to capture the conserved phase current. Tangent vectors δ   Φ in T Φ S are required to satisfy these conditions throughout the boundary analysis below.

The choice (18)-(19) is one polarisation among several admissible choices in the IWZ framework; we discuss its scope and uniqueness in Proposition 3.4 below.

Proposition 3.3(Explicit boundary density). In the polar decomposition (1), and under the boundary conditions (18)-(19) of Definition 3.5, the boundary symplectic density

satisfies the matching condition (17).

Proof. We compute the polar form of the presymplectic current and identify which terms reduce to δ α on the boundary.

Step 1: polar form of θ μ . Substituting Φ = t ^ e i θ into the field gradient gives

with the conjugate analogous. Substituting into the presymplectic potential current θ μ = ∇ μ Φ † δ   Φ + ∇ μ Φ δ   Φ † and using e − i θ e i θ = 1 , the imaginary cross-terms cancel between the two summands, leaving

This is the manifestly real polar expression for θ μ , decomposed cleanly into amplitude and phase sectors.

Step 2: polar form of ω μ . The presymplectic current is ω μ ( δ 1 , δ 2 ) = δ 1 θ μ ( δ 2 ) − δ 2 θ μ ( δ 1 ) . Acting with δ 1 on (21) and antisymmetrising in 1 ↔ 2 , the symmetric (diagonal) terms cancel and the surviving structure is

The first group is a pure-amplitude bracket, the second a pure-phase bracket, and the third is the mixed ( θ , t ^ ) bracket whose structure matches the proposed α in (20).

Step 3: contraction with the boundary normal. On ∂ ℳ , contracting (22) with the outward normal n ˜ μ gives, with ∇ n : = n ˜ μ ∇ μ ,

Step 4: identification with δ α . With α as in (20), the variation is

Within the polar polarisation—in which θ and t ^ are treated as the independent boundary variables and the field-space variation δ commutes with the variations δ 1 , δ 2 in the standard way—this evaluates on the configuration to a sum involving ∇ n θ and ∇ n t ^ (the normal derivatives that appear because δ α is computed against the bulk symplectic flux). Specifically, δ α produces precisely the third bracket of (23), − 2 ( ∇ n θ ) [ δ 1 θ   δ 2 t ^ − δ 2 θ   δ 1 t ^ ] , after the boundary conditions (18)-(19) are imposed. This is the term that exhibits the bilinear ( θ , t ^ ) structure essential to the cocycle analysis.

Step 5: cancellation of the remaining brackets under (18)-(19). The first bracket of (23) is the amplitude-sector contribution. Under the Neumann condition (18), δ ( ∇ n t ^ ) | ∂ Σ = 0 , so ( ∇ n δ i t ^ ) = δ i ( ∇ n t ^ ) = 0 on ∂ Σ , and the first bracket vanishes pointwise. The second bracket is the phase-sector contribution. Under the mixed condition (19), δ ( t ^ 2 ∇ n θ ) | ∂ Σ = 0 implies t ^ 2 δ i ( ∇ n θ ) + 2 t ^ ∇ n θ   δ i t ^ = 0 on ∂ Σ . Substituting δ i ( ∇ n θ ) = − 2 t ^ − 1 ( ∇ n θ ) δ i t ^ into the second bracket yields

which is + 4 ( ∇ n θ ) [ δ 1 θ   δ 2 t ^ − δ 2 θ   δ 1 t ^ ] . Combining with the third bracket of (23) (which is − 2 ( ∇ n θ ) [ ⋯ ] ), the total mixed-sector contribution under the boundary conditions is + 2 ( ∇ n θ ) [ δ 1 θ   δ 2 t ^ − δ 2 θ   δ 1 t ^ ] . This is the surviving term, and up to an overall normalisation constant absorbed into the conventions of Ω ˜ ∂ Σ , it is exactly δ α as computed in Step 4. Hence

which is the matching condition (17).

The decomposition into θ - and t ^ -sectors visible in (20) is the structural feature that drives the cocycle suppression argument in Section 6: the cocycle K inherits the same bilinear ( θ , t ^ ) -structure, and the dense-stratum condition δ θ = 0 thereby kills it directly.

The boundary 2-form Ω ˜ ∂ Σ is defined by the matching condition (17), which determines its variation, but does not by itself fix α uniquely: there is the well-known Iyer-Wald-Zoupas freedom [4][11] to shift the presymplectic potential by an exact piece, θ ↦ θ + d β for any local ( n − 2 ) -form β on ℳ . We do not attempt a general resolution of this ambiguity. The polar-amplitude boundary conditions of Definition 3.5 remove the freedom within the polar polarisation, in the sense made precise in the following proposition.

Proposition 3.4 (Removal of IWZ freedom within the polar polarisation). Within the polar-amplitude polarisation in which ( θ , t ^ ) are the independent boundary variables, and under the boundary conditions (18)-(19) of Definition 3.5, the boundary symplectic density α defined by Proposition 3.3 is unique up to terms that contribute trivially to the matching (17). In particular, the IWZ freedom within this polarisation is removed.

Proof. The proof of Proposition 3.3 (Steps 3-5) showed that under (18)-(19), the boundary contraction ω μ n ˜ μ | ∂ Σ reduces, modulo terms that vanish by the boundary conditions, to a multiple of the mixed bracket δ 1 θ   δ 2 t ^ − δ 2 θ   δ 1 t ^ . Any IWZ shift of α that is linear in amplitude- and phase-normal-derivative variations is thereby projected onto a vanishing contribution. Boundary modifications outside this linear class—in particular, polarisations that take different combinations of ( θ , t ^ , ∇ n θ , ∇ n t ^ ) as the independent boundary variables—are admissible in the IWZ framework but lead to different boundary symplectic densities; we do not address them here.

Remark 3.2 (Scope of Proposition 3.4). We are not claiming a general resolution of the Iyer-Wald-Zoupas ambiguity. The IWZ freedom permits a wide class of boundary modifications, and the proposition addresses one class—shifts of α within the polar-amplitude polarisation that are linear in amplitude- and phase-normal-derivative variations. Within this class, the boundary conditions (18)-(19) fix α to the form (20). Other choices of boundary polarisation are consistent with the formalism but lead to different boundary symplectic densities, with corresponding modifications of the cocycle K . The choice (20) is the one that admits the cleanest cocycle representation (30) and is suppressed cleanly on the dense stratum by the finite-energy selection rule.

The augmented presymplectic form is

Theorem 3.2(Hypersurface independence). Under (17), the augmented form Ω Σ aug is hypersurface independent: Ω Σ 2 aug = Ω Σ 1 aug for any two Cauchy hypersurfaces Σ 1 , Σ 2 .

Proof. Combine (16) and (17). The deformation of Ω Σ between Σ 1 and Σ 2 is exactly cancelled by the variation of Ω ˜ ∂ Σ :

where the final equality uses the matching condition integrated over ℬ (Stokes between ∂ Σ 1 and ∂ Σ 2 ).

The augmentation is therefore not optional but mathematically necessary whenever the boundary flux is non-vanishing.

The presymplectic form Ω Σ aug is antisymmetric and closed but generally degenerate; passing to the symplectic phase space requires quotienting by its kernel.

Definition 3.6 (Kernel of Ω Σ aug )

Phase-redundancy convention for the global U ( 1 ) direction. In this paper the global U ( 1 ) phase direction Φ ↦ e i λ Φ , with λ ∈ ℝ constant, is treated under a phase-redundancy convention for purposes of constructing the reduced phase space associated with the stiffness-induced boundary transition. This convention means that constant phase directions are not retained as independent dense-stratum degrees of freedom when the finite-action regulator suppresses them. The convention is a choice of reduced phase-space description for the present structural theorem, not a claim that every global U ( 1 ) symmetry in scalar field theory is physically gauge in all contexts. If global U ( 1 ) is instead treated as a physical charge symmetry, the same finite-action regulator suppresses constant dense-stratum phase variations, while the charge interpretation must be adjusted accordingly; the structural conclusion of the theorem is unchanged.

Definition 3.7(Reduced phase space). The reduced phase space is defined by quotienting the solution space by the genuine degeneracy distribution of the augmented presymplectic form,

Under the phase-redundancy convention adopted in this paper, the constant global phase direction is included in the quotient whenever it lies in the presymplectic kernel or is removed by the finite-action dense-stratum admissibility condition. Under an alternative reading in which global U ( 1 ) is treated as a physical charge symmetry, the quotient should be understood as removing only genuine kernel directions, with the associated charge retained on the regular stratum. Either reading gives the same structural conclusion below: dense-stratum admissibility eliminates phase variations supported in ℳ dense and enlarges the presymplectic degeneracy sector accordingly.

Theorem 3.3 (Symplectic reduction). Ω Σ aug descends to a non-degenerate symplectic form on

.

Proof. Antisymmetry and closure of Ω Σ aug are inherited from Ω Σ and Ω ˜ ∂ Σ . Non-degeneracy on the quotient is by construction: any direction along which Ω Σ aug pairs to zero is, by Definition 3.6, in the kernel and is identified to zero in the quotient. The result is a closed, non-degenerate, antisymmetric bilinear form, i.e., a symplectic form on

.

Remark 3.3(Recovery of standard covariant phase space). When ℳ dense = ∅ —e.g. when P ( x ) < P ⋆ everywhere, so Assumption 3 imposes no restriction on δ θ —the construction of this section reduces identically to the standard Lee-Wald [3] and Iyer-Wald [4] covariant phase-space formalism on a spacetime with boundary. The augmented form Ω Σ aug coincides with the standard presymplectic form, and

carries the full complement of phase-wave degrees of freedom together with a centrally extended boundary algebra. The unstratified theory is therefore the P ⋆ → ∞ limit of the construction; every formula of this paper specialises correctly to the Lee-Wald / Iyer-Wald result in that limit. The stratified theory is a strict extension, not a replacement.

The Poisson structure on

is determined by Ω Σ aug through the standard prescription. For a smooth functional

, the Hamiltonian vector field X F is defined by

existing and being unique by Theorem 3.3. The Poisson bracket of two functionals is

Proposition 3.5 (Poisson algebra). The bracket (26) is bilinear, antisymmetric, satisfies the Leibniz rule, and obeys the Jacobi identity.

Proof. Bilinearity and antisymmetry are immediate from the corresponding properties of Ω Σ aug . The Leibniz rule follows from the chain rule applied to δ ( F G ) = G δ F + F δ G in (25), giving X F G = G X F + F X G and hence { F G , H } = G { F , H } + F { G , H } . The Jacobi identity is equivalent to closure of Ω Σ aug via the standard identity

and d Ω Σ aug = 0 since Ω Σ aug is the variation of an action functional augmented by a closed boundary form.

In a coordinate chart on Σ adapted to a time foliation, with conjugate momentum π = ∂ t Φ † , the canonical equal-time bracket takes the standard form

with all other brackets vanishing modulo boundary terms. These relations establish the full canonical structure on each stratum.

Let g ∂ denote a Lie algebra of infinitesimal transformations acting on boundary data on ∂ ℳ . For ξ ∈ g ∂ , the induced variation of the field is denoted δ ξ Φ ; in the polar decomposition this decomposes as δ ξ Φ = 1 2 t ^ − 1 / 2 e i θ δ ξ t ^ + i t ^ e i θ δ ξ θ .

Definition 4.1(Hamiltonian generator). A functional

is a Hamiltonian generator of the boundary symmetry ξ ∈ g ∂ if

The existence of such a Q ξ is not automatic: the right-hand side of (28) must be an exact 1-form on

, equivalently the linearisation δ Q ξ must satisfy the integrability condition δ ( δ Q ξ ) = 0 .

Definition 4.2 (Integrability). The expression δ Q ξ is integrable on

if δ ( δ Q ξ ) = 0 as a 2-form on

.

Proposition 4.1 (Integrability of boundary charges). If Ω Σ aug is closed and δ ξ preserves the boundary conditions (18)-(19), then δ Q ξ is integrable, and hence Q ξ exists.

Proof. Computing δ ( δ Q ξ ) from (28) via Cartan’s magic formula,

The second term vanishes because Ω Σ aug is closed (Theorem 3.3). The first term is the Lie derivative of the augmented form along the symmetry direction; if δ ξ preserves the boundary polarisation (18)-(19), then ℒ δ ξ Ω Σ aug = 0 . Hence δ ( δ Q ξ ) = 0 and Q ξ exists by the Poincaré lemma applied to the (formally infinite-dimensional) phase space.

The boundary symmetries close under the bracket of vector fields on C : [ δ ξ , δ η ] = δ [ ξ , η ] . The corresponding algebra of charges, however, generally fails to close exactly—it closes only up to a 2-cocycle.

Proposition 4.2(Boundary algebra with central extension). Suppose [ δ ξ , δ η ] = δ [ ξ , η ] . Then under the Poisson bracket (26),

where K ( ξ , η ) is a bilinear, antisymmetric expression depending only on the boundary data and not on the bulk values of the field.

Proof. By (26) and (28),

where the second equality uses the definition of Q η as Hamiltonian for δ η . On the other hand, Q [ ξ , η ] is the Hamiltonian for δ [ ξ , η ] = [ δ ξ , δ η ] , so

using closure of Ω Σ aug and preservation of boundary conditions. Hence δ ξ Q η − Q [ ξ , η ] is a δ -closed quantity on

, i.e. a constant on each connected component, depending only on ξ and η and not on the field. We denote this constant by K ( ξ , η ) , giving (29). Bilinearity follows from linearity of the construction in ξ and η ; antisymmetry follows from antisymmetry of Ω Σ aug . The constant depends only on boundary data because the bulk part of Ω Σ aug generates the closed bracket exactly; the offset arises entirely from Ω ˜ ∂ Σ .

The term K ( ξ , η ) is the central extension of the boundary charge algebra: it is the failure of the map ξ ↦ Q ξ to be a Lie algebra homomorphism, and it arises because the boundary symplectic density Ω ˜ ∂ Σ contributes a piece that the bulk equations cannot absorb.

Theorem 4.1 (Explicit form of the cocycle). With the boundary symplectic density (20) of Proposition 3.3,

Proof. By Proposition 4.2, K ( ξ , η ) is the boundary contribution to { Q ξ , Q η } − Q [ ξ , η ] . Substituting the explicit boundary symplectic density (20) into Ω ˜ ∂ Σ ( δ ξ , δ η ) gives

and tracking the construction of Proposition 4.2 shows that this is exactly the central-extension contribution. The bulk part of Ω Σ aug produces Q [ ξ , η ] exactly via the standard Hamiltonian-action computation, leaving (30) as the residual.

The representation (30) exposes two structural features that we will use heavily in Section 6: K is bilinear in the phase-sector variation δ θ and the amplitude-sector variation δ t ^ , and it is supported entirely on ∂ Σ . Both features will be essential for the suppression argument.

Definition 4.3 (2-cocycle). A bilinear antisymmetric form K : g ∂ × g ∂ → ℝ is a 2-cocycle on g ∂ if

for all ξ , η , ζ ∈ g ∂ .

Theorem 4.2 (Cocycle condition). K as in (30) is a 2-cocycle on g ∂ .

Proof. Apply the Jacobi identity for the Poisson bracket of three charges Q ξ , Q η , Q ζ . Cyclic permutation gives, after using (29),

The Jacobi identity for the Poisson bracket (Proposition 3.5) makes the left-hand side vanish. The right-hand side, using (29) once more and the Jacobi identity in g ∂ (which gives [ [ ξ , η ] , ζ ] + cyclic = 0 ), reduces to K ( [ ξ , η ] , ζ ) + cyclic . Equating to zero gives the cocycle condition.

The boundary algebra is therefore represented up to a central extension defined by a 2-cocycle, in the standard sense of Lie algebra cohomology.

Remark 4.1(On the cohomological status of K ). We do not claim here that K represents a non-trivial cohomology class in H 2 ( g ∂ , ℝ ) . Establishing non-triviality would require fixing the symmetry algebra g ∂ explicitly and showing that K cannot be written as a coboundary K ( ξ , η ) = f ( [ ξ , η ] ) for any linear

functional f on g ∂ . The structural results of this paper require only the weaker and explicit fact that the displayed boundary cocycle (30) is bilinear in phase- and amplitude-sector variations, and therefore vanishes on the dense stratum once admissible phase variations are suppressed. Whether K also represents a non-trivial cohomology class—a question of independent interest, depending on the choice of g ∂ —is not needed for any conclusion below.

For technical control of ultraviolet or singular behaviour at ∂ Σ —e.g. if the boundary admits asymptotic limits or sharp corners—one may regulate the boundary symplectic density by a smooth, bounded multiplier R ( χ ; ϵ , Ξ 1 , Ξ 2 ) , with χ a coordinate along ∂ Σ and ϵ , Ξ 1 , Ξ 2 regulator parameters. The regulated edge form is

and the regulated augmented form is Ω Σ aug , ( R ) = Ω Σ + Ω ˜ ∂ Σ ( R ) . Multiplication of the integrand by a smooth bounded function preserves bilinearity, antisymmetry, and closure, so Ω Σ aug , ( R ) remains closed and hypersurface independent for any fixed regulator. The regulated cocycle K R depends continuously on the regulator parameters and reduces to K in the limit R → 1 . We use the regulated form only as a technical device: all algebraic statements in Section 6 are stated for the unregulated form and are stable under the regulator limit.

The trigger functional P [ Φ , g ] of Definition 2.2 can take any of a wide range of forms; the structural mechanism is independent of the particular illustrative choice, within the class of models satisfying Definition 2.2 and Assumptions 2-3. Three natural choices arise in physical contexts.

Curvature scalar (strong-gravity setting). The Kretschmann scalar

is a natural choice in strong-gravity environments. It is a scalar curvature invariant—not a complete invariant of the Riemann tensor (the full set of independent algebraic curvature invariants in four dimensions is larger; see e.g. the Carminati-McLenaghan invariants), but it does provide a coordinate-free measure of curvature magnitude that is non-vanishing wherever spacetime curvature is dynamically significant and is well-defined even on non-stationary backgrounds. The threshold P ⋆ then selects regions whose curvature scale exceeds a critical value, giving the dense stratum the precise geometric meaning of a strong-curvature interior. The use of the Kretschmann scalar here is illustrative; the formal construction requires only a scalar trigger P whose threshold set is a regular level surface, and other curvature invariants may be substituted.

Energy-momentum scalar (dense-matter setting). The square of the stress-energy tensor,

is a natural choice when the stratification is driven by matter density rather than spacetime geometry. It picks out regions of high energy density and, with a suitable threshold, models phase transitions in dense matter (e.g. between hadronic and quark phases in compact-star interiors).

Order-parameter scalar (condensed-matter setting). When Φ has the interpretation of an order parameter of a phase transition, P may be taken as a local function of Φ and its derivatives that distinguishes the phases—for instance P = | Φ | 2 or a more elaborate Landau-type scalar. The threshold P ⋆ then realises the phase boundary as a level set of the order parameter.

All subsequent results hold for any P satisfying Definition 2.2 together with Assumptions 2 and 3. The Kretschmann scalar is the canonical illustration in the gravitational setting; in what follows, the reader may safely substitute any choice from the above list (or any other admissible one) without altering a single proof.

Recall from (2) the partition ℳ = ℳ reg ∪ ℋ ∪ ℳ dense induced by P . The solution space inherits a corresponding stratification.

Definition 5.1 (Stratified solution space).

giving S = S reg ∪ S dense .

The Euler-Lagrange equations (13) are unmodified across this partition: S is the joint zero locus of E ( Φ ) = 0 and E ( Φ † ) = 0 globally on ℳ , irrespective of which stratum the field configuration falls into. The stratification affects only the tangent structure of S —which directions δ   Φ are admissible at which configurations—not the configurations themselves.

The relevant restriction on tangent directions is exactly that of Assumption 3, made concrete in Lemma 2.1. We re-state it here in the form most useful for the symplectic analysis.

Definition 5.2(Dense-time admissibility). A tangent vector δ   Φ ∈ T Φ S at a configuration Φ ∈ S dense is dense-time admissible if, in the polar decomposition (1),

On the regular stratum, no such restriction is imposed.

By Lemma 2.1, condition (32) is the unique consequence of the finite-action condition (3); we have not added any new content. Definition 5.2 is simply the form of the selection rule that makes its impact on the presymplectic structure most transparent.

Remark 5.1(Interpretation: selection rule, not axiom). We re-emphasise the point of Remark 2.1: the dense-time admissibility condition (32) is a derived consequence of the diverging stiffness κ ( P ) → ∞ , not a freely-imposed axiom. The mathematical content lies in Assumption 3 and Lemma 2.1. The Euler-Lagrange equations are not modified; only the finite-energy sector of the tangent space at each Φ ∈ S dense is restricted, and that restriction is forced by the action functional itself. The phase boundary ℋ is therefore not a singular dynamical surface but a locus where the finite-energy variation class changes rank.

Recall from (15), after substituting the polar decomposition, that the presymplectic current contains an amplitude-sector contribution proportional to δ t ^ and a phase-sector contribution proportional to δ θ . Inside ℳ dense , the dense-time admissibility condition kills the phase-sector contribution.