The companion paper [1] (hereafter referred to as Paper I) established a structural theorem about complex scalar field theory on a globally hyperbolic Lorentzian manifold ( ℳ , g ) with boundary, in the presence of a diffeomorphism-invariant scalar trigger functional P [ Φ , g ] that partitions ℳ across a level set ℋ = { P = P ⋆ } into a regular stratum ℳ reg = { P < P ⋆ } and a high-stiffness (dense) stratum ℳ dens = { P ≥ P ⋆ } . In polar variables Φ = t ^ e i θ , finite-action admissibility under a divergent phase-stiffness coefficient κ ( P ) → ∞ as P → P ⋆ + was shown to enforce δ θ = 0 on ℳ dens . This single energetic condition produced two algebraic effects on the augmented presymplectic form Ω Σ aug = Ω Σ + Ω ˜ ∂ Σ : enlargement of ker Ω Σ aug in the phase sector, and vanishing of the explicitly-represented boundary 2-cocycle K dens = 0 .

Paper I deliberately stopped short of four claims. We list them now, since each is addressed by a theorem or an explicit structural assumption in the present paper.

(1) The trigger functional was treated as prescribed. Paper I assumed P was given as an external scalar functional on ℳ ; its variations did not enter the symplectic computation. For physical applications—where P is typically a dynamical scalar built from the field Φ and the metric g μ ν —the variation δ P produces additional contributions to the presymplectic potential through δ κ ( P ) = κ ′ ( P ) δ P . Whether the finite-action selection rule survives these additional variations was not addressed.

(2) The stratification was described informally as “in the sense ofSjamaar-Lerman.” Paper I cited the Sjamaar-Lerman stratified symplectic reduction framework [2] and observed that the two-stratum decomposition of the covariant phase space had the right informal shape, but did not prove a precise field-theoretic analogue of the Sjamaar-Lerman theorem.

(3) The cocycle was called “explicitly represented” rather than “non-trivial.” The K of Paper I was given by an explicit boundary integral (linear in phase- and amplitude-sector variations), but it was not shown to define a non-trivial class in the second Lie algebra cohomology H 2 ( g ∂ , ℝ ) . Paper I disclaimed the cohomological non-triviality claim and noted that the structural arguments did not require it.

(4) The Phase BoundaryCharacterisationTheorem was a strictequivalence, but its converse direction relied on tacit assumptions about which kernel directions of Ω Σ aug counted. Specifically, the statement that ℋ is the unique locus of phase-sector kernel enlargement and cocycle vanishing was qualified to the “stiffness-induced” mechanism, but the precise hypotheses excluding accidental degeneracies (gauge, zeros of Φ, topological sectors, etc.) were not isolated.

The purpose of the present paper is to close each of these four gaps. Sections 3 through 8 address them in order.

Remark 1.1 (Physical interpretation of ℋ ). Throughout, ℋ is treated as a mathematical object: the regular level set of a scalar trigger functional. In the physical settings motivating Paper I and the present work, ℋ models phase-transition surfaces of various kinds—threshold surfaces in dense-matter or condensed-matter systems where a stiffness modulus diverges, and, in the gravitational setting, surfaces playing the role of black-hole horizons in covariant phase-space descriptions. The mathematical results below are stated and proved without reference to any specific physical interpretation; the boundary-local converse theorem (Theorem 8.1) gives, in the stated class, an algebraic characterisation that is independent of which physical surface ℋ models in a given application.

The principal new result is a boundary-local converse direction of the Phase Boundary Characterisation Theorem, made rigorous within a precisely-specified class of mechanisms. Once the necessary scaffolding is in place—the dynamical-trigger extension (Section 3), the global reduction theorem (Section 4), the cocycle non-triviality theorem (Section 6), and the no-accidental-degeneracy framework (Section 7)—the converse takes the following form, which we state here informally and prove rigorously as Theorem 8.1.

Main Converse Theorem (Boundary-Local Converse Phase BoundaryCharacterisation).Let the field theory satisfy Assumptions 7.2-7.3 of Section 8 (the pure stiffness-induced class together with regularity, Hamiltonian boundary-algebra, local-separation, and unique-threshold hypotheses). Then in a neighbourhood U of any point x 0 ∈ ℋ ∩ ∂ ℳ , restricted to boundary-accessible points x ∈ U ∩ ∂ Σ (made precise in Definition 8.1), the threshold hypersurface ℋ is locally characterised by the simultaneous algebraic signatures of phase-sector presymplectic rank reduction, formal kernel enlargement under Ω Σ aug , and dense-boundary 2-cocycle suppression—and the converse holds: any boundary-accessible point exhibiting these signatures together must lie on ℋ .

This converse should be read carefully. It is not a classification of all degeneracy loci of Ω Σ aug in arbitrary field theories; gauge directions, zeros of Φ, topological sectors, and other unrelated mechanisms can produce kernel directions whose loci are unrelated to ℋ . Nor is it a bulk classification: the cocycle is a boundary integral, and we do not extend the converse to bulk points without an additional propagation argument. The claim is rather that, within the restricted class of theories satisfying Assumptions 7.2-7.3, the algebraic signatures of phase suppression genuinely determine the threshold hypersurface at boundary-accessible points. The role of the assumptions is to delete the accidental degeneracies and to ensure that vanishing of a boundary integral implies pointwise vanishing distributionally; the work of the theorem is to show that, having done so, the algebraic data of Ω Σ aug recover ℋ ∩ ∂ Σ .

The results below are conditional closure results. The paper does not claim an unconditional classification of all possible degeneracy loci in covariant phase space. Rather, it identifies a precise class of pure stiffness-induced phase-sector mechanisms and proves that, within this class and under explicit regularity, Hamiltonian boundary-algebra, local-separation, unique-threshold, and no-accidental-degeneracy assumptions, the phase boundary is recoverable from its algebraic signatures at boundary-accessible points. Each assumption is stated explicitly and used substantively in at least one proof; the cumulative effect is to delete every alternative mechanism by which the algebraic signatures could occur, leaving the stiffness-induced mechanism as the unique source.

Paper I proves the local phase-boundary mechanism: finite-action admissibility suppresses phase variations on the dense stratum, enlarges the phase-sector presymplectic kernel, and suppresses the explicit boundary cocycle. The present paper does not re-prove that local mechanism as a new result; it is recalled in Section 2 (Theorem 2.1) and the relevant boundary inputs of Paper I are restated explicitly there for self-containedness (see the imported boundary data following Section 2). Instead, the present paper proves four closure statements that Paper I deliberately left open: persistence of the selection rule under admissible dynamical triggers, a global two-stratum presymplectic reduction, cohomological non-triviality of the regular-boundary cocycle under an explicit test-algebra criterion, and a boundary-local converse theorem under no-accidental-degeneracy hypotheses. The division of labour is therefore strict: every appeal below to “the local suppression theorem,” “the polar-amplitude boundary conditions,” or “the Hamiltonian integrability criterion” refers to an input recalled verbatim in Section 2, while every theorem carrying a number in Sections 3-8 is a new conditional result of the present paper.

Section 2 fixes notation and recalls the precise local result of Paper I. Section 3 treats dynamical trigger functionals. Section 4 constructs the global two-stratum presymplectic reduction. Section 5 sets up the boundary algebra and reviews the explicit cocycle. Section 6 proves the cohomological non-triviality theorem and constructs an explicit abelian amplitude-phase test subalgebra. Section 7 isolates the no-accidental-degeneracy assumptions. Section 8 states and proves the converse Phase Boundary Characterisation Theorem. Section 9 gives the precise distinction between dense-boundary and global boundary suppression. Section 10 concludes.

We adopt the notation of Paper I throughout. ( ℳ , g μ ν ) is a smooth, oriented, globally hyperbolic Lorentzian four-manifold with smooth boundary ∂ ℳ . We fix a foliation by smooth Cauchy hypersurfaces { Σ τ } τ ∈ ℝ with future-directed unit normal n μ , induced spatial metric h i j , and inverse h i j . We write D i for the Levi-Civita connection on ( Σ , h ) . The boundary of a Cauchy slice is ∂ Σ = Σ ∩ ∂ ℳ .

The dynamical field is a complex scalar Φ : ℳ → ℂ , with polar decomposition

defined on the open set where Φ ≠ 0 . We use ρ and t ^ interchangeably; both refer to the field’s modulus.

The Lagrangian density of the phase-suppressed model is

where the phase-stiffness coefficient κ : ℝ → ℝ + is smooth and positive, with κ ( P ) → ∞ as P → P ⋆ + on the dense side of the trigger threshold. For comparison with Paper I, the ordinary kinetic Lagrangian ∇ Φ † ∇ Φ − V ( | Φ | 2 ) corresponds to the case κ ≡ 1 ; our Lagrangian (2) is the polar-form generalisation in which the phase sector carries a position-dependent stiffness. The Euler-Lagrange equations remain smooth across ℋ for finite κ ; the divergence is taken in a limiting sense, justified by the finite-action argument below.

Several arguments below depend on two inputs from Paper I, cited there as Definition 3.5 and Proposition 4.1. For self-containedness we restate them here in the present notation; all subsequent uses of “the polar-amplitude boundary conditions” and “Hamiltonian admissibility” refer to these recalled statements.

Definition 2.1 (Polar-amplitude boundary conditions; PaperI, Definition3.5). In the polar variables (1), a variation δ is said to satisfy thepolar-amplitude boundary conditions at ∂ Σ if it preserves the normal-amplitude and phase-flux boundary data, that is,

where ∇ n denotes the outward normal derivative at ∂ Σ . Equivalently, admissible boundary variations preserve the mixed polar boundary structure used to define the augmented boundary symplectic density Ω ˜ ∂ Σ , so that the boundary contribution to Ω Σ aug is well defined and hypersurface-independent.

Definition 2.2 (Hamiltonian admissibility; PaperI, Proposition4.1). A boundary symmetry ξ ∈ g ∂ isHamiltonian admissibleif it preserves the polar-amplitude boundary conditions (3) and if the one-form ι δ ξ Ω Σ aug is exact on the reduced phase space, i.e., there exists a generator Q ξ with

Paper I, Proposition 4.1, establishes the integrability criterion in the form used below: given closure of Ω Σ aug (Paper I, Theorem 3.3) and preservation of (3), Cartan’s identity ℒ δ ξ = ι δ ξ d + d ι δ ξ reduces the existence of Q ξ to preservation of the boundary polarisation; this is the integrability requirement for boundary Hamiltonian generators in the sense of Wald and Zoupas [3]. Throughout, “admissible boundary symmetry” and “integrable Hamiltonian generator” refer to this recalled meaning.

These two statements are the only Paper I inputs invoked in Sections 5-8; isolating them here makes the proof dependencies of the present paper transparent without requiring the reader to consult Paper I.

Definition 2.3 (Finite-action admissibility on the dense stratum).A phase variation δ θ is finite-action admissible on Ω ⊂ Σ ∩ ℳ dens if

The integrand is positive-definite (the spatial metric h i j is Riemannian) and the zero-order term | δ θ | 2 removes the otherwise-surviving constant phase mode; equivalently, omitting the zero-order term and quotienting by the global U ( 1 ) phase rotation gives the same content.

Theorem 2.1 (Local finite-action phase suppression; PaperI). Let Ω ⊂ Σ ∩ ℳ dens be open, and assume ρ ≥ ρ 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. If δ θ n ∈ H loc 1 ( Ω ) satisfies the uniform bound

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

Proof. This is Lemma 2.1 of Paper I; we recall the proof for completeness. Let K ⋐ Ω be compact. Since ρ ≥ ρ 0 > 0 and κ n ( P ) → ∞ uniformly on K , the quantity m n ( K ) : = inf K κ n ( P ) ρ 2 satisfies m n ( K ) → ∞ . By the assumed uniform bound,

Hence δ θ n → 0 in L 2 ( K ) . Since K ⋐ Ω was arbitrary, the convergence is local. □

This theorem is the engine driving every result below.

We begin by formalising the finite-action condition of Definition 2.3 as a quadratic form on phase variations. This is the natural object on which the suppression argument acts, and stating it explicitly removes the need for first-variation language in what follows.

Definition 3.1 (Stiffness-weighted quadratic admissibility form). Let κ n ( P ) be a sequence of positive phase-stiffness coefficients with κ n ( P ) → ∞ uniformly on compact subsets of Ω ⊆ Σ ∩ ℳ dens . For K ⋐ Ω and a phase variation δ θ ∈ H 1 ( K ) , define

The divergent lower-bound coefficient on K is

The form Q n K is positive-definite (since h i j is Riemannian and the zero-order term is non-negative). Finite-action admissibility (Definition 2.3) is the condition that Q n K ( δ θ ) remains uniformly bounded as n → ∞ .

Definition 3.2 (Admissible dynamical trigger). A scalar functional P : C → C ∞ ( ℳ ) is an admissible dynamical trigger if there exist locally bounded smooth functions A ρ , A θ on ℳ , a tensor field A g μ ν , and a smooth vector field B μ ( δ ) depending linearly on the variation δ = ( δ ρ , δ θ , δ g μ ν ) such that

Each coefficient is required to be locally bounded on every compact subset of ℳ , and the boundary term B μ ( δ ) is required to satisfy the support or fall-off conditions imposed by the variational principle (e.g., compact support away from ∂ Σ , or vanishing at infinity).

The decomposition (9) captures the natural form taken by the variation of any local scalar functional built from ρ , θ , and g : a sum of pointwise multiplicative terms plus a divergence. Curvature-built triggers such as the Kretschmann scalar K = R μ ν ρ σ R μ ν ρ σ , energy-density triggers T μ ν T μ ν , and order-parameter triggers like | Φ | 2 all satisfy Definition 3.2.

Definition 3.3 (Subleadingdynamical-trigger correction). Let P [ Φ , g ] be an admissible dynamical trigger. The correction induced by δ κ n ( P ) = κ ′ n ( P ) δ P to the quadratic admissibility form is subleading relative to Q n K if, for every compact K ⋐ Ω ⊆ Σ ∩ ℳ dens , there exist sequences ϵ n ( K ) → 0 and constants C K (independent of n ) such that

where Δ Q n K denotes the contribution to the quadratic admissibility form on K induced by δ κ n ( P ) acting on − 1 2 ρ 2 h i j D i θ D j θ (the spatial restriction of the phase-stiffness density).

The bound (10) is on the unconstrained variations and uses the spatial integration on K ⋐ Σ ∩ ℳ dens , matching the spatial scope of the admissibility form Q n K . The first term on the right-hand side allows the dynamical-trigger correction to grow with δ θ , but only at a rate ϵ n ( K ) ⋅ m n ( K ) slower than the leading positive contribution m n ( K ) ‖ δ θ ‖ L 2 2 in Q n K . The second term controls the dependence on amplitude and metric variations.

Remark 3.1 (Conditions under which thesubleadingbound holds). The subleading condition (10) is not automatic in general; it requires structural cancellation between δ P and the phase-stiffness density. Two illustrative cases:

For the order-parameter trigger P = ρ 2 , one has δ P = 2 ρ δ ρ , so A θ = 0 and the dynamical-trigger correction has no δ θ dependence at leading order; the first term of (10) vanishes and only the second term contributes. For curvature triggers like P = R μ ν ρ σ R μ ν ρ σ , δ P depends only on δ g μ ν and its derivatives, again with no δ θ dependence at leading order; the same conclusion follows.

For triggers with A θ ≠ 0 (e.g., P depending explicitly on θ ), the subleading condition may require additional smallness of δ P or a specific growth rate on κ ′ n ( P ) / κ n ( P ) . The condition is a quantitative hypothesis to be verified in each application, not an automatic consequence of κ ′ n ( P ) not growing faster than κ n ( P ) .

Theorem 3.1 (Persistence of phase suppression for admissible dynamical triggers). Let P be an admissible dynamical trigger whose induced correction to the quadratic admissibility form is subleading in the sense of Definition 3.3. Let K ⋐ Σ ∩ ℳ dens with ρ ≥ ρ 0 > 0 on K , and let κ n ( P ) → ∞ uniformly on K . Suppose the total admissibility form (including the trigger correction) is uniformly bounded:

and sup n ‖ δ ρ n ‖ H 1 ( K ) , sup n ‖ δ g n ‖ H 1 ( K ) are finite. Then δ θ n → 0 in L 2 ( K ) . Consequently, the limiting dense-stratum admissible tangent space satisfies δ θ = 0 on ℳ dens .

Proof. The positive part of Q n K gives the lower bound

The dynamical-trigger correction satisfies, by Definition 3.3,

Combining,

The first term on the right is uniformly bounded by hypothesis (call the bound C * ). The second satisfies the displayed bound. Substituting and rearranging,

Since ϵ n ( K ) → 0 , the factor ( 1 − ϵ n ( K ) ) exceeds 1/2 for n large, and

Hence δ θ n → 0 in L 2 ( K ) . □

Remark 3.2 (Symplecticcorrection from a dynamical trigger). Theorem 3.1 establishes the persistence of phase suppression but does not claim that the presymplectic structure on the dense stratum is unchanged. The dynamical-trigger variation (9) contributes an additional term to the presymplectic potential current

of the form Θ trig μ ( δ ) = − 1 2 ρ 2 ∇ μ θ   ∇ α θ B α ( δ ) + ( lower order ) , with B α the boundary contribution of (9). On the dense stratum, δ θ = 0 kills any term containing δ θ as a factor; however, terms containing ∇ α θ (the background phase gradient, not its variation) and B α ( δ ρ , δ g ) may survive and contribute to the dense-stratum presymplectic structure. These are amplitude- and metric-sector contributions; they do not restore an independent phase direction. The detailed form for the case P = ρ 2 is recorded in Appendix.

Let S denote the space of smooth solutions Φ of the Euler-Lagrange equations of (2), satisfying the polar-amplitude boundary conditions (Definition 2.1; Paper I, Definition 3.5) at ∂ ℳ . The trigger functional P partitions S into two subsets according to whether ℳ dens is empty or non-empty for the given configuration:

We have S = S reg ⊔ S dens as sets, but the closure structure is non-trivial: S reg is open in S in the natural topology (since P < P ⋆ everywhere is a stable condition under small perturbations away from ℋ ), and S dens contains its boundary configurations where ℳ dens is just beginning to be non-empty.

The smoothness of each stratum is not automatic from these set-theoretic descriptions; it is a regularity hypothesis that must be stated explicitly. We isolate it now.

Assumption 4.1 (Stratum regularity). Equipped with the induced solution-space topology and the polar-amplitude boundary conditions of Paper I, the regular and dense sectors S reg and S dens are smooth Fréchet submanifolds (or smooth strata of a stratified Fréchet space) on which the presymplectic current and its boundary pullbacks are well-defined. The threshold set ℋ is a regular level set of P on ℳ in the sense that d P ≠ 0 on ℋ , providing the transversality required for the partition of solution space to be well-posed.

Assumption 4.1 is a functional-analytic regularity hypothesis on the solution space, parallel to the regularity hypotheses standard in the covariant phase-space literature [4]-[8]. We do not attempt to derive it in general; for specific Lagrangians and boundary conditions, smoothness of the solution space is a separate analytic question, typically handled by elliptic-regularity arguments adapted to the variational principle. Within this paper, the assumption is sufficient. To keep the analytic input separate from the proved consequence, however, we now identify one standard model class in which Assumption 4.1 is expected to hold, together with a sketch of the elliptic-regularity argument that anchors it.

Remark 4.1 (A model class anchoring Assumption4.1). One standard class in which Assumption 4.1 is expected to hold is the semilinear massive complex scalar equation on a globally hyperbolic manifold with smooth timelike boundary,

where N is smooth and of at most polynomial growth on the bounded field ranges considered, together with the polar-amplitude boundary conditions of Definition 2.1. The associated stratum-regularity statement is anchored by the following standard argument, which we sketch rather than carry out in full since it is not a new result of the present paper.

Step 1 (energy estimates). On each Cauchy slice Σ, linearising (13) about a smooth background solution gives a normally hyperbolic operator L whose principal part is □ g . For the polar-amplitude boundary conditions (3), the boundary value problem for L is well posed in the Sobolev scale: standard hyperbolic energy estimates (Gronwall applied to the natural energy associated to L ) [9] give continuous dependence of solutions on Cauchy and boundary data,

for k sufficiently large that H k ↪ C 2 by Sobolev embedding.

Step 2 (elliptic regularity on each slice). After fixing the Cauchy foliation, the spatial part of L is a second-order elliptic operator on ( Σ , h ) with the elliptic boundary conditions induced by (3) on ∂ Σ . Interior and boundary elliptic regularity (the standard H k → H k + 2 shift for elliptic systems with complementing boundary conditions) [10] upgrades the H k solutions of Step 1 to C ∞ solutions on each open stratum, provided the boundary data are smooth and the compatibility conditions at the corner ∂ Σ hold to all orders. The positive amplitude bound ρ ≥ ρ 0 > 0 used throughout keeps the polar change of variables (1) a smooth diffeomorphism, so amplitude and phase inherit the same regularity as Φ.

Step 3 (smooth strata via the implicit function theorem). Let P be a regular level-set functional with d P ≠ 0 on ℋ (Assumption 4.1). The map Φ ↦ min x ( P [ Φ , g ] ( x ) − P ⋆ ) is then C 1 on the Fréchet solution chart provided by Steps 1-2, with surjective differential at configurations where the minimum is attained transversally. The implicit-function theorem on the corresponding tame Fréchet chart (Nash-Moser in the smooth category) [11] yields smooth local strata S reg and S dens separated by the transversal threshold locus. This is the precise sense in which Assumption 4.1 is expected to follow from standard analysis for the model class (13): it is not an additional physical postulate, but the standard regularity output for smooth semilinear wave equations with smooth boundary data and a regular threshold set.

We emphasise that the present paper still assumes stratum regularity rather than proving it: the sketch above identifies the standard construction it rests on for a concrete model class, in direct response to the requirement that the analytic input be separated from, and anchored independently of, the proved consequences in Theorem 4.1. A complete proof of well-posedness and stratum smoothness for (13) with the boundary conditions (3) is a self-contained problem in the analysis of boundary-value problems for nonlinear wave equations and is outside the scope of a structural theorem in covariant phase space.

Definition 4.1 (Stratum-wise admissible tangent spaces). At Φ ∈ S reg , the regular admissible tangent space is

At Φ ∈ S dens , the dense admissible tangent space is

The strict inclusion T Φ S dens adm ⊊ T Φ S reg adm is the field-theoretic statement of the strata being of different rank.

The presymplectic potential current associated to (2) is, in polar variables,

The presymplectic current is the antisymmetrisation

with the amplitude- and phase-sector contributions given by

Each phase-sector term contains δ i θ as a factor.

The augmented presymplectic form on each stratum is

with α = δ 1 θ   δ 2 t ^ − δ 2 θ   δ 1 t ^ as in Paper I.

Lemma 4.1 (Stratum-wise hypersurface independence). On each stratum S reg and S dens , the augmented presymplectic form Ω Σ aug is independent of the choice of Cauchy hypersurface Σ, provided that the polar-amplitude boundary conditions (Definition 2.1; Paper I, Definition 3.5) are imposed.

Proof. The proof is the standard hypersurface-independence argument from Paper I (Theorem 3.2 there) applied stratum by stratum. On the regular stratum, the argument is the verbatim Paper I result. On the dense stratum, the bulk part Ω Σ of Ω Σ aug involves the phase-sector current ω θ μ , which contains δ i θ as a factor; on S dens , this factor vanishes by Definition 4.1, so the phase-sector contribution to the bulk integral vanishes pointwise. The remaining amplitude-sector contribution and the boundary integral satisfy the hypersurface-independence relation from Paper I unchanged. □

Theorem 4.1 (Global two-stratum presymplectic structure). Under Assumption 4.1 (stratum regularity), the positive-amplitude condition ρ ≥ ρ 0 > 0 on Σ ∩ ℳ dens , and the finite-action admissibility condition of Definition 2.3 (or Theorem 3.1 in the dynamical-trigger case), the solution space S carries a two-stratum presymplectic structure

with the following properties:

(i) By Assumption 4.1, the regular and dense sectors S reg and S dens are smooth strata of the solution space.

(ii) The augmented presymplectic form Ω Σ aug restricts to a closed antisymmetric bilinear form on each stratum, hypersurface-independent within that stratum.

(iii) The strict tangent inclusion T Φ S dens adm ⊊ T Φ S reg adm holds at every Φ admitting both regular and dense restrictions; explicitly, the phase-sector tangent direction is admissible on S reg but excluded from T Φ S dens adm .

(iv) For dense-stratum admissible variations, the phase-sector contribution to the presymplectic current vanishes on the dense region of the slice:

The full bulk phase-sector contribution on all of Σ vanishes only if Σ ⊆ ℳ dens , or if the admissible variations or integration domain are restricted to the dense region.

(v) The reduced phase space on each stratum is the presymplectic quotient by ker Ω Σ aug restricted to that stratum:

and the dense stratum has lower phase-sector rank than the regular stratum (Definition 4.2 below).

Proof. (i) This is Assumption 4.1; we do not prove smoothness of the strata from set-theoretic descriptions but invoke the assumption directly.

(ii) Antisymmetry and bilinearity of Ω Σ aug are inherited from the bulk current ω μ and the boundary density α . Closure on each stratum is the variational identity d Ω Σ aug = 0 , which holds wherever Ω Σ aug is defined as the variation of an action (Lee-Wald [4], Iyer-Wald [5]). Hypersurface independence within each stratum is Lemma 4.1.

(iii) The strict inclusion is the content of Definition 4.1, supplemented by Theorem 2.1 which derives the dense-stratum constraint δ θ = 0 on ℳ dens from finite-action admissibility.

(iv) For Φ ∈ S dens and dense-stratum admissible variations δ 1 , δ 2 , we have δ i θ = 0 on ℳ dens (by Definition 4.1). Each term in ω θ μ contains δ i θ as a multiplicative factor; hence ω θ μ ( δ 1 , δ 2 ) = 0 pointwise on Σ ∩ ℳ dens . On Σ ∩ ℳ reg , the phase variations δ i θ may be non-zero, and ω θ μ contributes; we make no claim about that contribution. The localised statement is what is needed for the rank-reduction conclusion of (v).

(v) The quotients are well-defined since ker Ω Σ aug | stratum is closed under the natural Fréchet topology. The strict inclusion in (iii) implies ker Ω Σ aug | S dens ⊋ ker Ω Σ aug | S reg

when both are interpreted as subspaces of the unconstrained tangent bundle (in the sense of description (b) of Paper I, Remark 2.2). The rank statement is Definition 4.2 below. □

To formalise the rank statement in clause (v) of Theorem 4.1 without invoking infinite-dimensional dimension comparisons, we record the following.

Definition 4.2 (Phase-sector rank). The phase-sector rankof a stratum at a point Φ is the number of independent phase-variation directions admitted by the finite-action tangent space at Φ. In the regular stratum, δ θ is an independent phase variation; in the dense stratum, finite-action admissibility imposes δ θ = 0 and removes this direction. The dense stratum therefore has lower phase-sector rankthan the regular stratum.

Proposition 4.1 (Phase-sector rank reduction). In the regular stratum, admissible tangent vectors in polar variables have the form

In the dense stratum, finite-action admissibility imposes δ θ = 0 on ℳ dens , so dense-stratum admissible tangent vectors have the restricted form

The dense stratum therefore has lower phase-sector rank than the regular stratum.

Proof. The regular-stratum expression follows from the polar decomposition. The dense-stratum restriction follows from Theorem 2.1. Since the independent δ θ direction is present in the regular stratum but excluded from the dense stratum, the phase-sector rank is strictly lower on the dense stratum. □

Remark 4.2 (Comparison withSjamaar-Lerman). The Sjamaar-Lerman theorem [2] concerns symplectic reduction by a Hamiltonian group action on a finite-dimensional symplectic manifold, where the orbit-type stratification of the zero level set of the moment map yields a stratified symplectic space whose strata carry compatible symplectic forms. It sits in the lineage of Marsden-Weinstein symplectic reduction [12], while the stratified structure of the space of solutions of a covariant field equation was first analysed in the Einstein case by Arms, Marsden and Moncrief [13]. Theorem 4.1 gives a field-theoretic analogue: the strata are determined not by orbit type but by an energetic admissibility condition, and the compatible structure on each stratum is presymplectic rather than symplectic until quotient by the stratum-specific kernel. The closure relation S reg ¯ ⊇ ℋ -boundary configurations (where ℳ dens first becomes non-empty) plays the role of the closure ordering in Sjamaar-Lerman. The analogy is structural rather than literal: we do not invoke the moment-map machinery, and the strata here are not orbit types of any obvious group action.

The assumptions used in the principal theorems—the dynamical-trigger persistence theorem (Theorem 3.1), the global reduction theorem (Theorem 4.1), the cocycle non-triviality theorem (Theorem 6.1), and the converse theorem (Theorem 8.1)—could in principle be mutually inconsistent. We now exhibit a single explicit configuration realising all of them simultaneously, establishing non-vacuity.

Consider the stiffness-coupled Mexican-hat scalar on the half-space

with action

Take the order-parameter trigger and threshold

with divergent stiffness on the dense side,

extended smoothly and positively to P < P ⋆ . On the shifted kink background

the threshold surface is the finite regular plane

Thus

and, because the boundary { y = 0 } is independent of the threshold coordinate x , the threshold has a nonempty boundary trace

with a correspondingly nonempty dense boundary

Since

the threshold is a regular level set and it meets the boundary transversally, so Assumption 4.1(transversality) and Assumption 7.1(vi) hold; on any slab { | x − x ⋆ | ≤ δ } the kink amplitude is bounded below, ρ 0 ≥ ρ 0 ( x ⋆ − δ ) > 0 , giving the positive-amplitude hypothesis. We now apply the four theorems to this configuration.

Theorem 3.1 (dynamical trigger). Here P = ρ 2 is dynamical, with δ P = 2 ρ δ ρ and hence A θ = 0 in Definition 3.2. As recorded in Remark 3.1, the induced correction Δ Q n K then carries no leading δ θ term, so the subleading bound (10) holds with ϵ n ( K ) ≡ 0 . Theorem 3.1 applies and gives δ θ = 0 on ℳ dens .

Theorem 4.1 (global reduction). The model (13)-type field equation here is the semilinear equation □   ρ − λ ρ ( ρ 2 − v 2 ) − ( … ) = 0 with the polar-amplitude boundary conditions of Definition 2.1 at y = 0 ; by Remark 4.1 the regular and dense regions are smooth strata, so Assumption 4.1 is realised and the two-stratum presymplectic structure of Theorem 4.1 is well defined, with T Φ S dens adm ⊊ T Φ S reg adm .

Theorem 6.1 (cocycle non-triviality). On ∂ Σ = { y = 0 } , with boundary coordinates ( x , z ) , choose a small neighbourhood U ∂ of a point in ℋ ∩ ∂ Σ and a non-vanishing tangential vector field, for example X = ∂ z . With t ^ = ρ 0 ( x ) 2 > 0 non-vanishing on U ∂ , take bump functions ϕ 1 , ϕ 2 ∈ C c ∞ ( U ∂ ) with overlapping but non-centred support along the z -flow, as in Proposition 6.1. This produces an abelian amplitude-phase nondegenerate subalgebra a ∂ satisfying the Hamiltonian admissibility of Definition 2.2, so by Theorem 6.1 the cocycle K is cohomologically non-trivial for the constructed g ∂ .

Theorem 8.1 (boundary-local converse). On the dense boundary

finite-action admissibility forces δ θ = 0 , so the dense-boundary cocycle contribution

vanishes pointwise. The single-component half-space background carries no Φ-zeros, no winding, no amplitude vanishing, and a uniquely fixed boundary polarisation on the slab considered, so Assumption 7.1 holds; the only divergent phase-sector penalty is the stiffness one at P → P ⋆ + , giving Assumption 7.3; and the flow-box bump construction supplies the local separating algebra of Assumption 7.4. At boundary-accessible threshold points

the local separating boundary algebra detects the transition from the regular boundary side x < x ⋆ to the dense boundary side x > x ⋆ . The configuration therefore lies in the pure stiffness-induced class (Definition 7.2), and Theorem 8.1 applies in a boundary neighbourhood of x 0 .

All four theorems thus hold simultaneously for one explicit model, so the hypotheses used in this paper are mutually compatible and non-vacuous in practice.

Assumption 6.1 (Hamiltonian boundary test algebra). The boundary test subalgebras a ∂ ⊆ g ∂ considered below consist of boundary symmetry generators that preserve the polar-amplitude boundary conditions (Definition 2.1) and admit integrable Hamiltonian generators with respect to the augmented presymplectic form Ω Σ aug , in the sense of Definition 2.2 (Paper I, Proposition 4.1).

Assumption 6.1 is a non-trivial admissibility condition: it requires the boundary symmetries used in the cocycle analysis to be honest Hamiltonian symmetries of the augmented covariant phase space, not merely formal infinitesimal transformations of boundary data. We verify it explicitly for the construction of Proposition 6.1 below.

Definition 6.1 (Amplitude--phase nondegeneracy). An abelian Lie subalgebra a ∂ ⊆ g ∂ satisfying Assumption 6.1 is amplitude-phase nondegenerate if there exist ξ , η ∈ a ∂ such that

Theorem 6.1 (Cohomological non-triviality of K ). Let g ∂ contain an abelian amplitude-phase nondegenerate subalgebra a ∂ satisfying Assumption 6.1. Then the explicit boundary cocycle K of (19) is not a coboundary on g ∂ , and hence defines a non-trivial class [ K ] ∈ H 2 ( g ∂ , ℝ ) .

Proof. Suppose, for contradiction, that K is a coboundary: there exists a linear functional B : g ∂ → ℝ such that K ( ξ , η ) = B ( [ ξ , η ] ) for every ξ , η ∈ g ∂ . Restrict to the subalgebra a ∂ . Since a ∂ is abelian, [ ξ , η ] = 0 for every ξ , η ∈ a ∂ , so

This contradicts the assumed amplitude-phase nondegeneracy of a ∂ , which provides ξ , η ∈ a ∂ with K ( ξ , η ) ≠ 0 . Therefore K is not a coboundary, and [ K ] ≠ 0 in H 2 ( g ∂ , ℝ ) . □

The criterion of Theorem 6.1 is non-vacuous only if there exists a g ∂ admitting such an a ∂ . We now exhibit one.

Proposition 6.1 (Existence of an amplitude--phase nondegenerate abelian Hamiltonian subalgebra). Suppose ∂ Σ admits a non-vanishing smooth vector field X on an open subset U ∂ ⊆ ∂ Σ , and that t ^ is non-vanishing on U ∂ . Then there exists a Lie algebra g ∂ of boundary symmetries containing an abelian amplitude-phase nondegenerate subalgebra a ∂ satisfying Assumption 6.1.

Proof. Let ϕ 1 , ϕ 2 ∈ C c ∞ ( U ∂ ) be two smooth, real-valued, compactly supported functions on U ∂ . Define their X -derivatives by the Lie derivative ϕ i ' : = ℒ X ϕ i , which is a coordinate-free derivative along X that exists wherever X does. Choose ϕ 1 , ϕ 2 such that

Such functions exist: applying the Leibniz rule for ℒ X , the integrand equals 2 ϕ 1 ℒ X ϕ 2 − ℒ X ( ϕ 1 ϕ 2 ) . If X is divergence-free with respect to d S (or, more generally, the measure d S is X -invariant on the support of ϕ i —true in flow-box coordinates around any non-vanishing X ), then ∫ U ∂ ℒ X ( ϕ 1 ϕ 2 ) d S = 0 by Stokes’ theorem on compact-support functions, and (22) reduces to 2 ∫ U ∂ ϕ 1 ℒ X ϕ 2 d S . This is generically non-zero: for instance, take ϕ 1 a bump function and ϕ 2 a translate of ϕ 1 along the X -flow with supports overlapping but not centred, so that the asymmetric pairing under ℒ X is non-vanishing.

Define two boundary symmetry generators ξ 1 , ξ 2 acting on the boundary data by

extended to act trivially on the bulk away from a neighbourhood of ∂ Σ . Each ξ i is parameterised by a fixed function ϕ i ∈ C c ∞ ( U ∂ ) , so the corresponding flow is the additive shift ( θ , t ^ ) ↦ ( θ + ϵ i ϕ i , t ^ + ϵ i ℒ X ϕ i ) . Additive shifts of boundary fields commute pairwise: [ ξ 1 , ξ 2 ] = 0 . Hence a ∂ : = span ℝ { ξ 1 , ξ 2 } is abelian.

Hamiltonian admissibility (Assumption 6.1). It remains to verify directly that each ξ i preserves the recalled polar-amplitude boundary conditions of Definition 2.1 and admits an integrable Hamiltonian in the sense of Definition 2.2; the earlier informal claim that this holds “trivially” is replaced by the following check. The variations generated by ξ i are

Each generator is extended off ∂ Σ by a collar construction in the inward geodesic normal coordinate; the normal derivative of the collar extension of a compactly supported boundary profile is freely specifiable at ∂ Σ , which is the freedom exploited below. For the second boundary condition of (3), expand using the Leibniz rule:

Rather than requiring the phase profile to be normally constant, choose the collar extension of ϕ i so that its normal derivative at the boundary satisfies

This choice is possible for any smooth compactly supported boundary profile because the normal derivative of the collar extension is freely specifiable at ∂ Σ , and t ^ ≠ 0 on U ∂ by hypothesis. Substituting,

The first boundary condition of (3) is preserved independently by taking the amplitude profile ℒ X ϕ i to have vanishing normal derivative at ∂ Σ :

The two prescriptions are consistent because the off-boundary collar extensions of the generated variations δ ξ i θ = ϕ i and δ ξ i t ^ = ℒ X ϕ i are chosen independently once their boundary values are fixed: ∇ n ϕ i | ∂ Σ is the normal derivative of the collar extension of the phase variation, prescribed as above to preserve the phase-flux datum, while ∇ n ( ℒ X ϕ i ) | ∂ Σ is the normal derivative of the separate collar extension of the amplitude variation, prescribed to vanish so as to preserve the normal-amplitude datum. These are normal derivatives of two distinct scalar extensions, not two derivatives of one fixed extension, so the prescriptions do not interfere; they are simultaneously realisable, and no assumption ∇ n θ = 0 or support restriction is needed. Thus each ξ i preserves both polar-amplitude boundary conditions (3). Given preservation of the boundary conditions and closure of Ω Σ aug (Paper I, Theorem 3.3, recalled in Definition 2.2), the integrability criterion (4) is met via Cartan’s identity, so the Hamiltonian generators Q ξ i exist and a ∂ satisfies Assumption 6.1.

Amplitude-phase nondegeneracy. The cocycle pairing on a ∂ is

Hence a ∂ is amplitude-phase nondegenerate. □

Corollary 6.1. Under the hypotheses of Proposition 6.1, the boundary cocycle K of (19) represents a non-trivial class in H 2 ( g ∂ , ℝ ) , with g ∂ as constructed there.

Proof. Combine Theorem 6.1 and Proposition 6.1. □

Remark 6.1. The non-triviality result depends on the choice of g ∂ . For Lie algebras admitting only commutators that pair trivially under the cocycle integrand (e.g., a non-abelian g ∂ all of whose commutators are vector fields not generating amplitude-phase pairings), the criterion of Theorem 6.1 need not apply. The proposition shows the criterion is realised; we do not claim it is realised universally.