Covariant Phase Loading as a Sector-Exchange Identity in the CHC Framework

The covariant CHC formulation uses a scalar global phase field $\HH$TeX\HH as the common phase-response object of the admitted branch. In the root scalar-tensor realization, and in the cosmological expansion-response branch where applicable, departures from the controlled GR+$\Lambda_{\rm eff}$TeX\Lambda_{\rm eff} envelope are organized by the dimensionless hierarchy variable

\XiH=\frac{\Meff^2|\nabla\HH|_{\rm bg}^2}{\LamXi^4},

where $|\nabla\HH|_{\rm bg}^2$TeX|\nabla\HH|_{\rm bg}^2 is the background-adapted nonnegative norm and $\LamXi$TeX\LamXi is the phase-gradient scale. The variable $\XiH$TeX\XiH records the state of the global phase-field gradient on the admitted sector. It does not, by itself, say how a material sector, an electromagnetic-excitation sector, a boundary layer, or a detector-facing registration layer loads that global phase field.

The complementary object is a sector scalar measuring the variational response of a sector action to $\HH$TeX\HH. For a sector action $\Sa[g,\Psia,\HH]$TeX\Sa[g,\Psia,\HH], define

\lpa=\frac{1}{\LamXi^4\sqrt{-g}}\frac{\delta\Sa}{\delta\HH}.

The associated vector density of sector exchange is

\Fload_{(a)}^{\nu}:=\nabla_\mu\Ta^{\mu\nu}.

The covariant phase-loading identity derived below states that, on the declared variational domain,

\boxed{
\nabla_\mu\Ta^{\mu\nu}=\LamXi^4\lpa\nabla^\nu\HH .
}

The equation says that the failure of a sector stress tensor to be separately conserved is parallel to the phase gradient and proportional to the sector phase loading. If a sector has no direct $\HH$TeX\HH dependence, then $\lambda_{(a)}=0$TeX\lambda_{(a)}=0 and the sector is separately conserved on shell. If a sector mass scale, rigidity coefficient, or boundary threshold enters a declared $\HH$TeX\HH-dependent action, the exchange with the phase field is carried by $\lambda_{(a)}$TeX\lambda_{(a)}. If only a reduced registration factor depends on $\HH$TeX\HH on a declared window, the object obtained is a reduced phase sensitivity rather than a Noether loading scalar unless an action-level variational representation with fixed normalization and compatible stress-tensor and phase-Euler conventions has been supplied.

Equation reference is therefore a sector-exchange identity. It does not introduce a conserved charge for an individual open sector. The chart-invariant exchange object is the loading one-form $\Load_{(a)\nu}$TeX\Load_{(a)\nu}, and the conserved object in a closed variational system is the total stress tensor after the phase-sector equation is imposed. Separate conservation of a sector is recovered only in the zero-loading or zero-gradient cases, or when additional exchange terms have been explicitly included in the sector definition.

The identity is a covariant identity of the variational system under diffeomorphism invariance, in the sense of the Noether--Wald covariant variational formalism [citation]. It is not an added force postulate, not a new detector Hamiltonian, and not a universal material model. Its role is to identify the sector-exchange formalism available to carrier, material-rigidity, boundary-response, tunneling-registration, and homogeneous-response sectors when the relevant sector is represented by a declared $\HH$TeX\HH-dependent action. Reduced-window quantities are retained as phase-sensitivity proxies unless an action-level variational representation with fixed normalization and compatible stress-tensor and phase-Euler conventions is supplied. This terminology fixes the equation as a sector-exchange identity: isolated-sector conservation is recovered only under zero loading, zero phase gradient, or a redefinition that includes the missing exchange sector.

Throughout, the metric signature is $(-,+,+,+)$TeX(-,+,+,+) and $\HH$TeX\HH is taken to be dimensionless on the chosen phase chart. The scale $\LamXi$TeX\LamXi then carries mass dimension one in natural units $c=\hbar=1$TeXc=\hbar=1, so that $\LamXi^4\lambda_{(a)}$TeX\LamXi^4\lambda_{(a)} has the dimension of an action-density Euler derivative with respect to $\HH$TeX\HH.

definition: Sector action and internal equations. A sector is specified by a diffeomorphism-invariant action

\Sa[g,\Psia,\HH]=\int_{\mathcal M}\dvol\,\Lag_{(a)}(g,\Psia,\nabla\Psia,\ldots,\HH,\nabla\HH,\ldots),

where $\Psia$TeX\Psia denotes the internal fields of the sector. The sector may depend on $\HH$TeX\HH and its derivatives. For the closed identity below, any additional non-dynamical background tensor is either absent or contributes no uncanceled Lie-variation term on the declared variational domain. The internal equations are

\frac{1}{\sqrt{-g}}\frac{\delta\Sa}{\delta\Psia}=0.

definition: Hilbert stress tensor. The sector stress tensor is defined by variation with respect to the covariant metric:

\Ta^{\mu\nu}:=\frac{2}{\sqrt{-g}}\frac{\delta\Sa}{\delta g_{\mu\nu}}.

Equivalently, variation with respect to the inverse metric gives

\delta_g\Sa=-\frac12\int_{\mathcal M}\dvol\,T_{(a)\mu\nu}\delta g^{\mu\nu}.

definition: Phase-loading scalar and loading one-form. The phase Euler density and dimensionless phase-loading scalar of sector $a$TeXa are

\Ecal_{\HH}^{(a)}:=\frac{1}{\sqrt{-g}}\frac{\delta\Sa}{\delta\HH},
\qquad
\lpa:=\frac{\Ecal_{\HH}^{(a)}}{\LamXi^4}.

The corresponding loading one-form is

\Load_{(a)\nu}:=\LamXi^4\lpa\nabla_\nu\HH.

The loading scalar depends on the phase chart, while the one-form $\Load_{(a)\nu}$TeX\Load_{(a)\nu} does not. This distinction matters because $\HH$TeX\HH is a phase coordinate rather than an absolute material label.

proposition: Phase-chart covariance. Let $\widetilde\HH=f(\HH)$TeX\widetilde\HH=f(\HH) with $f'(\HH)\ne0$TeXf'(\HH)\ne0 on a declared phase patch. If

\widetilde\lambda_{(a)}=\frac{1}{\LamXi^4\sqrt{-g}}\frac{\delta\Sa}{\delta\widetilde\HH},

then

\widetilde\lambda_{(a)}\nabla_\nu\widetilde\HH=\lambda_{(a)}\nabla_\nu\HH.

proof. The chain rule gives $\delta\Sa/\delta\widetilde\HH=(\dd\HH/\dd\widetilde\HH)\delta\Sa/\delta\HH$TeX\delta\Sa/\delta\widetilde\HH=(\dd\HH/\dd\widetilde\HH)\delta\Sa/\delta\HH. Since $\nabla_\nu\widetilde\HH=(\dd\widetilde\HH/\dd\HH)\nabla_\nu\HH$TeX\nabla_\nu\widetilde\HH=(\dd\widetilde\HH/\dd\HH)\nabla_\nu\HH, the product in reference follows.

The phase-loading identity is the sector form of the diffeomorphism Noether identity. The derivation uses compactly supported vector fields or boundary data for which the boundary term vanishes.

theorem: Covariant phase-loading identity. Let $\Sa[g,\Psia,\HH]$TeX\Sa[g,\Psia,\HH] be a diffeomorphism-invariant sector action satisfying the assumptions in reference. Suppose the internal equations reference hold, every admitted non-dynamical structure is either absent or contributes no uncanceled Lie-variation term on the declared variational domain, and the vector field used in the variation has compact support or boundary data for which the boundary term vanishes. Then the sector stress tensor and the phase-loading scalar obey

\nabla_\mu\Ta^{\mu\nu}=\LamXi^4\lpa\nabla^\nu\HH.

proof. Let $\xi^\mu$TeX\xi^\mu be a compactly supported vector field. Diffeomorphism invariance gives $\delta_\xi\Sa=0$TeX\delta_\xi\Sa=0. On the internal equations, the variation is

0=\delta_\xi\Sa
=\int_{\mathcal M}\dvol\left[\frac12\Ta^{\mu\nu}\mathcal L_\xi g_{\mu\nu}+\Ecal_\HH^{(a)}\mathcal L_\xi\HH\right].

For a scalar field and the metric,

\mathcal L_\xi\HH=\xi^\rho\nabla_\rho\HH,
\qquad
\mathcal L_\xi g_{\mu\nu}=\nabla_\mu\xi_\nu+\nabla_\nu\xi_\mu.

Using the symmetry of $\Ta^{\mu\nu}$TeX\Ta^{\mu\nu},

0=\int_{\mathcal M}\dvol\left[\Ta^{\mu\nu}\nabla_\mu\xi_\nu+\Ecal_\HH^{(a)}\xi^\nu\nabla_\nu\HH\right].

After integration by parts and dropping the compact-support boundary term,

0=\int_{\mathcal M}\dvol\left[-\nabla_\mu\Ta^{\mu\nu}+\Ecal_\HH^{(a)}\nabla^\nu\HH\right]\xi_\nu.

The arbitrariness of $\xi_\nu$TeX\xi_\nu gives $\nabla_\mu\Ta^{\mu\nu}=\Ecal_\HH^{(a)}\nabla^\nu\HH$TeX\nabla_\mu\Ta^{\mu\nu}=\Ecal_\HH^{(a)}\nabla^\nu\HH. Substitution of $\Ecal_\HH^{(a)}=\LamXi^4\lpa$TeX\Ecal_\HH^{(a)}=\LamXi^4\lpa yields reference.

remark: Derivative dependence on the phase. The Euler density $\Ecal_\HH^{(a)}$TeX\Ecal_\HH^{(a)} in reference is the full variational derivative. If the sector depends on $\nabla\HH$TeX\nabla\HH or higher derivatives, integrations by parts are already included in $\delta\Sa/\delta\HH$TeX\delta\Sa/\delta\HH. No separate correction is added to reference.

remark: Admitted background structures. If a sector uses a prescribed external tensor, foliation, interface embedding, or source profile, its Lie variation must be included in the declared variational problem. Otherwise the missing Lie-variation term appears as an additional force density and reference is not the complete exchange identity on that window.

The sector identity describes exchange between one sector and the phase field. It is not an additional conservation law for the isolated sector. A single $\HH$TeX\HH-dependent sector may fail to be separately conserved because its action depends on the phase coordinate, while the exchange vector is the phase-loading one-form $\Load_{(a)}^{\nu}$TeX\Load_{(a)}^{\nu}. A closed covariant system still conserves the total stress tensor after the phase equation is included.

proposition: Total covariant closure. Let

S_{\rm tot}[g,\HH,\{\Psi_{(a)}\}]=S_{\HH}[g,\HH]+\sum_a S_{(a)}[g,\Psi_{(a)},\HH]

be diffeomorphism invariant. Suppose all internal equations and the phase equation

\Ecal_\HH^{(\HH)}+\sum_a \Ecal_\HH^{(a)}=0

are imposed. Then

\nabla_\mu\left(T_{\HH}^{\mu\nu}+\sum_a T_{(a)}^{\mu\nu}\right)=0.

proof. Apply reference to the phase sector and to all other sectors. The sum gives

\nabla_\mu T_{\rm tot}^{\mu\nu}=\left(\Ecal_\HH^{(\HH)}+\sum_a\Ecal_\HH^{(a)}\right)\nabla^\nu\HH,

which vanishes by reference.

corollary: Separate conservation criterion. Under the hypotheses of reference, for the sector definition used in reference, a sector stress tensor is separately conserved on a declared window if and only if its loading one-form vanishes on that window,

\Load_{(a)}^{\nu}=\LamXi^4\lambda_{(a)}\nabla^\nu\HH=0.

If additional exchange terms are declared, the criterion applies to the enlarged sector stress tensor rather than to the original isolated sector. Thus $\lambda_{(a)}=0$TeX\lambda_{(a)}=0 is sufficient, and a phase-flat window with $\nabla^\nu\HH=0$TeX\nabla^\nu\HH=0 also suppresses the exchange even when a latent sector loading is nonzero.

proof. Equation reference identifies $\nabla_\mu T_{(a)}^{\mu\nu}$TeX\nabla_\mu T_{(a)}^{\mu\nu} exactly with $\Load_{(a)}^\nu$TeX\Load_{(a)}^\nu under the stated sector convention, proving the equivalence. Each stated sufficient case sets one factor of $\LamXi^4\lambda_{(a)}\nabla^\nu\HH$TeX\LamXi^4\lambda_{(a)}\nabla^\nu\HH to zero. No converse inference to $\lambda_{(a)}=0$TeX\lambda_{(a)}=0 is valid on a phase-flat window.

The root hierarchy variable $\XiH$TeX\XiH and the phase-loading scalar have different meanings. The former measures the global phase-field gradient state on the admitted branch; the latter measures a sector response to that phase coordinate. A simple invariant contraction relates them only after a background norm is chosen.

On a region where $|\nabla\HH|_{\rm bg}^2$TeX|\nabla\HH|_{\rm bg}^2 is the relevant nonnegative norm, the magnitude of the loading one-form satisfies

\frac{|\Load_{(a)}|_{\rm bg}}{\LamXi^4}
=|\lambda_{(a)}|\,|\nabla\HH|_{\rm bg}
=|\lambda_{(a)}|\frac{\LamXi^2}{\Meff}\,\XiH^{1/2}

when $\Meff$TeX\Meff is constant on the local branch. Thus the declared global phase-field branch may be phase-flat, $\XiH\to0$TeX\XiH\to0, even when a sector has a nonzero latent phase loading; the observable exchange is then suppressed by the phase gradient.

Scalar--tensor and dilaton models often define a scalar source or scalar charge through a model-specific matter coupling, such as a phase-dependent mass scale, coupling function, or conformal factor [citation]. The phase-loading scalar is adjacent to that class of objects but is not identical to a universal scalar charge.

First, $\lambda_{(a)}$TeX\lambda_{(a)} is defined only after a declared sector action, stress-tensor convention, and phase chart have been fixed. It is the normalized Euler response of that sector action to $\HH$TeX\HH, not a species charge assigned independently of the action. Second, $\lambda_{(a)}$TeX\lambda_{(a)} is chart dependent; the invariant object is the one-form $\Load_{(a)\nu}=\LamXi^4\lambda_{(a)}\nabla_\nu\HH$TeX\Load_{(a)\nu}=\LamXi^4\lambda_{(a)}\nabla_\nu\HH. Third, the same definition applies to electromagnetic-excitation sectors, material sectors, boundary actions, detector-facing interface actions, and reduced effective sectors whenever those objects are represented by declared variational actions. Thus phase loading is a taxonomy of sector response within the CHC phase chart rather than a single scalar charge density imported from one scalar-tensor model.

For the representative mass-loading model reference, one may write

\lambda_\psi
= -\frac{m^2(\HH)}{2\LamXi^4}\,\frac{\partial\log m^2}{\partial\HH}\,\psi^2 .

The logarithmic factor resembles a scalar coupling strength, while the full $\lambda_\psi$TeX\lambda_\psi includes the local field amplitude and the normalization by $\LamXi^4$TeX\LamXi^4. A different action, field content, or phase chart changes the scalar representative but leaves the loading one-form invariant under regular phase reparametrization. Consequently, two sectors may have comparable scalar-coupling parameters and still carry different phase-loading densities on the same window, and a reduced sensitivity proxy may resemble a scalar charge while failing to be a Noether loading scalar.

Equation reference is stronger than a proportionality statement: it predicts that the sector exchange vector is parallel to $\nabla^\nu\HH$TeX\nabla^\nu\HH on the declared variational domain. This gives a local extractor and an orthogonal residual on any regular-gradient patch.

definition: Regular-gradient patch. A regular-gradient patch is an open set $U\subset\mathcal M$TeXU\subset\mathcal M on which

\nabla_\rho\HH\nabla^\rho\HH\ne0.

proposition: Phase-loading extractor. On a regular-gradient patch satisfying the phase-loading identity,

\lambda_{(a)}=\frac{\nabla_\nu\HH\,\nabla_\mu\Ta^{\mu\nu}}{\LamXi^4\nabla_\rho\HH\nabla^\rho\HH}.

proof. Contract reference with $\nabla_\nu\HH$TeX\nabla_\nu\HH and divide by $\LamXi^4\nabla_\rho\HH\nabla^\rho\HH$TeX\LamXi^4\nabla_\rho\HH\nabla^\rho\HH.

definition: Orthogonal exchange residual. On a regular-gradient patch, define

P^\nu_{\ \rho}=\delta^\nu_{\ \rho}-\frac{\nabla^\nu\HH\nabla_\rho\HH}{\nabla_\alpha\HH\nabla^\alpha\HH},
\qquad
\mathcal R_{\perp,(a)}^{\nu}:=P^\nu_{\ \rho}\nabla_\mu\Ta^{\mu\rho}.

corollary: Parallel-loading rejection condition. If reference holds on a regular-gradient patch, then

\mathcal R_{\perp,(a)}^{\nu}=0.

Consequently, a declared window in which $\mathcal R_{\perp,(a)}^{\nu}$TeX\mathcal R_{\perp,(a)}^{\nu} cannot be made smaller than the stated residual tolerance is not described by a single scalar phase-loading identity on that window.

proof. Substitute reference into reference. Since $P^\nu_{\ \rho}\nabla^\rho\HH=0$TeXP^\nu_{\ \rho}\nabla^\rho\HH=0, the residual vanishes.

The Noether identity gives a forward implication from a sector action to an aligned exchange vector. A phenomenological relation $Q^\nu=\Lambda_\Xi^4\lambda\nabla^\nu\HH$TeXQ^\nu=\Lambda_\Xi^4\lambda\nabla^\nu\HH is not thereby known to arise from an action. The converse question is a local inverse problem of the calculus of variations.

Fix the metric and any spectator fields on a contractible spacetime domain, impose compactly supported variations, and work on a phase chart whose admissible field configurations form a star-shaped set. Let $\lambda[\HH]$TeX\lambda[\HH] be a local scalar differential function of finite order. Its Fr\'echet linearization in the direction $\eta$TeX\eta is

D_\lambda[\HH](\eta)
:=\left.\frac{\dd}{\dd\epsilon}\lambda[\HH+\epsilon\eta]\right|_{\epsilon=0}.

The formal adjoint $D_\lambda^*$TeXD_\lambda^* is defined by

\int_{\mathcal M}\dvol\,\zeta D_\lambda(\eta)
=\int_{\mathcal M}\dvol\,\eta D_\lambda^*(\zeta)

for compactly supported test functions, after all derivatives have been transferred by integration by parts.

theorem: Local variational realizability of a phase loading. Under the preceding locality, regularity, support, and star-shaped-domain assumptions, there exists a local functional $S_{(a)}[\HH]$TeXS_{(a)}[\HH] satisfying

\frac{1}{\sqrt{-g}}\frac{\delta S_{(a)}}{\delta\HH}
=\LamXi^4\lambda[\HH]

if and only if

D_\lambda=D_\lambda^*.

When the condition holds, one representative is the local homotopy functional

S_{(a)}[\HH]
=\LamXi^4\int_0^1\dd t\int_{\mathcal M}\dvol\,
\HH\lambda[t\HH],

up to a field-independent functional and a boundary divergence. If additional fields are varied rather than held fixed, the corresponding full matrix of Fr\'echet derivatives must satisfy the joint Helmholtz conditions.

proof. Necessity follows from symmetry of the second variation. If reference holds, then for compactly supported $\eta$TeX\eta and $\zeta$TeX\zeta,

\delta_\zeta\delta_\eta S_{(a)}-\delta_\eta\delta_\zeta S_{(a)}
=\LamXi^4\int_{\mathcal M}\dvol\,
\left[\eta D_\lambda(\zeta)-\zeta D_\lambda(\eta)\right]=0.

Using reference and arbitrariness of the variations gives $D_\lambda=D_\lambda^*$TeXD_\lambda=D_\lambda^*.

For sufficiency, vary reference in the direction $\eta$TeX\eta:

\delta_\eta S_{(a)}
=\LamXi^4\int_0^1\dd t\int\dvol
\left[\eta\lambda[t\HH]+\HH D_\lambda[t\HH](t\eta)\right]

=\LamXi^4\int_0^1\dd t\int\dvol\,
\eta\left[\lambda[t\HH]+tD_\lambda[t\HH](\HH)\right].

The second equality uses formal self-adjointness. The expression in brackets is $\dd\{t\lambda[t\HH]\}/\dd t$TeX\dd\{t\lambda[t\HH]\}/\dd t. Integration over $t$TeXt therefore gives

\delta_\eta S_{(a)}=\LamXi^4\int_{\mathcal M}\dvol\,\eta\lambda[\HH],

which is reference. The star-shaped assumption ensures that $t\HH$TeXt\HH remains in the phase patch. This homotopy proof is the scalar-field form of the Helmholtz criterion [citation].

corollary: Two-stage rejection test for an exchange law. Let a modeled exchange vector $Q^\nu$TeXQ^\nu be given on a regular-gradient patch. It can arise from a local scalar phase-loading action only if

Q^{[\mu}\nabla^{\nu]}\HH=0
\quad\text{and}\quad
D_{\lambda_Q}=D_{\lambda_Q}^*,
\qquad
\lambda_Q:=\frac{Q^\nu\nabla_\nu\HH}
{\LamXi^4\nabla_\rho\HH\nabla^\rho\HH}.

Under the hypotheses of reference, these two conditions are also locally sufficient for a scalar-only variational realization of the aligned exchange law.

proof. The first condition is equivalent to pointwise alignment on the regular-gradient patch, and the displayed contraction recovers its scalar coefficient. The second condition is necessary and sufficient for that coefficient to be an Euler--Lagrange derivative by reference. Combining the two statements proves the claim.

proposition: Alignment alone is insufficient. On a one-dimensional compact-support variational domain, the loading $\lambda[q]=\dot q$TeX\lambda[q]=\dot q defines an aligned formal exchange $Q=\LamXi^4\dot q\,\dd q$TeXQ=\LamXi^4\dot q\,\dd q, but it is not the Euler derivative of any local scalar-only action on that domain.

proof. The linearization is $D_\lambda=\dd/\dd t$TeXD_\lambda=\dd/\dd t. Integration by parts gives $D_\lambda^*=-\dd/\dd t$TeXD_\lambda^*=-\dd/\dd t, so the Helmholtz condition fails unless the operator vanishes. The nonexistence follows from the necessity direction of reference, even though the exchange is algebraically parallel to $\dd q$TeX\dd q.

For a compact phase target the theorem is applied within a regular phase chart. A global action additionally requires the local representatives to agree on chart overlaps up to boundary divergences; the local Helmholtz condition does not remove a possible global cohomological obstruction.

theorem: Joint local Helmholtz criterion. Let $\varphi^A$TeX\varphi^A $(1\le A\le N)$TeX(1\le A\le N) be jointly varied fields on a contractible domain, let $E_A[\varphi]$TeXE_A[\varphi] be local differential functions on a star-shaped configuration patch, and let $D_{AB}$TeXD_{AB} denote the Fr\'echet derivative of $E_A$TeXE_A with respect to $\varphi^B$TeX\varphi^B. There exists a local functional $S[\varphi]$TeXS[\varphi] with

\frac{1}{\sqrt{-g}}\frac{\delta S}{\delta\varphi^A}=E_A

if and only if the full operator matrix satisfies

D_{AB}=D_{BA}^{*}
\qquad\text{for all }A,B.

When the condition holds, a representative is

S[\varphi]=\int_0^1\dd t\int_{\mathcal M}\dvol\,
\varphi^AE_A[t\varphi],

up to a field-independent functional and a boundary divergence.

proof. Necessity is symmetry of the mixed second variations:

\delta_{\zeta}\delta_{\eta}S-\delta_{\eta}\delta_{\zeta}S
=\int\dvol\,\eta^A(D_{AB}-D_{BA}^*)\zeta^B=0.

Arbitrariness of compactly supported $\eta$TeX\eta and $\zeta$TeX\zeta gives reference. Conversely, vary reference. After using reference to transfer the linearized operator between the two variations, the integrand becomes

\eta^A\left[E_A[t\varphi]+tD_{AB}[t\varphi](\varphi^B)\right]
=\eta^A\frac{\dd}{\dd t}\{tE_A[t\varphi]\}.

Integration from $t=0$TeXt=0 to $t=1$TeXt=1 gives $\delta S=\int\dvol\,\eta^AE_A[\varphi]$TeX\delta S=\int\dvol\,\eta^AE_A[\varphi].

theorem: Global gluing criterion for local phase actions. Let $\{U_i\}$TeX\{U_i\} be a good cover of the declared spacetime and phase bundle, and let $\mathcal L_i$TeX\mathcal L_i be local Lagrangian top forms on $U_i$TeXU_i that give the same Euler--Lagrange expressions. Suppose on every overlap

\mathcal L_i-\mathcal L_j=\dd B_{ij}.

The local Lagrangians can be modified by boundary terms to form a single global Lagrangian if and only if the overlap primitive is a Cech coboundary modulo exact forms: there exist local forms $b_i$TeXb_i and overlap forms $C_{ij}$TeXC_{ij} such that

B_{ij}=b_i-b_j+\dd C_{ij}.

In that case $\widetilde{\mathcal L}_i=\mathcal L_i-\dd b_i$TeX\widetilde{\mathcal L}_i=\mathcal L_i-\dd b_i agree on overlaps and define the global Lagrangian.

proof. If reference holds, then

\widetilde{\mathcal L}_i-\widetilde{\mathcal L}_j
=\dd(B_{ij}-b_i+b_j)=\dd^2C_{ij}=0,

so the modified forms glue. Conversely, if a global Lagrangian $\widetilde{\mathcal L}$TeX\widetilde{\mathcal L} differs locally from $\mathcal L_i$TeX\mathcal L_i by $\dd b_i$TeX\dd b_i, then

\dd B_{ij}=\mathcal L_i-\mathcal L_j=\dd(b_i-b_j).

On a good overlap the closed difference $B_{ij}-b_i+b_j$TeXB_{ij}-b_i+b_j is exact by the Poincar\'e lemma, giving reference. Failure of this coboundary condition is the global cohomological obstruction left invisible by the local Helmholtz equations.

corollary: Compact-phase action rejection test. For a compact phase covered by regular charts, formal self-adjointness of every local loading operator is insufficient for a global PLE action. A proposed global lift is rejected if either the joint matrix condition reference fails or the overlap class represented by $B_{ij}$TeXB_{ij} is nontrivial under reference.

proof. Apply reference within each chart and reference to their overlap data.

The identity is independent of any particular microscopic sector. The examples below fix representative readings of $\lambda_{(a)}$TeX\lambda_{(a)} on declared branches.

Electromagnetic-excitation sector with no direct phase loading

Let an electromagnetic-excitation sector be described by an action $S_{\rm car}[g,A]$TeXS_{\rm car}[g,A] with no direct dependence on $\HH$TeX\HH. The Maxwell action in a fixed metric-coupled form is the standard example:

S_{\rm car}[g,A]=-\frac14\int\dvol\,F_{\mu\nu}F^{\mu\nu}.

Then

\lambda_{\rm car}=0,
\qquad
\nabla_\mu T_{\rm car}^{\mu\nu}=0

provided the carrier field equations hold. Local propagation still depends on the metric and on the declared propagation branch; the zero-loading statement concerns only direct variational dependence on $\HH$TeX\HH.

Phase-dependent mass loading

Scalar-dependent inertial scales occur in scalar--tensor and dilaton-type coupling problems [citation]. The following model fixes only the local variational reading of such a dependence. Consider a real scalar matter field $\psi$TeX\psi with phase-dependent inertial scale:

S_\psi=\int\dvol\left[-\frac12 g^{\mu\nu}\nabla_\mu\psi\nabla_\nu\psi-\frac12 m^2(\HH)\psi^2\right].

The phase Euler density is

\Ecal_\HH^{(\psi)}=-\frac12\frac{\dd m^2}{\dd\HH}\psi^2,

so that

\lambda_\psi=-\frac{1}{2\LamXi^4}\frac{\dd m^2}{\dd\HH}\psi^2,
\qquad
\nabla_\mu T_\psi^{\mu\nu}=-\frac12\frac{\dd m^2}{\dd\HH}\psi^2\nabla^\nu\HH.

Mass-bearing response is therefore encoded by a nonzero variational loading against the global phase field on the declared branch. The equation does not determine the function $m(\HH)$TeXm(\HH); it states how any declared $m(\HH)$TeXm(\HH) sector exchanges energy--momentum with the phase field.

Constitutive rigidity loading

A minimal rigidity sector with a phase-dependent kinetic coefficient has

S_\chi=\int\dvol\left[-\frac12 Z(\HH)g^{\mu\nu}\nabla_\mu\chi\nabla_\nu\chi-V(\chi,\HH)\right].

Then

\lambda_\chi=\frac{1}{\LamXi^4}\left[-\frac12 Z'(\HH)g^{\mu\nu}\nabla_\mu\chi\nabla_\nu\chi-\partial_\HH V(\chi,\HH)\right],

and

\nabla_\mu T_\chi^{\mu\nu}=\LamXi^4\lambda_\chi\nabla^\nu\HH.

This representative form includes both mode-stiffness loading through $Z'(\HH)$TeXZ'(\HH) and potential loading through $\partial_\HH V$TeX\partial_\HH V. It is a constitutive branch statement, not a universal material theory.

Boundary-localized response is naturally represented by distributional phase loading. The distributional reading follows the same variational logic as boundary stress and thin-shell constructions, provided the induced metric and junction data are part of the declared variational problem [citation]. Let $\Sigma$TeX\Sigma be a smooth hypersurface with induced metric $h$TeXh and boundary fields $\chi$TeX\chi. A localized boundary action

S_\Sigma=\int_\Sigma \dd^3y\,\sqrt{|h|}\,B(\HH,\chi)

has the spacetime Euler density

\Ecal_\HH^{(\Sigma)}=\delta_\Sigma\,\partial_\HH B(\HH,\chi),

where $\delta_\Sigma$TeX\delta_\Sigma is the invariant hypersurface delta distribution. Hence

\nabla_\mu T_\Sigma^{\mu\nu}=\delta_\Sigma\,\partial_\HH B(\HH,\chi)\nabla^\nu\HH

as a representative distributional identity whenever the boundary fields, induced-metric variations, and junction data are varied consistently and the resulting surface stress contribution is included in $T_\Sigma^{\mu\nu}$TeXT_\Sigma^{\mu\nu}. A detector opening, an interface sink, or a registration boundary may therefore be represented by phase loading concentrated on the accessible boundary, while the internal detector Hamiltonian remains outside this identity.

In a declared weak-tunneling spectroscopy window, reduced current formulae may separate barrier support from registration availability, as in standard tunneling treatments [citation]. A reduced current of the form

I_\alpha(V,\lambda_c)=\int \dd E\,\mathcal R_\alpha(E,V,\lambda_c)\exp[-2\mathcal A_{B,\alpha}(E,V,\lambda_c)]+\delta I_\alpha

fixes only the declared reduced window. If the local phase chart enters only through a control parameter $\lambda_c=\lambda_c(\HH)$TeX\lambda_c=\lambda_c(\HH) on that reduced window, the corresponding reduced logarithmic phase-sensitivity proxy is

\chitun
=\frac{\partial}{\partial\HH}
\log\!\bigg[
\frac{1}{I_0}
\int \dd E\,\mathcal R_\alpha(E,V,\lambda_c(\HH))
e^{-2\mathcal A_{B,\alpha}(E,V,\lambda_c(\HH))}
\bigg].

Here $I_0>0$TeXI_0>0 is fixed and independent of $\HH$TeX\HH on the declared reduced window; it renders the logarithm dimensionless and drops out of the $\HH$TeX\HH derivative. The quantity $\chitun$TeX\chitun is not the Noether loading scalar $\lambda_{(a)}$TeX\lambda_{(a)} unless an action-level variational representation with fixed normalization and compatible stress-tensor and phase-Euler conventions has been supplied. It is the reduced-window phase sensitivity of the support--registration factorization and not a replacement for the microscopic tunneling action.

The theorem-level object is $\lambda_{(a)}$TeX\lambda_{(a)} in reference. In a declared reduced window one may instead have access only to a positive response scalar $Y_\alpha(\HH)$TeXY_\alpha(\HH). Such a response defines the reduced phase-sensitivity proxy

\chi_\alpha^{\rm red}
:=\frac{\partial}{\partial\HH}\log\!\left(\frac{Y_\alpha(\HH)}{Y_{\alpha,0}}\right),
\qquad
Y_{\alpha,0}>0,

where $Y_{\alpha,0}$TeXY_{\alpha,0} is fixed and independent of $\HH$TeX\HH on that window. The proxy can be compared with a Noether loading scalar only after an action-level variational representation with fixed normalization and compatible stress-tensor and phase-Euler conventions has been supplied. Without that representation it remains a window-level phase sensitivity.

Tunneling support-action extraction

For a declared weak-tunneling spectroscopy window with current factorization reference, assume $I_\alpha-\delta I_\alpha>0$TeXI_\alpha-\delta I_\alpha>0 and a calibrated registration integral

\mathcal N_\alpha(V,\lambda_c)=\int \dd E\,\mathcal R_\alpha(E,V,\lambda_c)>0.

The reduced support-action estimator is

\widehat{\mathcal A}_{B,\alpha}
=-\frac12\log\!\left(\frac{I_\alpha-\delta I_\alpha}{\mathcal N_\alpha}\right).

In an opaque single-action window where

I_\alpha(V,\lambda_c)-\delta I_\alpha
=\mathcal N_\alpha(V,\lambda_c)
\exp[-2\mathcal A_{B,\alpha}(V,\lambda_c)]
[1+\varepsilon_\alpha],
\qquad |\varepsilon_\alpha|\ll1,

reference gives

\widehat{\mathcal A}_{B,\alpha}
=\mathcal A_{B,\alpha}-\frac12\log(1+\varepsilon_\alpha).

If the control parameter is a phase function $\lambda_c=\lambda_c(\HH)$TeX\lambda_c=\lambda_c(\HH), the reduced logarithmic sensitivity becomes

\chi_{\rm tun}^{\rm red}
=\frac{\partial\log\mathcal N_\alpha}{\partial\HH}
-2\frac{\partial\mathcal A_{B,\alpha}}{\partial\HH}
+\frac{\partial}{\partial\HH}\log(1+\varepsilon_\alpha).

The same reduced-window reading can be written in terms of reference by taking

Y_\alpha(\HH)=
\int \dd E\,\mathcal R_\alpha(E,V,\lambda_c(\HH))
e^{-2\mathcal A_{B,\alpha}(E,V,\lambda_c(\HH))}.

It coincides with the tunneling sensitivity $\chitun$TeX\chitun of reference after the declared normalization has been fixed. This extraction logic applies to scanning tunneling, thin-film tunnel junctions, field-emission barriers, semiconductor tunnel junctions, and superconducting tunneling spectroscopy only on the declared barrier and registration windows [citation]. It does not supply a bulk-conductor theory, a microscopic tunneling-time law, or a Josephson phase-dynamics theory.

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

The phase-loading identity fixes a variational exchange form for declared $\HH$TeX\HH-dependent sector actions. It is a sector exchange identity rather than an isolated-sector conservation law: total covariant conservation belongs to the closed phase-plus-sector system. Declared reduced-window functionals enter only through separately stated phase-sensitivity proxies unless an action-level variational representation with fixed normalization and compatible stress-tensor and phase-Euler conventions is supplied. The scalar $\lambda_{(a)}$TeX\lambda_{(a)} is not a universal scalar charge density; it is a normalized phase-Euler response scalar tied to a declared sector action and phase chart. The identity does not supply a calibrated branch-fraction instance, a covariance object, or a same-window empirical assignment for electromagnetic-excitation branch tomography. It also does not select a sector action, a calibration map, a detector model, or an observational window.

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

The phase-loading identity can be read as a sector-exchange formalism only where the assumptions of reference are satisfied. It does not license the following inferences:

- a universal origin theorem for all particle masses; - a Standard-Model Higgs or Yukawa identity; - a Maxwell/QED replacement; - a detector microdynamics or universal measurement-collapse law; - a bulk conductor transport theory; - a branch-independent Planck-scale prediction; - an all-observation cosmological inference closure; - an electromagnetic-excitation branch calibration instance or same-window empirical branch-fraction assignment.

The phase-loading description fails on a declared window if any of the following conditions holds.

- Non-covariant sector data. The sector action contains an undeclared non-dynamical background whose Lie variation is not included. - No stress-tensor convention. The stress tensor is not derived from a metric variation compatible with reference or an explicitly equivalent convention. - Internal off-shell leakage. The internal field equations fail on the window and the residual terms cannot be bounded separately. - Singular phase chart. The phase reparametrization has $f'(\HH)=0$TeXf'(\HH)=0 or is not one-to-one on the declared patch, while scalar loading rather than the loading one-form is treated as invariant. - Regular-gradient extraction failure. The patch has $\nabla_\rho\HH\nabla^\rho\HH=0$TeX\nabla_\rho\HH\nabla^\rho\HH=0 and no alternate branch estimator is declared. - Orthogonal residual failure. The measured or modeled exchange vector has a nonzero orthogonal component $\mathcal R_{\perp,(a)}^\nu$TeX\mathcal R_{\perp,(a)}^\nu beyond the declared tolerance. - Boundary inconsistency. A boundary-localized loading term is used without consistent boundary variation or junction data. - Reduced-window overread. A declared phase-sensitivity proxy is promoted from a reduced window to an unrestricted microscopic theory. - Electromagnetic-excitation branch overread. A variational loading identity is used to assign an electromagnetic-excitation branch calibration instance, branch-fraction value, or same-window empirical closure that was not separately declared. - Conservation-law overread. The sector exchange identity is treated as a new conservation law for an isolated sector, or total conservation is asserted without including the phase-sector equation and all declared exchange terms. - Scalar-charge collapse. The loading scalar is treated as a universal scalar charge density independent of the declared sector action, stress-tensor convention, phase chart, or normalization scale. - Proxy-to-loading overread. A reduced sensitivity such as $\chitun$TeX\chitun, $\chi_{\rm tun}^{\rm red}$TeX\chi_{\rm tun}^{\rm red}, or $\chi_\alpha^{\rm red}$TeX\chi_\alpha^{\rm red} is used as a Noether loading scalar without an action-level variational representation. - Inverse-variational failure. An aligned phenomenological exchange law is promoted to an action-level PLE sector even though the extracted $\lambda$TeX\lambda violates the Helmholtz condition reference, or only the scalar diagonal of a coupled Helmholtz matrix has been checked.

The closure test for the covariant phase loading is applied to a dimensionless observable vector $y\in\mathbb R^m$TeXy\in\mathbb R^m formed from fixed reference scales and the declared basket of Euler loading densities, exchange currents, reduced response kernels, and chart-overlap data. Let $a$TeXa range over the independent constitutive inputs comprising sector Lagrangians, phase charts, boundary representatives, and reduction maps.

proposition: Functional saturation, finite closure, and sector admissibility. Suppose the unrestricted prediction map $F:a\mapsto y$TeXF:a\mapsto y is continuously differentiable on a Banach space of constitutive inputs. If $D_aF$TeXD_aF is surjective and has a bounded right inverse at the calibration point, the unrestricted family is locally open in observable space and supplies no nonzero local equality restriction on $y$TeXy. Suppose instead that a single microscopic closure replaces $a$TeXa by finite parameters $\theta\in\mathbb R^p$TeX\theta\in\mathbb R^p, with profiled nuisance coordinates $\eta\in\mathbb R^q$TeX\eta\in\mathbb R^q. If

J=D_\eta F_{\rm cl}D_\theta F_{\rm cl},
 \qquad \operatorname{rank}J=r<m,

then there are $m-r$TeXm-r independent first-order restrictions

w^{\mathsf T}\delta y=0,
 \qquad w\in\ker J^{\mathsf T}.

If the rank is constant locally, these restrictions are tangent to a compatibility manifold of codimension $m-r$TeXm-r. For this sector, the finite closure is admissible only if the joint Helmholtz symmetry and the global Cech gluing condition hold for one sector action.

proof. Split surjectivity gives a bounded right inverse $R$TeXR with $D_aF\,R=I_m$TeXD_aF\,R=I_m. The Banach-space submersion theorem then makes $F$TeXF locally onto a neighborhood of the calibrated observable vector. Any smooth equality holding throughout that image must therefore vanish on an open set and contributes no model-specific local restriction. Under finite closure, the attainable first-order variations are exactly the column space of $J$TeXJ. Its orthogonal complement is $\ker J^{\mathsf T}$TeX\ker J^{\mathsf T}, whose dimension is $m-r$TeXm-r by rank--nullity, which proves reference. The constant-rank theorem supplies the stated local manifold. The sector condition is necessary because component-wise local primitives need not assemble into a jointly varied or globally defined action. Failure of that condition therefore rejects the proposed microscopic closure before parameter estimation can be counted as evidence for it.

Microscopic loading example..

In Paper CHC-MHB, the compact phase enters one hopping matrix $K(\theta)$TeXK(\theta) and the load is fixed as $\mathcal L_\theta=-\partial_\theta H$TeX\mathcal L_\theta=-\partial_\theta H. For its eigenbranches,

\langle\mathcal L_\theta\rangle_k
 =-\hbar\,\partial_\theta\Omega_k
 =\frac{2\hbar g}{3}\sin\!\left(\frac{\theta+2\pi k}{3}\right).

The same Hamiltonian derivative enters the retarded response. This finite action therefore realizes the common-primitive condition explicitly and forbids independent retuning of the static phase load and its susceptibility.

The phase-gradient variable $\Xi$TeX\Xi describes the state of the global phase field in the admitted sector; where a cosmological reading is involved, it tracks the corresponding expansion-response branch. The phase-loading scalar $\lambda_{(a)}$TeX\lambda_{(a)} describes how a declared sector loads that global phase field. Diffeomorphism invariance then fixes the sector exchange identity

\nabla_\mu T_{(a)}^{\mu\nu}=\LamXi^4\lambda_{(a)}\nabla^\nu\HH.

The equation is exact under the stated variational assumptions and is a sector exchange identity, not a new isolated-sector conservation law. Its chart-invariant content is the one-form $\LamXi^4\lambda_{(a)}\nabla_\nu\HH$TeX\LamXi^4\lambda_{(a)}\nabla_\nu\HH, while total covariant conservation is recovered for the closed phase-plus-sector system after the phase equation is imposed. Its regular-gradient projection gives an extractor for the loading scalar together with an orthogonal-residual failure condition.

As a sector-exchange identity, the result is downstream of the declared sector actions to which it is applied: it organizes their $\HH$TeX\HH-loading once those actions have been fixed. The inverse theorem closes the converse local question. A proposed exchange law must first be parallel to the phase gradient and must then pass formal self-adjointness of the extracted loading operator; alignment alone is insufficient. Reduced-window functionals enter only through phase-sensitivity proxies unless this variational test, fixed normalization, and compatible stress-tensor and phase-Euler conventions are supplied. The result does not by itself select a sector action, calibration map, detector model, or observational window.

Sectors with no direct phase dependence have zero direct phase loading. Mass-bearing, rigidity, boundary, detector-facing, and electromagnetic-excitation sectors carry theorem-level loading only when represented by declared $\HH$TeX\HH-dependent sector actions. Tunneling-reduced descriptions and other reduced-window descriptions supply only phase-sensitivity proxies unless action-level variational representations with fixed normalization and compatible stress-tensor and phase-Euler conventions are supplied. Inclusion of the phase equation restores total covariant conservation. The result supplies a common exchange identity for CHC sectors without replacing their field equations, their detector microdynamics, their gauge dynamics, or their window-specific empirical closures.

When several fields vary, every cross derivative must satisfy the joint adjoint condition, and compact-phase chart representatives must pass the global Čech gluing test. These conditions turn local alignment into a falsifiable action-level claim without presuming that an application has already supplied the required global primitive.

For a scalar sector with

S_\psi=\int\dvol\left[-\frac12 g^{\mu\nu}\nabla_\mu\psi\nabla_\nu\psi-U(\psi,\HH)\right],

one has

T_\psi^{\mu\nu}=\nabla^\mu\psi\nabla^\nu\psi-g^{\mu\nu}\left(\frac12\nabla_\rho\psi\nabla^\rho\psi+U(\psi,\HH)\right).

A direct calculation gives

\nabla_\mu T_\psi^{\mu\nu}
=(\Box\psi-\partial_\psi U)\nabla^\nu\psi-\partial_\HH U\nabla^\nu\HH.

On the scalar equation $\Box\psi-\partial_\psi U=0$TeX\Box\psi-\partial_\psi U=0,

\nabla_\mu T_\psi^{\mu\nu}=-\partial_\HH U\nabla^\nu\HH.

Since $\Ecal_\HH^{(\psi)}=-\partial_\HH U$TeX\Ecal_\HH^{(\psi)}=-\partial_\HH U, this is exactly reference.

The estimate below is a conditional stability statement for a declared regular-gradient window, not a numerical assignment and not an interval-arithmetic enclosure. All constants entering the bound must be supplied by the same window and norm convention. Let a regular-gradient window have a lower bound

|\nabla_\rho\HH\nabla^\rho\HH|\ge c_H>0.

If a modeled exchange vector $F_{(a)}^\nu$TeXF_{(a)}^\nu is known with an absolute error bound

\|F_{(a)}-\nabla_\mu T_{(a)}^{\mu\cdot}\|\le \epsilon_F

and $\nabla\HH$TeX\nabla\HH is known with error at most $\epsilon_H$TeX\epsilon_H, then the extractor reference is stable whenever $c_H$TeXc_H remains larger than the induced denominator error. To first order,

|\delta\lambda_{(a)}|
\le
\frac{|\nabla\HH|\epsilon_F+|F_{(a)}|\epsilon_H}{\LamXi^4 c_H}
+\frac{|F_{(a)}\cdot\nabla\HH|}{\LamXi^4 c_H^2}\,\delta c_H
+O(\epsilon^2).

Thus the extractor is not used on phase-flat or null-gradient patches without a separately declared branch estimator.

Funding and competing interests..

No external funding was received for this work. The author declares no competing interests.