CHC: A Restricted Covariant Expansion-Phase Scalar-Tensor Framework with Controlled General-Relativistic Recovery

Scalar-tensor departures from general relativity are tightly constrained by the multimessenger luminality bound following GW170817/GRB170817A, by solar-system light-deflection and time-delay measurements, and by current strong-field gravitational-wave consistency tests, including the GWTC-3 published analysis and the GWTC-4.0 O4a general, parameterized, and remnant-test papers [citation]. The -min action, the admissible nondegenerate branch, the reduced degree-of-freedom count, the conditional hyperbolicity assumptions, the controlled recovery theorem, and the exclusion criterion are specified below.

The construction uses a scalar global phase field $\HH(x)$TeX\HH(x) coupled covariantly to the metric and to a constrained auxiliary sector. In cosmological applications, the relevant sector of this field can describe an expansion response. On the admissible nondegenerate branch, however, the covariant core is exactly equivalent to GR plus one canonical scalar. This equivalence fixes what the root action does and does not imply. In particular, a small gradient does not by itself guarantee recovery if the field remains far from a stationary value of its potential. We therefore use the independent parameters

\Xi_{\nabla}\equiv\frac{\mathcal K_\chi}{\Lambda_\Xi^4},
\qquad
\Xi_V\equiv\frac{|V(\chi)-V(\chi_\star)|}{\Lambda_\Xi^4},
\qquad
\epsilon_{\CHC}\equiv\Xi_{\nabla}+\Xi_V,

where $\mathcal K_\chi$TeX\mathcal K_\chi is the background-adapted nonnegative norm of $\nabla\chi$TeX\nabla\chi. A separate covariant test-probe coupling is introduced below to determine which optical and clock laws can follow from the field content.

The assumptions below define the branch variables, admissibility conditions, and hierarchy conventions for the restricted branch. Only the consequences required for branch admissibility, stability, recovery, and hierarchy control are developed explicitly. The finite-access, bookkeeping, and empirical-discipline assumptions restrict the allowed branch and are not expanded into detector statistics or event-probability laws.

- Global phase field and expansion-response branch. The construction uses a scalar global phase field $\HH(x)$TeX\HH(x). Its target topology is part of the theory data: the noncompact branch uses a real target, whereas the compact branch uses $S^1_P$TeXS^1_P and requires every coefficient in the reduced action to descend to that circle. Its cosmological sector encodes an expansion-response branch where applicable, while local matter, boundary, media, and detector effects are described through local phase-field responses rather than through a separate material carrier. - Local covariance. Laws are tensorial and form-invariant under diffeomorphisms. - Phase-response encoding. Observable structure is represented through phase response to $\HH$TeX\HH on finite observation domains; the locally accessible information used in the construction need not coincide with the full global information state. - Controlled recovery. There is a stationary value $\HH_\star$TeX\HH_\star of the reduced potential. The metric equations reduce exactly to GR+$\Lambda_{\mathrm{eff}}$TeX\Lambda_{\mathrm{eff}} at $\HH=\HH_\star$TeX\HH=\HH_\star and continuously approach that limit when both the kinetic and potential displacement measures in reference vanish. - Minimal propagation. Beyond the two tensor polarizations of GR, \ carries at most one branch-local non-ghost scalar in the nondegenerate branch under the stated positivity and hyperbolicity assumptions. - Energetic consistency. The scalar sector contributes positive energy under the positivity conditions summarized in reference. - Information-energy bookkeeping. Any information-bearing realization used in the construction is required to satisfy thermodynamic bookkeeping compatible with the hierarchy variables. - Empirical admissibility. Parameter choices are subject to explicit falsification thresholds when independent probes are introduced. - Recovery hierarchy. Let $\chi$TeX\chi be the canonical field defined in reference and $V_\star=V(\chi_\star)$TeXV_\star=V(\chi_\star). With signature $(-,+,+,+)$TeX(-,+,+,+), define $\mathcal K_\chi=-g^{\mu\nu}\nabla_\mu\chi\nabla_\nu\chi$TeX\mathcal K_\chi=-g^{\mu\nu}\nabla_\mu\chi\nabla_\nu\chi on a homogeneous timelike background and $\mathcal K_\chi=g^{ij}\partial_i\chi\partial_j\chi$TeX\mathcal K_\chi=g^{ij}\partial_i\chi\partial_j\chi in a static spatial problem. The independent expansion parameters are

\Xi_{\nabla}=\frac{\mathcal K_\chi}{\Lambda_\Xi^4}, \qquad \Xi_V=\frac{|V(\chi)-V_\star|}{\Lambda_\Xi^4}.

For homogeneous configurations, $\Xi_{\nabla}=\Meff^2\dot{\HH}^{\,2}/\Lambda_\Xi^4$TeX\Xi_{\nabla}=\Meff^2\dot{\HH}^{\,2}/\Lambda_\Xi^4. The earlier symbol $\Xi$TeX\Xi is retained only as shorthand for $\Xi_{\nabla}$TeX\Xi_{\nabla} in formulae that depend exclusively on the scalar gradient; it is not by itself a recovery criterion.

We use the metric signature $(-,+,+,+)$TeX(-,+,+,+) and $\Box\equiv\nabla_\mu\nabla^\mu$TeX\Box\equiv\nabla_\mu\nabla^\mu throughout. For the Lorentzian kinetic contractions entering the action and field equations, we write

X_{\HH}\equiv g^{\mu\nu}\nabla_\mu\HH\nabla_\nu\HH,
\qquad
X_{\PPhi}\equiv g^{\mu\nu}\nabla_\mu\PPhi\nabla_\nu\PPhi.

The nonnegative kinetic norm $\mathcal K_\chi$TeX\mathcal K_\chi used in $\Xi_{\nabla}$TeX\Xi_{\nabla} is the background-adapted quantity defined in Assumption L9.

We posit the covariant action

S = \int \dd^4x\,\sqrt{-g}\,\Big[\frac{\Mpl^2}{2}R + \Lag_{\CHC}(g,\HH,\PPhi,\OOm) + \Lag_{\mathrm{m}}(g,\Psi)\Big] + S_{\mathrm{GHY}} + S^{\CHC}_{\mathrm{bdry}},

with

\Lag_{\CHC} = -\frac{\Mh^2}{2}X_{\HH} - \frac{\Mp^2}{2}X_{\PPhi} - \UH(\HH,\PPhi) + \OOm\big(\PPhi-f(\HH)\big).

For Neumann or mixed scalar data we use the boundary term

S^{\CHC}_{\mathrm{bdry}} = + \int_{\partial M}\!\dd^3x\,\sqrt{|h|}\,\Big(\Mh^2\,\HH\,n^\mu\nabla_\mu\HH + \Mp^2\,\PPhi\,n^\mu\nabla_\mu\PPhi\Big),

which implements the scalar-side boundary Legendre transform for the adopted sign convention. For pure Dirichlet data on $(\HH,\PPhi)$TeX(\HH,\PPhi) one omits $S^{\CHC}_{\mathrm{bdry}}$TeXS^{\CHC}_{\mathrm{bdry}}.

The Euler-Lagrange equations are

\Mpl^2 G_{\mu\nu} = T^{\mathrm{m}}_{\mu\nu} + T^{\CHC}_{\mu\nu}, 

\Mh^2 \Box \HH - \partial_{\HH}\UH - \OOm f'(\HH) = 0, 

\Mp^2 \Box \PPhi - \partial_{\PPhi}\UH + \OOm = 0, 

\PPhi - f(\HH) = 0.

The -sector stress tensor is

T^{\CHC}_{\mu\nu}
= \Mh^2\nabla_\mu\HH\nabla_\nu\HH + \Mp^2\nabla_\mu\PPhi\nabla_\nu\PPhi 

\quad - g_{\mu\nu}\!\left[\frac{\Mh^2}{2}X_{\HH} + \frac{\Mp^2}{2}X_{\PPhi} + \UH(\HH,\PPhi) - \OOm\big(\PPhi-f(\HH)\big)\right].

Admissible -min branch

We work throughout in the nondegenerate -min branch defined by the following conditions:

- the auxiliary constraint is enforced as $\PPhi=f(\HH)$TeX\PPhi=f(\HH) on the domain of interest, - the reduced kinetic prefactor

\Meff^2(\HH) \equiv \Mh^2 + \Mp^2 f'(\HH)^2

obeys $\Meff^2(\HH)\ge m_0^2>0$TeX\Meff^2(\HH)\ge m_0^2>0, - the reduced potential $U_{\mathrm{eff}}(\HH)\equiv U(\HH,f(\HH))$TeXU_{\mathrm{eff}}(\HH)\equiv U(\HH,f(\HH)) has $U''_{\mathrm{eff}}(\HH_\star)>0$TeXU''_{\mathrm{eff}}(\HH_\star)>0 near the recovery point, and - the reduced chart is used only on domains where the chosen branch parametrization remains regular. Zeros of $f'(\HH)$TeXf'(\HH) are not singular in the reduced kinetic coefficient by themselves; they matter only if an alternative inversion or auxiliary-field chart is invoked.

Alternative local charts may be introduced near isolated zeros of $f'(\HH)$TeXf'(\HH) if needed, but no such reparametrization is required for the reduced theory used below.

proposition: Reduced propagating content on the -min branch. On the -min branch, the scalar sector reduces to a single effective field $\HH$TeX\HH with Lagrangian density

\Lag_{\CHC}^{\rm eff} = -\frac{1}{2}\Meff^2(\HH)X_{\HH} - U_{\rm eff}(\HH),
\qquad U_{\rm eff}(\HH)=U\big(\HH,f(\HH)\big).

After the standard diffeomorphism constraints of GR are handled, the propagating content is the two tensor polarizations of GR plus one scalar degree of freedom.

proof. Variation with respect to $\OOm$TeX\OOm gives the algebraic equation $\PPhi=f(\HH)$TeX\PPhi=f(\HH). Hence $\nabla_\mu\PPhi=f'(\HH)\nabla_\mu\HH$TeX\nabla_\mu\PPhi=f'(\HH)\nabla_\mu\HH, and direct substitution in reference gives

-\frac12\big[\Mh^2+\Mp^2f'(\HH)^2\big]X_{\HH}-U\big(\HH,f(\HH)\big).

The multiplier has no kinetic term and the constraint removes the independent $\PPhi$TeX\PPhi direction. Equivalently, eliminating $\OOm$TeX\OOm from the two scalar equations yields

\Meff^2\Box\HH+\frac12(\Meff^2)'X_{\HH}-U_{\rm eff}'=0,

which is the Euler--Lagrange equation of the displayed reduced action. Thus only one scalar Cauchy datum pair remains. The metric sector retains the two physical tensor polarizations after its four diffeomorphism constraints and four gauge conditions are imposed.

theorem: Canonical-field equivalence. On every connected field interval $I$TeXI on which $\Meff^2(\HH)\ge m_0^2>0$TeX\Meff^2(\HH)\ge m_0^2>0, define

\chi(\HH)=\int_{\HH_0}^{\HH}\Meff(u)\,\dd u,
\qquad
V(\chi)=U_{\rm eff}\big(\HH(\chi)\big).

The reduced CHC action on $I$TeXI is exactly the Einstein--canonical-scalar action

S_{\rm red}=\int\dd^4x\sqrt{-g}\left[\frac{\Mpl^2}{2}R-\frac{1}{2}g^{\mu\nu}\nabla_\mu\chi\nabla_\nu\chi-V(\chi)+\Lag_{\rm m}(g,\Psi)\right]
+S_{\rm GHY}+S^{\chi}_{\rm bdry}.

The equivalence is local and invertible throughout $I$TeXI.

proof. The lower bound on $\Meff$TeX\Meff implies $\dd\chi/\dd\HH=\Meff>0$TeX\dd\chi/\dd\HH=\Meff>0. The inverse-function theorem therefore supplies a smooth inverse $\HH(\chi)$TeX\HH(\chi) on $I$TeXI. The chain rule gives $\nabla_\mu\chi=\Meff\nabla_\mu\HH$TeX\nabla_\mu\chi=\Meff\nabla_\mu\HH and hence $(\nabla\chi)^2=\Meff^2X_{\HH}$TeX(\nabla\chi)^2=\Meff^2X_{\HH}. Substitution in the reduced action proves the identity. Because the transformation is invertible and contains no derivatives, it preserves the Euler--Lagrange solution space, the constraint count, and the principal characteristics.

The word phase acquires global mathematical content only after its target topology has been specified. We therefore distinguish the noncompact branch $\HH:M\to\mathbb R$TeX\HH:M\to\mathbb R from the compact branch $\HH:M\to S^1_P$TeX\HH:M\to S^1_P, where $S^1_P=\mathbb R/P\mathbb Z$TeXS^1_P=\mathbb R/P\mathbb Z has coordinate period $P>0$TeXP>0. On the compact branch, $f$TeXf, $\Meff$TeX\Meff, and $U_{\rm eff}$TeXU_{\rm eff} are required to be single-valued periodic functions of $\HH$TeX\HH. This is target-space compactness of the four-dimensional scalar and is distinct from compactification of an additional spacetime dimension.

theorem: Local canonical equivalence and global compact inequivalence. Let $\HH:M\to S^1_P$TeX\HH:M\to S^1_P, let $\Meff\in C^1(S^1_P)$TeX\Meff\in C^1(S^1_P) satisfy $\Meff\ge m_0>0$TeX\Meff\ge m_0>0, and let $U_{\rm eff}$TeXU_{\rm eff} be single valued on $S^1_P$TeXS^1_P. Define the invariant target circumference

L_\chi:=\oint_{S^1_P}\Meff(\HH)\,\dd\HH.

Then every target patch admits the canonical coordinate of reference. Globally, however, the canonical coordinate obeys

\chi\sim\chi+L_\chi,

so the reduced scalar is a canonical map into $S^1_{L_\chi}$TeXS^1_{L_\chi} and is not globally equivalent to a scalar with target $\mathbb R$TeX\mathbb R. No regular local field redefinition removes this distinction.

proof. On every simply connected target patch, the one-form $\Meff(\HH)\dd\HH$TeX\Meff(\HH)\dd\HH is the differential of a strictly increasing coordinate $\chi$TeX\chi, so the proof of reference applies unchanged. Following the coordinate once around the target changes its lift by the period integral reference; hence the transition law is reference. The arc-length coordinate therefore identifies the target with a circle of circumference $L_\chi$TeXL_\chi. A regular field redefinition is a target diffeomorphism and preserves the fundamental group. Since $\pi_1(S^1)=\mathbb Z$TeX\pi_1(S^1)=\mathbb Z whereas $\pi_1(\mathbb R)=0$TeX\pi_1(\mathbb R)=0, no global diffeomorphism can identify the two targets.

For a closed spatial curve $\gamma$TeX\gamma, the pullback of the angular one-form defines

n_\gamma:=\frac1P\oint_\gamma\dd\HH\in\mathbb Z.

Equivalently, the canonical arc-length one-form has period

\oint_\gamma\dd\chi=n_\gamma L_\chi.

theorem: Sharp winding-energy bound. Let $\gamma$TeX\gamma be a spatial circle of proper length $\ell>0$TeX\ell>0, parametrized by arc length $s$TeXs, and let the compact phase field restricted to $\gamma$TeX\gamma have winding $n$TeXn. Its one-dimensional Dirichlet energy satisfies

E_\gamma[\HH]
:=\frac12\int_0^\ell\left(\frac{\dd\chi}{\dd s}\right)^2\dd s
\ge \frac{n^2L_\chi^2}{2\ell}.

Equality holds if and only if $\dd\chi/\dd s=nL_\chi/\ell$TeX\dd\chi/\dd s=nL_\chi/\ell almost everywhere. On a product spatial geometry $S^1_\ell\times N$TeXS^1_\ell\times N with a field independent of $N$TeXN, the full gradient energy is $\operatorname{Vol}(N)E_\gamma$TeX\operatorname{Vol}(N)E_\gamma.

proof. By reference and Cauchy--Schwarz,

n^2L_\chi^2
=\left(\int_0^\ell\frac{\dd\chi}{\dd s}\,\dd s\right)^2
\le \ell\int_0^\ell\left(\frac{\dd\chi}{\dd s}\right)^2\dd s.

Division by $2\ell$TeX2\ell proves reference. Equality in Cauchy--Schwarz holds exactly when $\dd\chi/\dd s$TeX\dd\chi/\dd s is constant almost everywhere. The product-space statement follows by direct integration over $N$TeXN.

corollary: Homotopy preservation and obstruction to the constant GR branch. For a smooth one-parameter family of compact phase configurations with no phase flux through the boundary and no singular event, $n_\gamma$TeXn_\gamma is constant. If $n_\gamma\ne0$TeXn_\gamma\ne0 for at least one closed spatial curve, the configuration cannot be continuously deformed within that class to the constant-field GR+$\Lambda_{\rm eff}$TeX\Lambda_{\rm eff} solution.

proof. A smooth time evolution is a homotopy of maps into $S^1$TeXS^1. The degree on a closed curve is homotopy invariant, so $n_\gamma$TeXn_\gamma cannot change. A constant map has zero degree on every closed curve and therefore does not belong to a nonzero-winding component.

theorem: Topological obstruction to a phase-only defect core. Let $D^2$TeXD^2 be a closed spatial disk and let $\HH:\partial D^2\to S^1_P$TeX\HH:\partial D^2\to S^1_P have degree $n\ne0$TeXn\ne0. There is no continuous extension $\widehat{\HH}:D^2\to S^1_P$TeX\widehat{\HH}:D^2\to S^1_P with $\widehat{\HH}|_{\partial D^2}=\HH$TeX\widehat{\HH}|_{\partial D^2}=\HH. Consequently a codimension-two configuration with nonzero compact-phase winding cannot possess a regular core within the phase-only target. A regular completion must leave the $S^1_P$TeXS^1_P target at the core, introduce a boundary or puncture, or enlarge the field space.

proof. If an extension existed, the boundary map would be the composition of the inclusion $\partial D^2\hookrightarrow D^2$TeX\partial D^2\hookrightarrow D^2 with $\widehat{\HH}$TeX\widehat{\HH}. On fundamental groups this composition factors through $\pi_1(D^2)=0$TeX\pi_1(D^2)=0, so the induced homomorphism $\pi_1(S^1)\to\pi_1(S^1_P)$TeX\pi_1(S^1)\to\pi_1(S^1_P) would be trivial. Its degree would therefore be zero, contradicting $n\ne0$TeXn\ne0.

theorem: Compactness does not select constitutive data. Let $\HH_1,\ldots,\HH_N$TeX\HH_1,\ldots,\HH_N be distinct points of $S^1_P$TeXS^1_P and let $a_1,\ldots,a_N\in\mathbb R$TeXa_1,\ldots,a_N\in\mathbb R. There exists a smooth real $P$TeXP-periodic function $c(\HH)$TeXc(\HH) satisfying $c(\HH_j)=a_j$TeXc(\HH_j)=a_j for every $j$TeXj. Hence target compactness and any finite list of calibration values do not determine a probe function, mass function, or other constitutive coefficient.

proof. Put $z_j=\exp(2\pi i\HH_j/P)$TeXz_j=\exp(2\pi i\HH_j/P). Complex polynomial interpolation gives a polynomial $q(z)$TeXq(z) with $q(z_j)=a_j$TeXq(z_j)=a_j. On the unit circle define the Laurent polynomial

p(z)=\frac12\left[q(z)+\overline{q(1/\overline z)}\right].

For $|z|=1$TeX|z|=1, one has $1/\overline z=z$TeX1/\overline z=z, so $p(z)$TeXp(z) is real and $p(z_j)=a_j$TeXp(z_j)=a_j. The function $c(\HH)=p(\exp(2\pi i\HH/P))$TeXc(\HH)=p(\exp(2\pi i\HH/P)) is smooth and $P$TeXP-periodic. Infinitely many further solutions follow by adding any smooth periodic function vanishing at all $\HH_j$TeX\HH_j.

Compact scalar targets and winding sectors are established field-theoretic structures [citation]. The CHC-specific result is narrower: the same arc-length transformation that proves exact local equivalence to a canonical scalar also exposes the invariant circumference and the global obstruction. It adds no local propagating degree of freedom. A singular defect core is not resolved by a phase-only target; such a core requires an enlarged field space, boundary data, or ultraviolet completion.

Hyperbolicity and local well-posedness

Introduce $X_\mu\equiv\nabla_\mu\HH$TeXX_\mu\equiv\nabla_\mu\HH and work in a standard strongly hyperbolic gravitational reduction. Using standard hyperbolic formulations of Einstein's equations together with well-posed scalar-tensor EFT reductions as external anchors [citation], the linearized principal symbol block-diagonalizes as

\mathcal{P}(\xi)=\mathrm{diag}\!\big(\mathcal{P}_{\mathrm{GR}}(\xi),\,\Meff^2\xi^2\big),\qquad \xi^2\equiv g^{\mu\nu}\xi_\mu\xi_\nu,

provided the branch conditions above hold.

theorem: Conditional local Cauchy theory on the nondegenerate branch. Assume the -min branch conditions and a generalized harmonic reduction of the gravitational equations. For initial data in $H^s\times H^{s-1}$TeXH^s\times H^{s-1} with $s>5/2$TeXs>5/2 that satisfy the Einstein constraints, the reduced vacuum or minimally coupled Einstein--CHC system has a unique local solution, up to diffeomorphism, with continuous dependence on the data for as long as the solution remains inside the nondegenerate field interval.

proof. By reference, the reduced equations are those of Einstein gravity coupled to a canonical scalar. In generalized harmonic coordinates the metric equation has principal part $-\tfrac12g^{\rho\sigma}\partial_\rho\partial_\sigma g_{\mu\nu}$TeX-\tfrac12g^{\rho\sigma}\partial_\rho\partial_\sigma g_{\mu\nu}, while the scalar equation has principal part $g^{\rho\sigma}\partial_\rho\partial_\sigma\chi$TeXg^{\rho\sigma}\partial_\rho\partial_\sigma\chi. A standard first-order reduction therefore has a block-diagonal principal symbol whose two blocks share the metric null cone and possess a complete real characteristic basis. It is strongly hyperbolic on every Lorentzian background on which the gauge reduction is regular. The coefficients are smooth functions of the fields on the chosen interval, so the quasilinear hyperbolic existence theorem applies for $s>n/2+1=5/2$TeXs>n/2+1=5/2 in three spatial dimensions [citation]. Constraint propagation follows from the contracted Bianchi identity and the scalar equation. Invertibility of $\HH\leftrightarrow\chi$TeX\HH\leftrightarrow\chi transfers the result back to the CHC variables.

The theorem concerns the reduced leading-order -min system on the stated branch. Higher-derivative EFT operators are organized perturbatively and are not promoted here to an independent well-posed PDE claim.

Energy positivity and stability

The unconstrained two-field Hessian is not the physical stability matrix because $\delta\PPhi=f'(\HH_0)\delta\HH$TeX\delta\PPhi=f'(\HH_0)\delta\HH on the constraint surface. Stability must be tested after reduction.

proposition: Reduced energy criterion. Let $\chi_0$TeX\chi_0 be a stationary point of $V$TeXV. The scalar perturbation has positive kinetic and gradient energy if and only if $\Meff^2(\HH_0)>0$TeX\Meff^2(\HH_0)>0. It is linearly non-tachyonic if, in addition,

m_\chi^2=V''(\chi_0)=\frac{U_{\rm eff}''(\HH_0)}{\Meff^2(\HH_0)}\ge0.

If $V$TeXV is bounded below on the field interval, the full reduced scalar Hamiltonian is bounded below there.

proof. In a local orthonormal frame, the canonical action of reference gives

\mathcal H_\chi=\frac12\dot\chi^2+\frac12|\boldsymbol\nabla\chi|^2+V(\chi).

The transformation exists precisely when $\Meff^2>0$TeX\Meff^2>0, proving the kinetic and gradient statement. Expanding $V(\chi_0+\delta\chi)$TeXV(\chi_0+\delta\chi) at $V'(\chi_0)=0$TeXV'(\chi_0)=0 gives $\mathcal H^{(2)}=\tfrac12\dot{\delta\chi}^{\,2}+\tfrac12|\nabla\delta\chi|^2+\tfrac12V''(\chi_0)\delta\chi^2$TeX\mathcal H^{(2)}=\tfrac12\dot{\delta\chi}^{\,2}+\tfrac12|\nabla\delta\chi|^2+\tfrac12V''(\chi_0)\delta\chi^2. At a stationary point the chain rule gives $V''=U_{\rm eff}''/\Meff^2$TeXV''=U_{\rm eff}''/\Meff^2. The remaining assertions follow directly from this quadratic form and from the lower bound on $V$TeXV.

Higher operators are organized by a cutoff $\Lambda_{\HH}$TeX\Lambda_{\HH}. -min is restricted to a luminality-safe admissible basis, motivated by the post-GW170817 requirement that tensor modes remain luminal to very high accuracy [citation].

definition: Luminality-safe operator basis. The admissible operator class $\mathfrak{I}_{\mathrm{lum}}$TeX\mathfrak{I}_{\mathrm{lum}} is the smallest diffeomorphism-covariant set of local operators that is closed under integrations by parts, local field redefinitions, and use of the leading equations of motion, while excluding curvature-dressed scalar-gradient terms that would split the tensor kinetic and gradient coefficients on smooth backgrounds. In particular, operators of the forms

R(\nabla\HH)^2,\qquad R_{\mu\nu}\nabla^\mu\HH\nabla^\nu\HH,\qquad G^{\mu\nu}\nabla_\mu\HH\nabla_\nu\HH

are excluded from the admissible -min basis unless their tensor-sector contribution cancels identically.

Within this admissible basis,

\Delta\Lag_{\CHC}^{\mathrm{EFT}} = \sum_i \frac{c_i}{\Lambda_{\HH}^{\Delta_i-4}}\,\mathcal{O}_i,

\mathcal{O}_i \in \Big\{ (\nabla\HH)^4,\; (\nabla\HH)^2(\nabla\PPhi)^2,

 (\nabla_\mu\HH\nabla_\nu\HH)(\nabla^\mu\PPhi\nabla^\nu\PPhi),\; \nabla^2\HH(\nabla\HH)^2,\ldots\Big\}.

-min is formulated within this admissible operator class. The loop-level remarks recorded in reference remain within the same restricted basis and are not used as independent field equations.

The derivative expansion requires $\Xi_{\nabla}\ll1$TeX\Xi_{\nabla}\ll1 and characteristic momenta below the cutoff. This condition controls higher-gradient operators; it is logically independent of the potential-displacement condition $\Xi_V\ll1$TeX\Xi_V\ll1 required for GR recovery.

theorem: Exact GR branch and quantitative recovery. Let $\chi_\star$TeX\chi_\star satisfy $V'(\chi_\star)=0$TeXV'(\chi_\star)=0 and set $\Lambda_{\rm eff}=V(\chi_\star)/\Mpl^2$TeX\Lambda_{\rm eff}=V(\chi_\star)/\Mpl^2. Then $\chi=\chi_\star$TeX\chi=\chi_\star is an exact solution of the scalar equation, and the metric equation is exactly

G_{\mu\nu}+\Lambda_{\mathrm{eff}}g_{\mu\nu}=\Mpl^{-2}T^{\mathrm{m}}_{\mu\nu}.

If $V$TeXV is continuous near $\chi_\star$TeX\chi_\star, then on every compact neighborhood contained in that domain there are constants $C_1,C_2>0$TeXC_1,C_2>0 such that, in any local orthonormal frame,

\left|T^{\chi}_{ab}+V(\chi_\star)\eta_{ab}\right|
\le C_1|\partial\chi|^2+C_2|V(\chi)-V(\chi_\star)|.

Consequently the scalar stress converges uniformly to a cosmological-constant stress along any family for which both $\Xi_{\nabla}$TeX\Xi_{\nabla} and $\Xi_V$TeX\Xi_V vanish uniformly. Vanishing $\Xi_{\nabla}$TeX\Xi_{\nabla} alone is insufficient.

proof. The canonical scalar equation is $\Box\chi-V'(\chi)=0$TeX\Box\chi-V'(\chi)=0. A constant $\chi_\star$TeX\chi_\star solves it exactly when $V'(\chi_\star)=0$TeXV'(\chi_\star)=0. Its stress tensor is $T^\chi_{\mu\nu}=-V(\chi_\star)g_{\mu\nu}$TeXT^\chi_{\mu\nu}=-V(\chi_\star)g_{\mu\nu}, which proves the Einstein equation. For a nearby field,

T^\chi_{ab}+V_\star\eta_{ab}
=\partial_a\chi\partial_b\chi-\eta_{ab}\!\left[\frac12(\partial\chi)^2+V(\chi)-V_\star\right].

The triangle inequality in the chosen frame gives the stated estimate directly; the frame-dependent numerical constants are uniform on the selected compact set. The bound gives uniform convergence when both control quantities vanish uniformly. A constant configuration away from a stationary point has $\Xi_{\nabla}=0$TeX\Xi_{\nabla}=0 but either violates the scalar equation or carries a different vacuum stress, proving the final assertion.

The core action minimally couples $\Lag_{\rm m}$TeX\Lag_{\rm m} to $g_{\mu\nu}$TeXg_{\mu\nu}. Therefore its local freely falling matter cone is the metric cone. A distinct optical or clock response requires an interaction term. In the negligible-backreaction limit, that interaction can be parameterized without choosing a detector microphysics. The conformal--disformal class is established in scalar--tensor geometry [citation]; the result below fixes its precise restricted realization and admissibility conditions in the present variables rather than claiming a new general metric class.

theorem: Local scalar-gradient representation of a probe metric. Let $\widetilde g_{\mu\nu}$TeX\widetilde g_{\mu\nu} be a symmetric rank-two tensor that is constructed locally and algebraically from a Lorentzian metric $g_{\mu\nu}$TeXg_{\mu\nu}, a scalar $\chi$TeX\chi, and its first derivative, with no additional vector or tensor field. Then, wherever $\nabla\chi$TeX\nabla\chi is nonzero,

\widetilde g_{\mu\nu}=C(\chi,Y)g_{\mu\nu}+D(\chi,Y)\nabla_\mu\chi\nabla_\nu\chi,
\qquad Y=g^{\rho\sigma}\nabla_\rho\chi\nabla_\sigma\chi.

Writing $C=A^2>0$TeXC=A^2>0 and $D/C=B/\Lambda_\Xi^4$TeXD/C=B/\Lambda_\Xi^4, the metric is invertible with the same Lorentzian signature as $g_{\mu\nu}$TeXg_{\mu\nu} on the connected recovery branch if

1+\frac{B Y}{\Lambda_\Xi^4}>0.

proof. At a point, covariance permits scalar coefficients depending on the scalar invariants $\chi$TeX\chi and $Y$TeXY. With only $g_{\mu\nu}$TeXg_{\mu\nu} and the covector $v_\mu=\nabla_\mu\chi$TeXv_\mu=\nabla_\mu\chi available, the only symmetric covariant rank-two tensors are $g_{\mu\nu}$TeXg_{\mu\nu} and $v_\mu v_\nu$TeXv_\mu v_\nu; contractions of additional copies of $v$TeXv reduce to powers of $Y$TeXY multiplying these two tensors. This proves the representation. The matrix determinant lemma and the rank-one inverse formula give

\det\widetilde g=A^8\det g\left(1+\frac{B Y}{\Lambda_\Xi^4}\right),

\widetilde g^{\mu\nu}=A^{-2}\left[g^{\mu\nu}-
\frac{B/\Lambda_\Xi^4}{1+B Y/\Lambda_\Xi^4}
\nabla^\mu\chi\nabla^\nu\chi\right].

Starting from $B=0$TeXB=0 and varying continuously, the inertia cannot change before an eigenvalue crosses zero. The displayed positivity condition excludes that crossing and hence preserves Lorentzian signature.

corollary: Homogeneous null and clock response. In a local $g$TeXg-orthonormal frame with homogeneous $\chi(t)$TeX\chi(t), let $\Xi_{\nabla}=\dot\chi^2/\Lambda_\Xi^4$TeX\Xi_{\nabla}=\dot\chi^2/\Lambda_\Xi^4 and evaluate $A$TeXA and $B$TeXB on the local background. Then

\dd\widetilde s^2=A^2\left[-(1-B\Xi_{\nabla})\dd t^2+\dd\boldsymbol{x}^{\,2}\right],

so a comoving probe clock and a probe-null ray obey

\frac{\dd\widetilde\tau}{\dd t}=A\sqrt{1-B\Xi_{\nabla}},
\qquad
\frac{c_{\rm probe}^2}{c_g^2}=1-B\Xi_{\nabla}.

For $B\Xi_{\nabla}\ll1$TeXB\Xi_{\nabla}\ll1, the fractional clock and speed shifts are $-B\Xi_{\nabla}/2$TeX-B\Xi_{\nabla}/2 to leading order. The minimal branch $A=1$TeXA=1, $B=0$TeXB=0 has no such shift. If $B(\chi_\star)>0$TeXB(\chi_\star)>0, its magnitude may be absorbed into the definition of $\Lambda_\Xi$TeX\Lambda_\Xi, leaving the normalization $B(\chi_\star)=1$TeXB(\chi_\star)=1 used by the homogeneous response branch.

proof. For $g_{ab}=\mathrm{diag}(-1,1,1,1)$TeXg_{ab}=\mathrm{diag}(-1,1,1,1) and $\partial_a\chi=(\dot\chi,0,0,0)$TeX\partial_a\chi=(\dot\chi,0,0,0), substitution into reference gives the line element. Setting $\dd\boldsymbol{x}=0$TeX\dd\boldsymbol{x}=0 yields the clock relation; setting $\dd\widetilde s^2=0$TeX\dd\widetilde s^2=0 yields the null speed. Taylor expansion of $\sqrt{1-z}$TeX\sqrt{1-z} at $z=0$TeXz=0 gives the leading shifts. The final normalization follows by replacing $\Lambda_\Xi^4$TeX\Lambda_\Xi^4 with $\Lambda_\Xi^4/B(\chi_\star)$TeX\Lambda_\Xi^4/B(\chi_\star).

Weak-field and time-transfer regime

On the exact recovery solution $\chi=\chi_\star$TeX\chi=\chi_\star, the scalar stress is purely cosmological and the minimally coupled weak-field solution is the GR solution; hence $\gamma=\beta=1$TeX\gamma=\beta=1 and the standard deflection and Shapiro delay follow. Away from that solution, neither $\gamma-1$TeX\gamma-1 nor $\beta-1$TeX\beta-1 is fixed by covariance or by $\Xi_{\nabla}$TeX\Xi_{\nabla} alone. They must be computed from a specified potential, boundary data, and probe coupling. The admissibility conditions are therefore the directly testable inequalities

|\gamma(\vartheta)-1|<\delta_\gamma,
\qquad
|\beta(\vartheta)-1|<\delta_\beta,
\qquad
|B\Xi_{\nabla}|<\delta_{c^2}

for one common parameter vector $\vartheta$TeX\vartheta and independently specified experimental tolerances. Solar VLBI and time-delay measurements constrain the first two quantities [citation]; multimessenger propagation constrains the last when the electromagnetic and gravitational cones differ [citation]. No unsolved coefficient is presented as a prediction.

Homogeneous-FRW branch

For a flat FRW metric with scale factor $a(t)$TeXa(t) and homogeneous $\HH(t)$TeX\HH(t),

3\Mpl^2 H_{\FRW}^2 = \rho_{\mathrm{m}}+\rho_{\mathrm{rad}}+\rho_{\HH\PPhi}^{\mathrm{eff}},
\qquad
\rho_{\HH\PPhi}^{\mathrm{eff}} = \frac{\Mh^2}{2}\dot{\HH}^2+\frac{\Mp^2}{2}\dot{\PPhi}^2+\UH(\HH,\PPhi),
\qquad \PPhi=f(\HH).

Imposing the branch constraint and using $\dot\chi=\Meff\dot\HH$TeX\dot\chi=\Meff\dot\HH yields

\rho_\chi=\frac12\dot\chi^2+V(\chi),
\qquad
p_\chi=\frac12\dot\chi^2-V(\chi).

proposition: Necessary and sufficient plateau condition. If $\rho_\chi>0$TeX\rho_\chi>0, then

1+w_\chi=\frac{\rho_\chi+p_\chi}{\rho_\chi}
=\frac{\dot\chi^2}{\tfrac12\dot\chi^2+V(\chi)}.

Thus $w_\chi\to-1$TeXw_\chi\to-1 if and only if $\dot\chi^2/\rho_\chi\to0$TeX\dot\chi^2/\rho_\chi\to0. Convexity of $V$TeXV alone does not imply the plateau.

proof. Substitution of the displayed density and pressure gives the identity. Since $\rho_\chi>0$TeX\rho_\chi>0, the right-hand side tends to zero exactly when $\dot\chi^2/\rho_\chi$TeX\dot\chi^2/\rho_\chi does. A convex potential permits initial data with arbitrarily large kinetic energy, which proves that convexity without the dynamical ratio is insufficient.

The covariant core is excluded on a proposed domain if no single parameter set simultaneously (i) satisfies the nondegenerate branch and positivity conditions of reference, (ii) preserves the local Cauchy structure of reference, (iii) satisfies the kinetic and potential recovery conditions of reference, and (iv) remains consistent with the weak-field and homogeneous-FRW relations of reference.

For the response branch, the same parameter vector must also keep $\widetilde g$TeX\widetilde g Lorentzian and satisfy the independently measured bounds on $A$TeXA, $B\Xi_{\nabla}$TeXB\Xi_{\nabla}, clock ratios, and propagation delays. A statistically significant violation of any one of these necessary conditions rejects that parameter vector. Fitting a coefficient with the same datum used for evaluation is not a test and is excluded from the likelihood analysis.

The closure test for the covariant compact-phase core 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 weak-field, propagation, clock, winding-energy, and compact-object observables. Let $a$TeXa range over the independent constitutive inputs comprising the potential, probe-metric coefficients, higher-derivative coefficients, and defect-core data.

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 a nonzero winding configuration is not assigned a regular phase-only core, and every claimed radiative restriction satisfies the operator-basis and spurion-charge tests.

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 a circle-valued phase of nonzero boundary degree has no continuous disk extension, while an excluded divergent operator cannot be absorbed by renormalizing the admitted basis. Failure of that condition therefore rejects the proposed microscopic closure before parameter estimation can be counted as evidence for it.

Explicit compact-holonomy realization..

Paper CHC-MHB supplies a finite microscopic realization in which a compact coordinate survives local basis changes only through the holonomy of a three-mode loop. If $\Omega_j$TeX\Omega_j are the three one-excitation frequencies, the shift- and scale-free combination

\frac{3\sqrt6\prod_j(\Omega_j-\bar\Omega)}
 {\bigl[\sum_j(\Omega_j-\bar\Omega)^2\bigr]^{3/2}}
 =\cos\theta

is exact, while every apparent phase on an open chain is removable. This is a constructive witness that global compact data can have observable content after local canonicalization. It is not a cosmological solution and does not identify the laboratory control phase with the root scalar.

The nondegenerate CHC core is locally exactly GR plus one canonical scalar written in a constrained two-field chart. This equivalence is a theorem, not an analogy: it fixes the degree-of-freedom count, characteristic cone, Hamiltonian criterion, and local Cauchy problem. On the compact target branch, the same transformation produces a canonical circle of circumference $L_\chi$TeXL_\chi rather than the real line. Its winding sectors are globally inequivalent to the noncompact scalar and satisfy the sharp lower bound reference. The exact constant-field GR+$\Lambda_{\rm eff}$TeX\Lambda_{\rm eff} branch occurs at a stationary scalar value and belongs to the zero-winding component. Controlled recovery requires both small kinetic stress and small potential displacement; the old one-parameter small-gradient criterion is not sufficient.

Optical and clock modifications arise only after an interaction with probes is specified. The covariant scalar-gradient representation theorem restricts that interaction to a conformal-disformal metric at first-derivative order and yields the homogeneous propagation and clock formulae exactly. Its coefficient is an empirical coupling, not a consequence of the root action. This distinction supplies a coherent route from the covariant core to downstream measurements while preserving a sharp null branch: $A=1$TeXA=1 and $B=0$TeXB=0 recover standard local propagation identically.

Nonzero winding cannot be regularized inside a phase-only disk, and compactness does not select arbitrary constitutive values. Any claim of radiative stability must additionally pass the divergent operator-basis criterion of reference; a preserved spurion charge constrains harmonics but does not fix their coefficients.

Solar-system implication..

At $\chi=\chi_\star$TeX\chi=\chi_\star the scalar stress is $-V_\star g_{\mu\nu}$TeX-V_\star g_{\mu\nu} and the minimally coupled post-Newtonian solution is precisely the GR solution with a negligible local cosmological-constant term. Any nonzero correction requires a nonconstant scalar solution or a nonminimal probe coefficient. Its magnitude is therefore obtained by solving the boundary-value problem and cannot be inferred from $b/\ell_{\HH}$TeXb/\ell_{\HH} alone. Solar-grazing deflection and time-delay data constrain that solved metric together with the response metric of reference.

Auxiliary stationary-symmetric compact-object diagnostic..

The quantity $\Xith$TeX\Xith is retained only as an auxiliary stationary-symmetric compact-object diagnostic, to be used diagnostically only on the admitted branch. It does not enter the exclusion criterion and is not promoted to a compact-object theorem or inference law. With horizon generator $\chi^\mu$TeX\chi^\mu and surface gravity $\kappa$TeX\kappa, define

\Xith \equiv \frac{\chi^\mu\nabla_\mu\HH}{\kappa}-(\Xith)_\star.

Under $\chi^\mu\to\lambda\chi^\mu$TeX\chi^\mu\to\lambda\chi^\mu one has $\kappa\to\lambda\kappa$TeX\kappa\to\lambda\kappa, so $(\chi\cdot\nabla\HH)/\kappa$TeX(\chi\cdot\nabla\HH)/\kappa is invariant. In admitted stationary-symmetric settings, one may test whether a suitable choice of $(\Xith)_\star$TeX(\Xith)_\star aligns $\Xith=0$TeX\Xith=0 with the marginally outer trapped surface in the recovery limit. The quantity $\Xith$TeX\Xith remains diagnostic-only in this admitted stationary-symmetric class and does not by itself define a compact-object theorem or inference scheme. Away from that admitted stationary-symmetric class, no further claim is made.

proposition: Constraint reduction on the admissible branch. The conjugate momentum of $\OOm$TeX\OOm yields the primary constraint $\Pi_{\OOm}\approx0$TeX\Pi_{\OOm}\approx0, whose preservation gives the holonomic constraint $C_2\equiv\PPhi-f(\HH)\approx0$TeXC_2\equiv\PPhi-f(\HH)\approx0. Preservation of $C_2$TeXC_2 imposes the corresponding normal-momentum constraint, and its preservation fixes $\OOm$TeX\OOm on the admitted branch. These conditions remove the multiplier pair and the field direction normal to $C_2=0$TeXC_2=0, leaving the single effective field $\HH$TeX\HH with kinetic prefactor $\Meff^2(\HH)$TeX\Meff^2(\HH). The equivalent Lagrangian reduction is obtained by substituting $\PPhi=f(\HH)$TeX\PPhi=f(\HH) together with its differentiated relation.

proof. Because $\dot\OOm$TeX\dot\OOm is absent, $\Pi_{\OOm}\approx0$TeX\Pi_{\OOm}\approx0. Hamiltonian preservation of this primary constraint gives $C_2\approx0$TeXC_2\approx0. On a spacelike slice, preservation of $C_2$TeXC_2 along the unit normal gives, up to the common density factor,

C_3\equiv\frac{\Pi_{\PPhi}}{\Mp^2}-f'(\HH)\frac{\Pi_{\HH}}{\Mh^2}\approx0.

The constraint bracket in the scalar normal direction is proportional to

\frac{1}{\Mp^2}+\frac{f'(\HH)^2}{\Mh^2}>0.

Consequently preservation of $C_3$TeXC_3 determines the multiplier field $\OOm$TeX\OOm rather than producing an additional propagating datum; preservation of $\Pi_{\OOm}$TeX\Pi_{\OOm} then pairs with that determination. Equivalently, $C_2=0$TeXC_2=0 and $C_3=0$TeXC_3=0 impose $\PPhi=f(\HH)$TeX\PPhi=f(\HH) and $n^\mu\nabla_\mu\PPhi=f'(\HH)n^\mu\nabla_\mu\HH$TeXn^\mu\nabla_\mu\PPhi=f'(\HH)n^\mu\nabla_\mu\HH. Substitution makes the scalar kinetic Hessian one-dimensional with entry $\Meff^2$TeX\Meff^2, and the bound $\Meff^2\ge m_0^2$TeX\Meff^2\ge m_0^2 makes the reduced Legendre map invertible. Thus exactly one scalar canonical pair remains.

proposition: Reduced Hamiltonian criterion. Let $\mathcal{H}_{AB}=\partial^2\Lag/\partial\dot\varphi^A\partial\dot\varphi^B$TeX\mathcal{H}_{AB}=\partial^2\Lag/\partial\dot\varphi^A\partial\dot\varphi^B for $\varphi^A=(g,\HH,\PPhi,\OOm,\Psi)$TeX\varphi^A=(g,\HH,\PPhi,\OOm,\Psi). On the nondegenerate branch where the reduced kinetic matrix is positive and the constraint algebra removes the auxiliary field, the Hamiltonian is bounded below modulo the usual first-class gravitational constraints. This is the standard no-ghost criterion for the reduced branch.

proof. After the constraint reduction, reference gives the scalar canonical momentum $\pi_\chi=\sqrt h\,n^\mu\nabla_\mu\chi$TeX\pi_\chi=\sqrt h\,n^\mu\nabla_\mu\chi. The scalar Hamiltonian density on a spacelike slice is

\mathcal H_\chi=N\left(\frac{\pi_\chi^2}{2\sqrt h}+\frac{\sqrt h}{2}h^{ij}\partial_i\chi\partial_j\chi+\sqrt h\,V\right)+N^i\pi_\chi\partial_i\chi.

For positive lapse the quadratic part is nonnegative, and a lower bound on $V$TeXV supplies the lower bound after quotienting by the diffeomorphism constraints. Conversely $\Meff^2<0$TeX\Meff^2<0 reverses the sign of the kinetic term, so positivity fails.

proposition: Leading-order principal-cone alignment on the admissible branch. If the conditions of reference hold, then the reduced scalar principal symbol is proportional to $\Meff^2 g^{\mu\nu}\xi_\mu\xi_\nu$TeX\Meff^2 g^{\mu\nu}\xi_\mu\xi_\nu. Thus, at leading principal order, the scalar characteristic set coincides with the metric null cone on the admitted branch. In particular, the admitted -min branch introduces no independent cone-widening operator or superluminal principal characteristic at this order.

proof. The reduced scalar equation is $\Meff^2\Box\HH+\tfrac12(\Meff^2)'X_{\HH}-U_{\rm eff}'=0$TeX\Meff^2\Box\HH+\tfrac12(\Meff^2)'X_{\HH}-U_{\rm eff}'=0. Only the first term contains second derivatives, so the principal symbol is $\Meff^2g^{\mu\nu}\xi_\mu\xi_\nu$TeX\Meff^2g^{\mu\nu}\xi_\mu\xi_\nu. Since $\Meff^2>0$TeX\Meff^2>0, its zero set is exactly $g^{\mu\nu}\xi_\mu\xi_\nu=0$TeXg^{\mu\nu}\xi_\mu\xi_\nu=0. The canonical transformation of reference gives the same conclusion directly.

proposition: Radiative-closure criterion for a restricted basis. Let $\mathfrak B=\{\mathcal O_i\}$TeX\mathfrak B=\{\mathcal O_i\} be the declared local operator basis, understood modulo total derivatives and operators proportional to the admitted leading equations of motion. At a fixed loop order, the restriction is radiatively closed if and only if the divergent effective action can be written

\Gamma_{\mathrm{div}}
 =\sum_i\delta c_i\int\dd^dx\,\sqrt{-g}\,\mathcal O_i
 +\Gamma_{\mathrm{EOM}}+\Gamma_{\partial}.

If an independent symmetry-allowed operator outside $\operatorname{span}\mathfrak B$TeX\operatorname{span}\mathfrak B occurs with nonzero divergent coefficient, the restricted theory is not radiatively closed at that order. No choice of finite values for the admitted coefficients can remove this obstruction.

proof. Renormalization within the restriction changes only the coefficients multiplying admitted operators, together with field redefinitions generated by equation-of-motion operators and boundary counterterms. Thus reference is sufficient to absorb the divergence without enlarging the basis. Conversely, decompose the divergent local functional in the quotient of local operators by equations of motion and total derivatives. A nonzero component outside the span of $\mathfrak B$TeX\mathfrak B cannot be cancelled by changing any $c_i$TeXc_i, because those changes remain inside that span. Finite counterterms do not cancel a nonzero regulator pole at the same order. The outside component therefore forces enlargement of the operator basis.

corollary: Spurion-charge stability at first breaking order. Suppose compact-phase breaking is represented by one complex spurion $z$TeXz of charge $-1$TeX-1 under $\HH/f\mapsto\HH/f+\alpha$TeX\HH/f\mapsto\HH/f+\alpha, $z\mapsto e^{-\mathrm i\alpha}z$TeXz\mapsto e^{-\mathrm i\alpha}z, and the regulator and subtraction scheme preserve this spurionic symmetry. Then every real local coefficient at first order in $z$TeXz has the form

c_A(\HH,z)=c_A^{(0)}+2\operatorname{Re}\!\left(\beta_A z e^{\mathrm i\HH/f}\right)+O(|z|^2).

In particular, radiative corrections may renormalize $\beta_A$TeX\beta_A but cannot generate an independent first-order harmonic of different compact charge.

proof. At first order in $z$TeXz and $\bar z$TeX\bar z, spurion invariance permits a Fourier monomial only when its phase charge cancels the spurion charge. The invariant monomials are therefore $z e^{\mathrm i\HH/f}$TeXz e^{\mathrm i\HH/f} and its complex conjugate. Reality fixes their coefficients to be conjugate, which gives reference. A symmetry-preserving counterterm obeys the same charge selection rule.

Positivity criterion..

Forward-limit analyticity and unitarity constrain EFT Wilson coefficients through positivity bounds, but in the presence of gravity the standard flat-space argument must be treated with care because of the massless $t$TeXt-channel graviton pole. The admissible -min basis therefore uses positivity only as a supporting filter. Standard flat-space positivity logic follows Adams et al. [citation], while gravitational refinements, scalar-tensor causality bounds, and the massless spin-2 subtlety are discussed by Tokuda, Aoki, and Hirano, Alberte et al., Hong, Wang, and Zhou, and Alviani, Falkowski, and Marinellis [citation].

Restricted tensor-luminality closure..

Assume diffeomorphism covariance, the admissible operator-basis restriction of Definition reference, and a renormalization scheme that preserves that restriction. Then no admitted operator in the selected basis splits the tensor kinetic and gradient coefficients on smooth backgrounds. Within that restricted basis one therefore retains