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 global phase field encodes the expansion-response branch where applicable; it is not a fixed background stage on which local physics rides. On the admissible nondegenerate branch, the reduced system contains a single effective scalar beyond GR, with departures organized by the background-adapted hierarchy parameter

\Xi \equiv \frac{\Meff^2\GradHbg}{\Lambda_\Xi^4}.

Here $\GradHbg$TeX\GradHbg denotes the background-adapted nonnegative norm used to define $\Xi$TeX\Xi, while $\Lambda_\Xi$TeX\Lambda_\Xi sets the phase-gradient hierarchy scale. The Lorentzian kinetic contractions entering the action are defined in reference. The following analysis develops only the covariant field equations, the admissible branch, the EFT hierarchy, the controlled recovery structure, and the branch-level exclusion criterion.

Detector-facing, information-access, source-model, distance/clock, and compact-object inference layers are outside the restricted branch analyzed here.

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 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 exists a stable regime in which the metric equations reduce to GR+$\Lambda_{\mathrm{eff}}$TeX\Lambda_{\mathrm{eff}} up to $\mathcal{O}(\GradHbg)$TeX\mathcal{O}(\GradHbg) under the conditions listed in reference. - 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. - Phase-gradient hierarchy. Departures are organized by the hierarchy variable

\Xi \equiv \frac{\Meff^2\GradHbg}{\Lambda_\Xi^4},

where $\Lambda_\Xi$TeX\Lambda_\Xi is the hierarchy scale controlling the phase-gradient expansion and $\GradHbg$TeX\GradHbg denotes the positive norm appropriate to the background under consideration. With the metric signature $(-,+,+,+)$TeX(-,+,+,+) adopted below, one uses $\GradHbg\equiv -g^{\mu\nu}\nabla_\mu\HH\nabla_\nu\HH$TeX\GradHbg\equiv -g^{\mu\nu}\nabla_\mu\HH\nabla_\nu\HH on homogeneous timelike backgrounds and $\GradHbg\equiv g^{ij}\partial_i\HH\,\partial_j\HH$TeX\GradHbg\equiv g^{ij}\partial_i\HH\,\partial_j\HH in static spatial problems. In particular, for homogeneous FRW-like backgrounds with $\HH=\HH(t)$TeX\HH=\HH(t),

\Xi = \Meff^2\dot{\HH}^{2}/\Lambda_\Xi^4 \ge 0.

When a process probes the global phase field with characteristic variation length $\ell_{\HH}$TeX\ell_{\HH} on a scale $L$TeXL, the small-gradient branch is $L/\ell_{\HH}\ll 1$TeXL/\ell_{\HH}\ll 1, and leading small-gradient envelopes are organized by powers of $(L/\ell_{\HH})^2$TeX(L/\ell_{\HH})^2. The formal limit $\Xi\to 0$TeX\Xi\to 0 is a hierarchy limit only.

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 hierarchy norm $\GradHbg$TeX\GradHbg used in $\Xi$TeX\Xi remains 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\big(\HH,f(\HH)\big) + \cdots .

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.

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, a positive-definite reduced Hessian near the recovery point, and a standard strongly hyperbolic gauge reduction for the gravitational sector. Then the reduced first-order formulation of the coupled GR+$\,$TeX\,\ system is strongly hyperbolic, with real diagonalizable principal symbol. Local existence, uniqueness, and continuous dependence in $H^s$TeXH^s then follow from the standard local theory for quasilinear hyperbolic systems [citation].

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

Linearizing around a homogeneous background $(\HH_0,\PPhi_0=f(\HH_0))$TeX(\HH_0,\PPhi_0=f(\HH_0)) gives the quadratic energy density

\mathcal{E}^{(2)}_{\CHC} =
\frac{\Mh^2}{2}\big[(\partial_t\delta\HH)^2 + |\nabla\delta\HH|^2\big]
+\frac{\Mp^2}{2}\big[(\partial_t\delta\PPhi)^2 + |\nabla\delta\PPhi|^2\big]
+\frac{1}{2}\delta\HH  \delta\PPhi
\mathbf{M}
\delta\HH
 \delta\PPhi,

with Hessian $\mathbf{M}_{ij}=\partial_i\partial_j\UH\vert_0$TeX\mathbf{M}_{ij}=\partial_i\partial_j\UH\vert_0. Sufficient positivity conditions are

\Mh^2>0,\qquad \Mp^2>0,\qquad \partial_{\PPhi\PPhi}U>0,\qquad \det\mathbf{M}>0.

On the reduced branch, one equivalently requires

U_{\mathrm{eff}}''(\HH_0)=U_{HH}+2f'U_{H\Phi}+(f')^2U_{\Phi\Phi}+f''U_{\Phi}\Big|_{(\HH_0,\,\PPhi=f(\HH_0))}>0,

together with $\Meff^2(\HH_0)>0$TeX\Meff^2(\HH_0)>0. These conditions bound the scalar Hamiltonian from below and exclude ghost and gradient instabilities on the stated branch.

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.

For a probe of characteristic size $L$TeXL in a region where the global phase field has variation length $\ell_{\HH}$TeX\ell_{\HH}, the small-gradient branch is $L/\ell_{\HH}\ll1$TeXL/\ell_{\HH}\ll1. Weak-field envelopes and homogeneous-FRW response quoted below are therefore organized by $(L/\ell_{\HH})^2$TeX(L/\ell_{\HH})^2, consistent with the underlying hierarchy parameter $\Xi$TeX\Xi.

theorem: Controlled GR recovery on the small-gradient branch. Assume the -min branch, $\nabla\HH\to0$TeX\nabla\HH\to0, and a local minimum of $U(\HH,\PPhi)$TeXU(\HH,\PPhi) on the constraint surface $\PPhi=f(\HH)$TeX\PPhi=f(\HH) with positive-definite reduced Hessian. Then the metric equations reduce to Einstein gravity with an effective cosmological constant,

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

equivalently $G_{\mu\nu}+\Lambda_{\mathrm{eff}}g_{\mu\nu}=\Mpl^{-2}T^{\mathrm{m}}_{\mu\nu}$TeXG_{\mu\nu}+\Lambda_{\mathrm{eff}}g_{\mu\nu}=\Mpl^{-2}T^{\mathrm{m}}_{\mu\nu}, up to $\mathcal{O}(\GradHbg)$TeX\mathcal{O}(\GradHbg). In the same regime, \ corrections are organized by the small hierarchy parameter $\Xi$TeX\Xi and vanish continuously with the phase gradient.

Weak-field and time-transfer regime

Precision solar-system tests require that the local phase-field environment lie in the small-gradient branch, with characteristic probe scale $L\sim b$TeXL\sim b much smaller than the variation length $\ell_{\HH}$TeX\ell_{\HH} of the global phase field. In that regime, the leading \ corrections are quadratic in the small ratio $b/\ell_{\HH}$TeXb/\ell_{\HH} and do not shift the PPN sector at linear order in the phase gradient:

\gamma - 1 = \mathcal{O}\!\big(\GradHbg\big), 

\beta - 1 = \mathcal{O}\!\big(\GradHbg\big).

Correspondingly,

\alpha_{\CHC} = \frac{4GM}{c^2 b}\,\Big[1+\varepsilon_\alpha(\nabla\HH;b)\Big],
\qquad
|\varepsilon_\alpha|\lesssim C_\alpha\left(\frac{b}{\ell_{\HH}}\right)^2,

\Delta t_{\CHC} = \Delta t_{\GR}\,\Big[1+\varepsilon_t(\nabla\HH;b)\Big],
\qquad
|\varepsilon_t|\lesssim C_t\left(\frac{b}{\ell_{\HH}}\right)^2.

Solar VLBI deflection and time-delay measurements therefore require the coefficients entering $\varepsilon_\alpha$TeX\varepsilon_\alpha and $\varepsilon_t$TeX\varepsilon_t to remain below current observational sensitivity on solar-system backgrounds [citation]. Recent weak-field analyses of luminality-safe scalar-tensor sectors exhibit the same requirement that admissible parameters recover standard solar-system behavior on slowly varying backgrounds [citation]. These relations are weak-field envelopes on slowly varying backgrounds and do not constitute a full source model or a global PN reconstruction.

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 yields

\rho_{\HH\PPhi}^{\rm eff}=\frac{\Meff^2}{2}\dot{\HH}^2+U\big(\HH,f(\HH)\big),
\qquad
p_{\HH\PPhi}^{\rm eff}=\frac{\Meff^2}{2}\dot{\HH}^2-U\big(\HH,f(\HH)\big).

When $\HH$TeX\HH evolves into a slowly varying regime with locally convex $U_{\mathrm{eff}}(\HH)=U(\HH,f(\HH))$TeXU_{\mathrm{eff}}(\HH)=U(\HH,f(\HH)), the homogeneous-FRW response approaches a plateau with $w_{\mathrm{eff}}\simeq -1$TeXw_{\mathrm{eff}}\simeq -1. Only this homogeneous-FRW branch and its small-$\Xi$TeX\Xi organization are retained below; no observational distance or clock law is introduced.

-min is excluded if no single admissible parameter set simultaneously (i) satisfies the nondegenerate branch and positivity conditions of reference, (ii) preserves the conditional local Cauchy structure of reference, (iii) reduces to GR+$\Lambda_{\mathrm{eff}}$TeX\Lambda_{\mathrm{eff}} in the small-gradient regime of reference, and (iv) remains consistent with the weak-field and homogeneous-FRW envelopes of reference.

A single failure of the admitted branch, stability, recovery, or envelope conditions is sufficient to reject the formulation on the stated branch. The auxiliary stationary-symmetric compact-object diagnostic recorded in reference does not enter this exclusion criterion.

We have specified a restricted covariant scalar-tensor formulation based on a global phase field, whose cosmological sector encodes an expansion-response branch where applicable, together with a constrained auxiliary sector and a nondegenerate admissible branch. Under the stated branch assumptions and adopted hyperbolic reduction, the reduced system has the propagating content described above and admits controlled recovery to GR+$\Lambda_{\mathrm{eff}}$TeX\Lambda_{\mathrm{eff}} in the small phase-gradient regime. Departures are organized by the hierarchy variable $\Xi$TeX\Xi and by probe-to-global-phase-field variation ratios.

The result established here is branch-local. It fixes the action, reduced branch, conditional local Cauchy structure, restricted EFT hierarchy, recovery theorem, and exclusion criterion for the stated formulation. The compact-object quantity $\Xith$TeX\Xith remains an auxiliary stationary-symmetric diagnostic and is not part of the exclusion criterion. Source models, detector response, distance/clock inference, and compact-object inference are not developed here.

Illustrative solar-grazing light-deflection bound..

For a static spherical lens of mass $M$TeXM and impact parameter $b$TeXb, the weak-field envelope reads

\alpha_{\CHC}=\alpha_{\GR}\big[1+\varepsilon_\alpha(\nabla\HH;b)\big],
\qquad
|\varepsilon_\alpha|\lesssim C_\alpha\left(\frac{b}{\ell_{\HH}}\right)^2.

For solar-grazing rays with $b\simeq R_\odot$TeXb\simeq R_\odot, current light-deflection bounds therefore require $b/\ell_{\HH}\ll1$TeXb/\ell_{\HH}\ll1 for $C_\alpha=\mathcal{O}(1)$TeXC_\alpha=\mathcal{O}(1), i.e. a background that is nearly homogeneous on scales much larger than the probed impact parameter. These relations are weak-field envelopes on slowly varying backgrounds and do not constitute a full source model or a global PN reconstruction.

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, and preservation of the corresponding multiplier equation gives the algebraic branch constraint $C_2\equiv\PPhi-f(\HH)\approx0$TeXC_2\equiv\PPhi-f(\HH)\approx0. On the nondegenerate admitted branch one solves this algebraic constraint and substitutes $\PPhi=f(\HH)$TeX\PPhi=f(\HH) in the reduced action. This removes the auxiliary direction from the reduced scalar sector and leaves the single effective field $\HH$TeX\HH with kinetic prefactor $\Meff^2(\HH)$TeX\Meff^2(\HH). No independent Hamiltonian gauge-fixing equivalence is asserted here beyond this branch reduction.

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.

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.

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

Z_T^{(K)}(\mathrm{bg})=Z_T^{(G)}(\mathrm{bg}),
\qquad
c_T^2=1.

This is a basis-closure statement and not a general UV-completion theorem.

Loop-level consequence..

Within the admissible basis, loop-induced deformations of weak-field observables remain at least quadratic in the small-gradient parameters used in the main text. Any stronger statement about arbitrary higher-curvature completions or all-sector symmetry protection lies beyond the present formulation.

Funding and competing interests..

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