Phase-Decoupling of Bound Structures in an Expanding Universe

We import only the cosmological coarse-grained specialization of the CHC hierarchy object on the declared slow-drive window, written here as $\Xi_{\mathrm{cos}}(L,\omega)$TeX\Xi_{\mathrm{cos}}(L,\omega). The root hierarchy object $\Xi$TeX\Xi is not redefined here, and no propagation-side or clock-inference structure enters the analysis. The question is how a fixed slow-drive-window criterion captures the non-tracking of normalized internal readouts in a background that itself undergoes cosmological expansion. That separation is stated here by comparing a coarse-grained cosmological phase drive with a proxy-based binding measure on a common slow-drive window.

The standard dynamical explanation is that the Hubble acceleration $H_0^2L$TeXH_0^2L on the scale $L$TeXL is negligible compared with local binding accelerations, so the bound system is effectively unaffected [citation]. The same operational content is formulated below as a comparison between a coarse-grained cosmological phase drive and a fixed rigidity measure on a common slow-drive window.

Related work: local systems, cosmological expansion, and effective rigidity

The question of how (and whether) the global FRW expansion influences local, bound systems has a long history in general relativity and cosmology. Classic exact-solution approaches include embedding a compact object into an expanding universe in a way that clarifies which degrees of freedom ``feel'' the Hubble flow. Two widely discussed examples are the McVittie spacetime, an early exact solution describing a central mass in an expanding background [citation], and the Einstein--Straus ``vacuole'' construction (the ``Swiss-cheese'' model), which matches a Schwarzschild region to an expanding FRW exterior [citation].

Modern treatments emphasize that the influence of expansion on local dynamics depends on the observable, the coordinate choice, and the operational procedure used to infer distances and times. A broad review of attempts to estimate expansion effects on local dynamics, including both Newtonian approximations and exact solutions, is given in [citation]. An early local-systems analysis in the Fermi normal frame is given in [citation]. Concrete analyses of the interplay between cosmological expansion and local attraction in bound systems and in exact embedding spacetimes are discussed in [citation]. Shorter expository treatments using Einstein--Straus-type intuition to argue that local dynamics is essentially decoupled from expansion (within well-defined regimes) appear in, e.g., [citation].

A clean pedagogical discussion of the question ``what expands in an expanding Universe?'' is given by Price and Romano [citation], who analyze a simple bound system in an expanding background and show how ``bound systems do not expand'' is an operational statement that depends on the observable used to characterize size. Recent local-observable analyses sharpen this point further by computing expansion-induced shifts for resonators and light-signal exchange directly in McVittie and Kottler backgrounds [citation]. Related operational discussions emphasize that local physics can be matched consistently to cosmological descriptions once the observable and coordinatization are fixed [citation]. This aligns with the present use of fixed response observables to state criterion-defined non-tracking on a common slow-drive window.

These models are not revisited here as alternatives to GR. Instead, it uses them as reference examples for the empirical fact being explained: there is an operational separation between cosmological expansion and internal sizes of bound structures. The central ingredient here is a single phase-geometric criterion that expresses this separation as a competition between a coarse-grained cosmological phase drive and a binding-induced phase rigidity. This reframes ``binding beats Hubble expansion'' into an inequality among dimensionless phase-geometric measures that can, in principle, be estimated from rigidity and response observables.

Laboratory analogue systems also motivate the choice of response quantities used here. Controlled quantum-fluid, cold-atom, and synthetic-lattice platforms now realize expanding-background response and curved-spacetime observables in the laboratory, including rapidly expanding Bose--Einstein condensates [citation], action-level phonon redshift and Hubble-friction response in expanding BECs [citation], direct measurements of Hubble attenuation and amplification in expanding and contracting cold-atom universes [citation], configurable quantum-field simulators in curved spacetime [citation], experimental observation of curved light-cones in a quantum field simulator [citation], rotating curved-spacetime signatures from a giant quantum vortex [citation], analogue cosmological particle creation in quantum fluids [citation], entangled-pair targets in BEC analog expanding universes [citation], entanglement in a cold-atom analog of cosmological preheating [citation], synthetic mechanical lattices for scalar fluctuations in expanding universes [citation], cosmological particle production in a quantum field simulator treated as a quantum-mechanical scattering problem [citation], and spin systems that realize quantum field theories in curved spacetimes [citation]. These platforms do not supply the underlying microphysics, but they show that phase rigidity and response kernels can be treated as transportable observables in controlled media rather than as already completed microscopic closure. Recent excitation-spectrum, phase-coherence, total phase-fluctuation, curved-light-cone, and coherent momentum-coupling measurements further show that the stiffness and response observables used in the analogue discussion can be tracked directly in laboratory families [citation].

The problem can thus be expressed in phase-geometric terms. Cosmological expansion appears as a coarse-grained phase drive, while bound systems are characterized by large internal phase rigidity. The relevant question is therefore not whether binding ``wins'' in a qualitative sense, but whether one can formulate a single dimensionless criterion that compares the cosmological phase drive with the internal rigidity of the bound structure.

The analysis is restricted to the deep-decoupling regime; competition windows $\Xi_{\mathrm{cos}}\sim\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}$TeX\Xi_{\mathrm{cos}}\sim\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}, boundary-response competition, robustness/null-test layers, and matter constitutive closure lie outside the present scope.

reference formulates the phase-decoupling criterion in terms of the coarse-grained cosmological phase drive $\Xi_{\mathrm{cos}}$TeX\Xi_{\mathrm{cos}}, the proxy-based binding measure $\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}$TeX\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}, and the response kernel $\chi_{\mathrm{ph}}$TeX\chi_{\mathrm{ph}}. reference applies this criterion to atoms and crystalline solids, while reference develops three tractable toy realizations of the same suppression class: a harmonic oscillator, a rotor chain, and a continuum phase mode.

Coarse-grained phase field around a bound structure

Consider a bound structure $\mathcal{B}$TeX\mathcal{B} localized around some worldline or world tube in the FRW background. Condensed-matter systems provide the main examples below; virialized gravitational systems are discussed separately only through the same hierarchy inequality. We decompose the phase field as

\HH(x)
=
\HH_{\mathrm{cos}}(x)
+
\delta\HH_{\mathcal{B}}(x),

where $\HH_{\mathrm{cos}}$TeX\HH_{\mathrm{cos}} is a slowly varying cosmological background and $\delta\HH_{\mathcal{B}}$TeX\delta\HH_{\mathcal{B}} encodes the localized deformation due to the bound structure. This decomposition is the carrier partition adopted for the analysis below, and no partition-free map statement is asserted.

We introduce a coarse-graining volume $V_L$TeXV_L of characteristic size $L$TeXL around $\mathcal{B}$TeX\mathcal{B} and define a coarse-grained cosmological phase-gradient measure

\Xi_{\mathrm{cos}}(L)
\equiv
\frac{M_{\mathrm{eff}}^2}{\Lambda_\Xi^4}
\left\langle (\nabla\HH_{\mathrm{cos}})^2 \right\rangle_{V_L}.

This quantity measures the effective cosmological phase curvature across the scale $L$TeXL on which the bound system resides. Here it is used only as a coarse-grained drive variable for the decoupling inequality, not as a complete microscopic model of the cosmological sector. When time scales are relevant, we refine it to $\Xi_{\mathrm{cos}}(L,\omega)$TeX\Xi_{\mathrm{cos}}(L,\omega), emphasizing the frequency $\omega$TeX\omega of the cosmological drive (typically $\omega=\omega_{\mathrm{drv}}$TeX\omega=\omega_{\mathrm{drv}} with $\omega_{\mathrm{drv}}\sim H_0$TeX\omega_{\mathrm{drv}}\sim H_0 or smaller on the fixed slow-drive window).

Phase rigidity functional

At the effective level, the Hamiltonian for a bound structure can be schematically written as

\hat{H}_{\mathcal{B}}
=
\hat{H}_{\mathrm{kin}}[\HH]
+
\hat{H}_{\mathrm{pot}}[\HH],

where the effective rigidity functional may be organized by a small-$\Xi$TeX\Xi kinetic scaling, schematically written in the metric-like form $g^{\mu\nu}_{\mathrm{eff}}\sim(1-\Xi)g^{\mu\nu}$TeXg^{\mu\nu}_{\mathrm{eff}}\sim(1-\Xi)g^{\mu\nu}, while the potential part reflects local binding interactions. This scaling is used only as an effective organizer inside the present rigidity criterion and is not promoted here to an independent propagation-side metric law. We assume that $\hat{H}_{\mathcal{B}}$TeX\hat{H}_{\mathcal{B}} admits a well-defined ground state $\ket{0}$TeX\ket{0} with energy $E_0[\HH]$TeXE_0[\HH].

To quantify how rigid the internal phase configuration of $\mathcal{B}$TeX\mathcal{B} is with respect to long-wavelength phase strains, we consider a slowly varying phase twist of characteristic scale $L$TeXL, implemented by a deformation $\delta\HH$TeX\delta\HH whose gradients satisfy $k\sim 1/L$TeXk\sim 1/L. We then define the phase rigidity functional as

\mathcal{R}_{\mathrm{ph}}(L)
\equiv
\left.
\frac{\delta^2 E_0[\HH_{\mathrm{cos}}+\delta\HH]}
{\delta(\partial_i \delta\HH)\,\delta(\partial_i \delta\HH)}
\right|_{\delta\HH\to 0,\;k\sim 1/L}.

Intuitively, $\mathcal{R}_{\mathrm{ph}}(L)$TeX\mathcal{R}_{\mathrm{ph}}(L) measures how energetically costly it is to shear or twist the internal phase configuration of $\mathcal{B}$TeX\mathcal{B} over the scale $L$TeXL.

In a highly rigid bound system (e.g.\ a crystal lattice), $\mathcal{R}_{\mathrm{ph}}$TeX\mathcal{R}_{\mathrm{ph}} is large; in a less rigid medium, it is smaller. The corresponding dimensionless quantity is obtained below by normalizing formal binding-induced phase measure $\Xi_{\mathrm{bind}}$TeX\Xi_{\mathrm{bind}}.

The functional $\mathcal{R}_{\mathrm{ph}}$TeX\mathcal{R}_{\mathrm{ph}} measures the long-wavelength energetic cost of phase twisting in the chosen bound system. Operational use below relies on a proxy-based measure built from long-wavelength rigidity observables rather than on a completed microscopic Hamiltonian for the full bound-structure family.

Phase-response kernel

We introduce a linear response relation between a small perturbation of the cosmological global phase-field state and the induced internal phase change in $\mathcal{B}$TeX\mathcal{B}:

\delta\HH_{\mathcal{B}}(k,\omega)
=
\chi_{\mathrm{ph}}(k,\omega)\,
\delta\HH_{\mathrm{cos}}(k,\omega),

where $k\sim 1/L$TeXk\sim 1/L parametrizes the spatial scale and $\omega$TeX\omega the frequency scale of the cosmological drive. Equation reference is the admitted response-update rule on the admitted family/window: it specifies the linear-response update from the cosmological carrier partition to the localized branch response and does not introduce detector-side local-commit dynamics or an optional background convention.

The phase-response kernel $\chi_{\mathrm{ph}}(k,\omega)$TeX\chi_{\mathrm{ph}}(k,\omega) is constrained by the structure of $\hat{H}_{\mathcal{B}}$TeX\hat{H}_{\mathcal{B}}. Its magnitude is controlled by the competition between the external phase-gradient energy and the internal phase rigidity.

To make this precise, we introduce a dimensionless formal binding-induced phase measure:

\Xi_{\mathrm{bind}}^{(\mathrm{formal})}(L,\omega)
\equiv
\frac{\mathcal{R}_{\mathrm{ph}}(L,\omega)}
{\Lambda_\Xi^4/M_{\mathrm{eff}}^2}.

The formal quantity $\Xi_{\mathrm{bind}}^{(\mathrm{formal})}(L,\omega)$TeX\Xi_{\mathrm{bind}}^{(\mathrm{formal})}(L,\omega) is defined from the phase rigidity $\mathcal{R}_{\mathrm{ph}}(L,\omega)$TeX\mathcal{R}_{\mathrm{ph}}(L,\omega). Its operational use below is restricted to the proxy-assigned measure $\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}(L,\omega)$TeX\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}(L,\omega) constrained by standard rigidity/relaxation observables that govern long-wavelength response. No stand-alone operational criterion is assigned below to $\Xi_{\mathrm{bind}}^{(\mathrm{formal})}(L,\omega)$TeX\Xi_{\mathrm{bind}}^{(\mathrm{formal})}(L,\omega) without that proxy realization. On the fixed slow-drive window it is useful to separate the externally imposed frequency $\omega_{\mathrm{drv}}$TeX\omega_{\mathrm{drv}} from the internal proxy scale $\omega_{\mathrm{bind}}^{(\mathrm{proxy})}(L)$TeX\omega_{\mathrm{bind}}^{(\mathrm{proxy})}(L) (or $t_{\mathrm{dyn}}^{-1}$TeXt_{\mathrm{dyn}}^{-1} for the virialized extension). The comparison is then carried out at $\omega=\omega_{\mathrm{drv}}$TeX\omega=\omega_{\mathrm{drv}}, while the binding side is anchored by the internal proxy scale. We therefore define the proxy-based binding measure on the slow-drive window by

\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}(L,\omega_{\mathrm{drv}};\omega_{\mathrm{bind}}^{(\mathrm{proxy})})
:=
\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}(L,\omega)\big|_{\omega=\omega_{\mathrm{drv}}},

with the normalization anchored by the internal rigidity scale $\omega_{\mathrm{bind}}^{(\mathrm{proxy})}$TeX\omega_{\mathrm{bind}}^{(\mathrm{proxy})}. We emphasize two regimes, corresponding to the two classes of systems highlighted in the initial phase-decoupling puzzle:

- (CM regime: solids and molecules). For a solid, the energy cost of a long-wavelength strain is controlled by elastic moduli. In standard elasticity theory, a uniform shear strain $\varepsilon$TeX\varepsilon in a volume $V\sim L^3$TeXV\sim L^3 costs $\Delta E\sim \tfrac{1}{2}\mu\,\varepsilon^2 V$TeX\Delta E\sim \tfrac{1}{2}\mu\,\varepsilon^2 V, where $\mu$TeX\mu is a shear modulus (or more generally $C_{ijkl}$TeXC_{ijkl}) [citation]. For a phase twist of amplitude $\theta$TeX\theta across size $L$TeXL, one may identify $\varepsilon\sim \theta/L$TeX\varepsilon\sim \theta/L and obtain the scaling equation R_ph^(solid)(L) mu L, equation up to geometry factors. Equivalently, characteristic phonon scales (e.g.\ Debye-scale frequencies $\omega_D$TeX\omega_D) provide a proxy for the relevant low-energy stiffness [citation]. - (Virialized regime: planetary and galactic systems). For a virialized gravitational system, a natural rigidity proxy is the inverse dynamical time $t_{\mathrm{dyn}}^{-1}$TeXt_{\mathrm{dyn}}^{-1} that sets the internal response rate to slow driving. Operationally, the ``size'' response of the system to an external long-wavelength drive is suppressed when the drive frequency satisfies $\omega\ll t_{\mathrm{dyn}}^{-1}$TeX\omega\ll t_{\mathrm{dyn}}^{-1}. We therefore treat $t_{\mathrm{dyn}}^{-1}$TeXt_{\mathrm{dyn}}^{-1} as the analog of a binding frequency in the hierarchy estimates, providing an empirical route to bounding $\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}(L,\omega_{\mathrm{drv}};t_{\mathrm{dyn}}^{-1})$TeX\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}(L,\omega_{\mathrm{drv}};t_{\mathrm{dyn}}^{-1}) at large scales [citation].

These proxy relations are intentionally conservative: they are not proposed as exact equalities, but as operational routes to estimate the scale of $\mathcal{R}_{\mathrm{ph}}(L,\omega)$TeX\mathcal{R}_{\mathrm{ph}}(L,\omega) from standard response observables, and thereby to bound $\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}$TeX\Xi_{\mathrm{bind}}^{(\mathrm{proxy})} without requiring a completed microscopic Hamiltonian. The response-kernel language used below is aligned with standard linear-response theory, where susceptibilities are defined as operational ratios of induced response to applied drive in a controlled perturbation setting [citation].

The previous proxy discussion identifies which classes of observables constrain $\mathcal{R}_{\mathrm{ph}}$TeX\mathcal{R}_{\mathrm{ph}}. We now state a minimal mapping protocol, consistent with the toy-model logic and with standard response practice, that specifies how $\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}$TeX\Xi_{\mathrm{bind}}^{(\mathrm{proxy})} and $\chi_{\mathrm{ph}}$TeX\chi_{\mathrm{ph}} are operationally bounded on the fixed window.

- (O1) Solids: low-frequency strain response as a proxy for phase rigidity. In a solid, the energy cost of a long-wavelength deformation is governed by elastic moduli. If an external drive induces an effective long-wavelength phase strain (represented here by a slow perturbation of $\HH_{\mathrm{cos}}$TeX\HH_{\mathrm{cos}} at $k\sim 1/L$TeXk\sim 1/L and $\omega=\omega_{\mathrm{drv}}$TeX\omega=\omega_{\mathrm{drv}} on the fixed slow-drive window, with $\omega_{\mathrm{drv}}\sim H_0$TeX\omega_{\mathrm{drv}}\sim H_0 in the present-epoch estimates), then the induced internal response is controlled by the same stiffness tensor that governs mechanical response. The operational statement of ``phase locking'' in this regime is therefore that, in the low-frequency limit, the ratio of induced internal response to applied drive amplitude (the measurable $\chi_{\mathrm{ph}}$TeX\chi_{\mathrm{ph}} defined below) is suppressed by a stiffness scale proportional to $\mu$TeX\mu (or more generally $C_{ijkl}$TeXC_{ijkl}) [citation]. Equation reference captures the minimal scaling needed for the hierarchy argument: $\mathcal{R}_{\mathrm{ph}}^{(\mathrm{solid})}(L)\sim \mu L$TeX\mathcal{R}_{\mathrm{ph}}^{(\mathrm{solid})}(L)\sim \mu L.

Virialized gravitational systems may be tested by the same hierarchy logic using the dynamical proxy $t_{\mathrm{dyn}}\sim L/v$TeXt_{\mathrm{dyn}}\sim L/v (equivalently $t_{\mathrm{dyn}}^{-1}\sim \sqrt{G\rho}$TeXt_{\mathrm{dyn}}^{-1}\sim \sqrt{G\rho} for mean density $\rho$TeX\rho), which governs the response to slow external driving [citation]. The associated gating inequality

\omega_{\mathrm{drv}}\ll t_{\mathrm{dyn}}^{-1}
\quad\Rightarrow\quad
\abs{\chi_{\mathrm{ph}}(k\sim 1/L,\omega=\omega_{\mathrm{drv}})}\ \text{is strongly suppressed},

is recorded only for the virialized case and is not used below for atoms or crystalline solids.

Proxy family and admissibility conditions

On the declared window, the analysis uses four fixed items: the carrier partition $\HH=\HH_{\mathrm{cos}}+\delta\HH_{\mathcal B}$TeX\HH=\HH_{\mathrm{cos}}+\delta\HH_{\mathcal B}, the admitted response-update rule reference, the rigidity-map pair $(\mathcal O_{\mathrm{rig}},\mathsf{Map}_{\mathrm{rig}})$TeX(\mathcal O_{\mathrm{rig}},\mathsf{Map}_{\mathrm{rig}}), and the reference comparison observable $\chi_{\mathrm{ph}}$TeX\chi_{\mathrm{ph}}. All hierarchy, suppression, and non-tracking claims below are read only on that fixed object set.

Operational use of $\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}$TeX\Xi_{\mathrm{bind}}^{(\mathrm{proxy})} requires one proxy family. We write

\mathcal F_{\mathrm{proxy}}
=
(\mathcal P,\mathcal O_{\mathrm{rig}},\mathcal O_{\mathrm{resp}},L,\mathcal W_\omega,\mathsf{Map}_{\mathrm{rig}},\mathsf{Est},\delta_{\mathrm{lin}},\delta_{\mathrm{bg}}),

where $\mathcal P$TeX\mathcal P is the platform family, $\mathcal O_{\mathrm{rig}}$TeX\mathcal O_{\mathrm{rig}} is the rigidity proxy observable, $\mathcal O_{\mathrm{resp}}$TeX\mathcal O_{\mathrm{resp}} is the normalized response observable used to define $\chi_{\mathrm{ph}}$TeX\chi_{\mathrm{ph}}, $L$TeXL is the coarse-graining scale, $\mathcal W_\omega$TeX\mathcal W_\omega is the slow-drive window, $\mathsf{Map}_{\mathrm{rig}}$TeX\mathsf{Map}_{\mathrm{rig}} is the map from the rigidity proxy to $\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}$TeX\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}, $\mathsf{Est}$TeX\mathsf{Est} is the common estimator/baseline convention, and $\delta_{\mathrm{lin}},\delta_{\mathrm{bg}}$TeX\delta_{\mathrm{lin}},\delta_{\mathrm{bg}} are the linear-response and background-stability tolerances. The pair $(\mathcal O_{\mathrm{rig}},\mathsf{Map}_{\mathrm{rig}})$TeX(\mathcal O_{\mathrm{rig}},\mathsf{Map}_{\mathrm{rig}}) is the rigidity-map pair used to assign $\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}$TeX\Xi_{\mathrm{bind}}^{(\mathrm{proxy})} on the admitted family/window. All claims below are read only on this declared proxy family, its fixed slow-drive window, its rigidity-map pair, and its common estimator convention.

A proxy family is used only if, on the fixed window,

\sup_{\omega\in\mathcal W_\omega}
\frac{\omega}{\omega_{\mathrm{bind}}^{(\mathrm{proxy})}}
\le \eta_{\mathrm{ad}}\ll 1,
\qquad
r_{\mathrm{lin}}(\mathcal W_\omega)\le \delta_{\mathrm{lin}},
\qquad
r_{\mathrm{bg}}(\mathcal W_\omega)\le \delta_{\mathrm{bg}},

with the same $\mathsf{Est}$TeX\mathsf{Est} and baseline convention applied across the fixed family. If reference fails, $\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}$TeX\Xi_{\mathrm{bind}}^{(\mathrm{proxy})} and $\chi_{\mathrm{ph}}$TeX\chi_{\mathrm{ph}} are not assigned on that window. When reference holds, the same admitted proxy family remains the only fixed object set for the hierarchy, suppression, and non-tracking claims below.

The denominator $\Lambda_\Xi^4/M_{\mathrm{eff}}^2$TeX\Lambda_\Xi^4/M_{\mathrm{eff}}^2 is the natural gradient-energy scale that appears in the definition of $\Xi$TeX\Xi; this choice makes $\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}$TeX\Xi_{\mathrm{bind}}^{(\mathrm{proxy})} a dimensionless stiffness measure that is directly comparable, on the same footing, to the coarse-grained cosmological drive $\Xi_{\mathrm{cos}}(L,\omega)$TeX\Xi_{\mathrm{cos}}(L,\omega) in the phase-decoupling inequality. At the effective level we expect a response of the schematic form

\chi_{\mathrm{ph}}(k,\omega)
\sim
\frac{\Xi_{\mathrm{cos}}(k,\omega)}
{\Xi_{\mathrm{cos}}(k,\omega) + \Xi_{\mathrm{bind}}^{(\mathrm{proxy})}(k,\omega)}.

Equation reference is a compact linear-response summary of the competition between external cosmological driving and internal binding-induced rigidity; it is not asserted as an exact identity, and its detailed form can depend on the microscopic realization, dissipation, and the choice of coarse-graining. The schematic structure reference is expected to emerge whenever (i) there exists a single dominant low-frequency internal response channel at $(k,\omega)$TeX(k,\omega), (ii) the external drive couples linearly to that channel, and (iii) the internal relaxation is fast compared to the drive period in the phase-decoupled regime (so that the adiabatic response is controlled by a stiffness scale rather than by resonant dynamics). These are precisely the conditions realized in the toy constructions in reference and are the standard conditions under which linear-response susceptibilities reduce to stiffness/relaxation-controlled ratios [citation]. In particular, reference is a two-scale interpolation between the limiting regimes $\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}\gg\Xi_{\mathrm{cos}}$TeX\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}\gg\Xi_{\mathrm{cos}} and $\Xi_{\mathrm{cos}}\gg\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}$TeX\Xi_{\mathrm{cos}}\gg\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}, not a microphysically universal formula. It is used only as a heuristic interpolator organizing the limiting response classes; the hierarchy results below depend only on the proxy family, the admissibility conditions, and the regime criterion reference. Operationally, $\chi_{\mathrm{ph}}(k,\omega)$TeX\chi_{\mathrm{ph}}(k,\omega) is taken to be a dimensionless response-amplitude ratio in linear response: the (Fourier) amplitude of a normalized internal readout divided by the amplitude of the imposed long-wavelength drive at the same $(k,\omega)$TeX(k,\omega). Concretely, let $S_{\mathcal{B}}(t)$TeXS_{\mathcal{B}}(t) be an internal readout that tracks the long-wavelength geometry of the bound structure (for example a strain mode in a solid or a size readout in a virialized system), with nonzero background value $S_{\mathcal{B},0}$TeXS_{\mathcal{B},0}, and define

\delta s_{\mathcal{B}}(k,\omega):=\frac{\delta S_{\mathcal{B}}(k,\omega)}{S_{\mathcal{B},0}},
\qquad
\chi_{\mathrm{ph}}(k,\omega)
\equiv
\frac{\abs{\delta s_{\mathcal{B}}(k,\omega)}}{\abs{\delta\HH_{\mathrm{cos}}(k,\omega)}}.

The gating enforces linear response (small perturbations, stable background, and controlled bandwidth around $\omega$TeX\omega). The normalized response ratio $\chi_{\mathrm{ph}}(k,\omega)$TeX\chi_{\mathrm{ph}}(k,\omega) is the reference comparison observable on the same admitted family/window. Together with the carrier partition, the admitted response-update rule, and the rigidity-map pair, it completes the fixed object set used for the suppression and non-tracking statements below. In the limit $\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}\gg\Xi_{\mathrm{cos}}$TeX\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}\gg\Xi_{\mathrm{cos}} we expect $\chi_{\mathrm{ph}}\to 0$TeX\chi_{\mathrm{ph}}\to 0 (phase locking), while in the opposite limit $\Xi_{\mathrm{cos}}\gg\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}$TeX\Xi_{\mathrm{cos}}\gg\Xi_{\mathrm{bind}}^{(\mathrm{proxy})} we expect $\chi_{\mathrm{ph}}\to 1$TeX\chi_{\mathrm{ph}}\to 1 (phase tracking of the cosmological background).

lemma: Operational non-tracking on a normalized readout. On the carrier partition and under the admitted response-update rule reference, let $S_{\mathcal B}$TeXS_{\mathcal B} be an internal size or strain readout on a linear-response window $\mathcal W$TeX\mathcal W, with nonzero background value $S_{\mathcal B,0}$TeXS_{\mathcal B,0}, and let $\delta s_{\mathcal B}=\delta S_{\mathcal B}/S_{\mathcal B,0}$TeX\delta s_{\mathcal B}=\delta S_{\mathcal B}/S_{\mathcal B,0}. If

\abs{\chi_{\mathrm{ph}}(k\sim 1/L,\omega)}\le \varepsilon\ll 1
\qquad \text{on } \mathcal W,

then the normalized variation of the readout is suppressed relative to the imposed long-wavelength drive on the same window:

\abs{\delta s_{\mathcal B}(k,\omega)}
=
\abs{\chi_{\mathrm{ph}}(k,\omega)}\,\abs{\delta\HH_{\mathrm{cos}}(k,\omega)}
\le
\varepsilon\,\abs{\delta\HH_{\mathrm{cos}}(k,\omega)}.

Equivalently,

\frac{\abs{\delta S_{\mathcal B}(k,\omega)}}{S_{\mathcal B,0}}
\le
\varepsilon\,\abs{\delta\HH_{\mathrm{cos}}(k,\omega)}.

Hence the internal readout does not operationally track the cosmological drive on $\mathcal W$TeX\mathcal W.

proof. Equation reference gives $\abs{\delta s_{\mathcal B}(k,\omega)} = \abs{\chi_{\mathrm{ph}}(k,\omega)}\,\abs{\delta\HH_{\mathrm{cos}}(k,\omega)}$TeX\abs{\delta s_{\mathcal B}(k,\omega)} = \abs{\chi_{\mathrm{ph}}(k,\omega)}\,\abs{\delta\HH_{\mathrm{cos}}(k,\omega)}. Combining this with reference yields reference; reference is just the definition of $\delta s_{\mathcal B}$TeX\delta s_{\mathcal B}.

proposition: Non-identifiability of the schematic response ratio. If $\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}$TeX\Xi_{\mathrm{bind}}^{(\mathrm{proxy})} is not fixed independently, the interpolator in reference cannot predict $\chi_{\mathrm{ph}}$TeX\chi_{\mathrm{ph}}. For every prescribed response $\chi_\star\in(0,1)$TeX\chi_\star\in(0,1) and every $\Xi_{\mathrm{cos}}>0$TeX\Xi_{\mathrm{cos}}>0, there is a positive binding proxy that reproduces $\chi_\star$TeX\chi_\star.

proof. Set

\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}
=\Xi_{\mathrm{cos}}\frac{1-\chi_\star}{\chi_\star}.

This quantity is positive for $\chi_\star\in(0,1)$TeX\chi_\star\in(0,1). Direct substitution into reference, when that schematic relation is taken as an equality on the comparison window, gives

\frac{\Xi_{\mathrm{cos}}}
{\Xi_{\mathrm{cos}}+\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}}
=\chi_\star.

Thus an observed response can always be absorbed into the proxy. An independent proxy construction and a fixed response coefficient are necessary before the relation has predictive empirical content.

Phase-decoupling criterion for bound structures

We now formulate the criterion that captures the qualitative structure just discussed. All claims in this theorem cluster are read only on the carrier partition and under the admitted response-update rule on the admitted proxy family/window. Competition-regime tests, robustness/null-test sealing, and matter constitutive closure are outside the present scope.

Phase-decoupling criterion. Let $\mathcal{B}$TeX\mathcal{B} be a bound structure of size $L$TeXL on an admitted proxy family and fixed slow-drive window, with proxy-based binding measure $\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}(L,\omega)$TeX\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}(L,\omega), and let $\Xi_{\mathrm{cos}}(L,\omega)$TeX\Xi_{\mathrm{cos}}(L,\omega) denote the coarse-grained cosmological phase drive across the same scale and frequency. If the low-frequency response on that window belongs to the stiffness-dominated class summarized by reference, then the regime

\Xi_{\mathrm{cos}}(L,\omega)
\ll
\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}(L,\omega),

defines the phase-decoupled window, in the sense that:

- the induced phase response satisfies $\abs{\chi_{\mathrm{ph}}(k\sim 1/L,\omega)}\ll 1$TeX\abs{\chi_{\mathrm{ph}}(k\sim 1/L,\omega)}\ll 1; - any normalized internal strain or size readout obeys the suppression law reference on the same window; - the declared normalized internal readout is non-tracking relative to the imposed long-wavelength background on that window, so the structure is operationally phase-decoupled in the sense of the present criterion.

When $\Xi_{\mathrm{cos}}(L,\omega)\gtrsim\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}(L,\omega)$TeX\Xi_{\mathrm{cos}}(L,\omega)\gtrsim\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}(L,\omega), significant phase responsiveness is expected and the competition regime falls outside the present analysis.

Interpretation in terms of Hubble expansion

In FRW coordinates, the physical separation between comoving points scales as $a(t)$TeXa(t). At the effective-metric level used only for interpretation, one may regard the internal response of a bound structure as governed by a phase-geometric renormalization of the local dynamics. In the phase-decoupled regime reference, the response of the chosen size observables is dominated by the binding-induced sector on the deep-decoupling window, so internal distances remain effectively fixed to leading order even as the ambient FRW metric expands.

Operationally, the present criterion classifies bound structures as non-tracking on the fixed window when their internal phase geometry belongs to a high-rigidity basin in the $\HH$TeX\HH configuration space and is insensitive to the slow drift of $\HH_{\mathrm{cos}}$TeX\HH_{\mathrm{cos}} there.

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

*Corollary (Present-epoch hierarchy estimate for atoms and crystalline solids)

For the present-epoch slow-drive window $\omega_{\mathrm{drv}}\sim H_0$TeX\omega_{\mathrm{drv}}\sim H_0, and still on the carrier partition with the same admitted response-update rule, the condensed-matter families use the fixed internal scales $\omega_{\mathrm{bind}}^{(\mathrm{proxy})}\in\{\omega_{\mathrm{atom}},\omega_D\}$TeX\omega_{\mathrm{bind}}^{(\mathrm{proxy})}\in\{\omega_{\mathrm{atom}},\omega_D\}. The hierarchy $\omega_{\mathrm{drv}}\ll \omega_{\mathrm{bind}}^{(\mathrm{proxy})}$TeX\omega_{\mathrm{drv}}\ll \omega_{\mathrm{bind}}^{(\mathrm{proxy})} implies

\Xi_{\mathrm{cos}}(L,\omega_{\mathrm{drv}})
\ll
\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}(L,\omega_{\mathrm{drv}};\omega_{\mathrm{bind}}^{(\mathrm{proxy})})

on the fixed window. Representative atomic proxy scales and the declared crystalline-solid proxy family therefore lie in the deep phase-decoupled regime of the present criterion on that slow-drive window. This hierarchy estimate is read only on the declared present-epoch slow-drive window and the corresponding fixed condensed-matter proxy families.

In this section we apply the phase-decoupling criterion to atoms and crystalline solids. Laboratory-fluid subclasses are mentioned only briefly and are not used below. The scale estimates in this section are read only on admitted proxy families and their fixed slow-drive windows; they do not widen the proxy-family reading of the criterion. Our goal is not to compute exact values of $\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}$TeX\Xi_{\mathrm{bind}}^{(\mathrm{proxy})} but to show qualitatively that the inequality $\Xi_{\mathrm{cos}}\ll\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}$TeX\Xi_{\mathrm{cos}}\ll\Xi_{\mathrm{bind}}^{(\mathrm{proxy})} holds in these condensed-matter families with overwhelming margin. Virialized gravitational estimates are collected separately in reference.

Atomic and molecular scales

At atomic scales, binding energies are set by electromagnetic interactions with characteristic frequencies $\omega_{\mathrm{atom}}\sim 10^{15}\,\mathrm{s^{-1}}$TeX\omega_{\mathrm{atom}}\sim 10^{15}\,\mathrm{s^{-1}}. The associated phase rigidity is enormous compared to any cosmological driving frequency on the fixed slow-drive window ($\omega_{\mathrm{drv}}\sim H_0$TeX\omega_{\mathrm{drv}}\sim H_0 or smaller). Thus, for $L\sim 10^{-10}\,\mathrm{m}$TeXL\sim 10^{-10}\,\mathrm{m} and $\omega=\omega_{\mathrm{drv}}$TeX\omega=\omega_{\mathrm{drv}} we have

\Xi_{\mathrm{cos}}(L,\omega_{\mathrm{drv}})
\ll
\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}(L,\omega_{\mathrm{drv}};\omega_{\mathrm{atom}}),

with the binding side anchored by the atomic proxy scale $\omega_{\mathrm{atom}}$TeX\omega_{\mathrm{atom}}. The representative atomic proxy scale is therefore deeply in the phase-decoupled regime on the fixed window.

Proxy hierarchy estimator..

For the atomic proxy family, define

\widehat{\mathcal H}_{\mathrm{atom}}
\equiv
\left(\frac{H_0}{\omega_{\mathrm{atom}}}\right)^2.

Taking a representative atomic binding frequency $\omega_{\mathrm{atom}}\sim 10^{15}\,\mathrm{s^{-1}}$TeX\omega_{\mathrm{atom}}\sim 10^{15}\,\mathrm{s^{-1}} and a present-epoch Hubble rate of order $10^{-18}\,\mathrm{s^{-1}}$TeX10^{-18}\,\mathrm{s^{-1}} [citation], one finds

\widehat{\mathcal H}_{\mathrm{atom}}\sim 10^{-66}.

On the fixed proxy map, $\widehat{\mathcal H}_{\mathrm{atom}}\ll 1$TeX\widehat{\mathcal H}_{\mathrm{atom}}\ll 1 is sufficient evidence that the same family lies deep in the proxy-based rigidity-dominated regime on the slow-drive window. Like the solid-state indicator introduced below, $\widehat{\mathcal H}_{\mathrm{atom}}$TeX\widehat{\mathcal H}_{\mathrm{atom}} is an order-of-magnitude hierarchy diagnostic on that proxy map, not a direct measurement of $\Xi_{\mathrm{cos}}$TeX\Xi_{\mathrm{cos}} or $\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}$TeX\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}.

Crystalline solids

In crystalline solids, phonon modes encode the low-energy phase dynamics of the lattice. The Debye frequency $\omega_D$TeX\omega_D and elastic moduli set a large phase rigidity scale. For cosmic driving frequencies on the fixed slow-drive window $\omega=\omega_{\mathrm{drv}}\lesssim H_0$TeX\omega=\omega_{\mathrm{drv}}\lesssim H_0 and macroscopic lengthscales $L$TeXL up to laboratory scales, the same inequality holds:

\Xi_{\mathrm{cos}}(L,\omega_{\mathrm{drv}})
\ll
\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}(L,\omega_{\mathrm{drv}};\omega_D),

with the binding side anchored by the Debye-scale proxy $\omega_D$TeX\omega_D, so the same fixed slow-drive-window criterion places the declared crystalline-solid proxy family in the non-tracking regime.

Hierarchy indicator..

For the solid-state proxy map, define

\widehat{\mathcal H}_{\mathrm{solid}}
\equiv
\left(\frac{H_0}{\omega_D}\right)^2.

Taking a representative Debye frequency $\omega_D\sim 10^{13}\,\mathrm{s^{-1}}$TeX\omega_D\sim 10^{13}\,\mathrm{s^{-1}} and a present-epoch Hubble rate of order $10^{-18}\,\mathrm{s^{-1}}$TeX10^{-18}\,\mathrm{s^{-1}} [citation], one finds

\widehat{\mathcal H}_{\mathrm{solid}}\sim 10^{-62}.

This quantity is used only as an order-of-magnitude hierarchy indicator on the fixed proxy map and is not promoted to an observational proof for crystalline-solid matter in general.

Crystalline-solid proxy family

Consider a family $\mathfrak F_{\mathrm{solid}}$TeX\mathfrak F_{\mathrm{solid}} of chemically stable crystalline laboratory solids on a fixed coarse-graining scale $L$TeXL and slow-drive window $\mathcal W_\omega$TeX\mathcal W_\omega. We write

\mathcal F_{\mathrm{proxy}}^{(\mathrm{solid})}
=
(\mathfrak F_{\mathrm{solid}},\mathcal O_{\mathrm{rig}}^{(\mathrm{solid})},\mathcal O_{\mathrm{resp}}^{(\mathrm{solid})},L,\mathcal W_\omega,\mathsf{Map}_{\mathrm{rig}}^{(\mathrm{solid})},\mathsf{Est}_{\mathrm{solid}},\delta_{\mathrm{lin}},\delta_{\mathrm{bg}}),

with rigidity proxy $\mathcal O_{\mathrm{rig}}^{(\mathrm{solid})}=\{\mu,\omega_D\}$TeX\mathcal O_{\mathrm{rig}}^{(\mathrm{solid})}=\{\mu,\omega_D\} and normalized response readout $\mathcal O_{\mathrm{resp}}^{(\mathrm{solid})}=s_{\mathcal B}$TeX\mathcal O_{\mathrm{resp}}^{(\mathrm{solid})}=s_{\mathcal B}, where $s_{\mathcal B}$TeXs_{\mathcal B} is the long-wavelength strain readout or the fractional size readout obtained from $S_{\mathcal B}$TeXS_{\mathcal B} with the common estimator convention $\mathsf{Est}_{\mathrm{solid}}$TeX\mathsf{Est}_{\mathrm{solid}}. A minimal monotone map consistent with the hierarchy argument is

\mathsf{Map}_{\mathrm{rig}}^{(\mathrm{solid})}:
(\mu,\omega_D)
\longmapsto
\Xi_{\mathrm{bind}}^{(\mathrm{proxy}),\mathrm{solid}}(L,\omega_{\mathrm{drv}})
=
C_{\mathrm s}(L)\left(\frac{\omega_D}{\omega_{\mathrm{drv}}}\right)^2,

with $C_{\mathrm s}(L)>0$TeXC_{\mathrm s}(L)>0 a dimensionless calibration factor held fixed across the family. Equation reference is not proposed as a universal microscopic law; it is a monotone proxy assignment used only on the fixed slow-drive window. The window is used only if

\sup_{\omega\in\mathcal W_\omega}\frac{\omega}{\omega_D}\le \eta_{\mathrm{ad}}\ll 1,
\qquad
r_{\mathrm{lin}}(\mathcal W_\omega)\le \delta_{\mathrm{lin}},
\qquad
r_{\mathrm{bg}}(\mathcal W_\omega)\le \delta_{\mathrm{bg}},

with the same $\mathsf{Est}_{\mathrm{solid}}$TeX\mathsf{Est}_{\mathrm{solid}} and baseline convention applied throughout. Changing $L$TeXL, $\mathcal W_\omega$TeX\mathcal W_\omega, $\mathsf{Map}_{\mathrm{rig}}^{(\mathrm{solid})}$TeX\mathsf{Map}_{\mathrm{rig}}^{(\mathrm{solid})}, or $\mathsf{Est}_{\mathrm{solid}}$TeX\mathsf{Est}_{\mathrm{solid}} defines a different family rather than a retuning inside the same proxy assignment. For the present-epoch slow-drive window one therefore has

\Xi_{\mathrm{cos}}(L,\omega_{\mathrm{drv}})
\ll
\Xi_{\mathrm{bind}}^{(\mathrm{proxy}),\mathrm{solid}}(L,\omega_{\mathrm{drv}})
\qquad (\mathfrak F_{\mathrm{solid}}\ \text{fixed}),

so the family lies in the deep-decoupling regime. By the operational lemma above, the same normalized readout obeys

\frac{\abs{\delta S_{\mathcal B}(k,\omega_{\mathrm{drv}})}}{S_{\mathcal B,0}}
=
\abs{\delta s_{\mathcal B}(k,\omega_{\mathrm{drv}})}
\ll
\abs{\delta\HH_{\mathrm{cos}}(k,\omega_{\mathrm{drv}})}
\qquad (\mathfrak F_{\mathrm{solid}},\ \mathcal W_\omega,\ \mathsf{Est}_{\mathrm{solid}}\ \text{fixed}),

so this crystalline family does not track the cosmological drive on that window.

*Remark on laboratory-fluid subclasses Laboratory-fluid subclasses may be treated by separate family-specific proxy families that fix a low-frequency compressibility or relaxation observable together with a normalized response readout on the fixed window. Because those proxy families are less uniform across platforms than the atomic and crystalline-solid cases, they are not used below.

*Virialized systems Virialized gravitational systems are not part of the atom and crystalline-solid examples considered here. The same hierarchy inequality can be checked for those systems using dynamical-time proxies, and the corresponding estimates are collected in Appendix reference.

*Summary across scales

To summarize the scale hierarchy, it is useful to display representative sizes, binding frequencies, and qualitative regimes in a single table. Here $H_0$TeXH_0 denotes the Hubble rate today.

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

The table places the declared condensed-matter proxy families in the phase-decoupled regime on the fixed slow-drive window. Virialized systems satisfy the same inequality under the dynamical-time proxy and are recorded separately in reference.

We now turn to explicit toy realizations of the effective rigidity and response quantities. The constructions below exhibit consistent realizations of $\mathcal{R}_{\mathrm{ph}}$TeX\mathcal{R}_{\mathrm{ph}}, toy-model binding measures, and $\chi_{\mathrm{ph}}(k,\omega)$TeX\chi_{\mathrm{ph}}(k,\omega) in tractable systems. These realizations are read only as illustrations of the declared proxy-family/window structure above and do not redefine the main object set used by the criterion. They are illustrative models rather than material-specific microscopic closures, and the central decoupling criterion does not depend on any single toy realization. The toy-specific quantities $\Xi_{\mathrm{bind}}^{(\mathrm{HO})}$TeX\Xi_{\mathrm{bind}}^{(\mathrm{HO})}, $\Xi_{\mathrm{bind}}^{(\mathrm{chain})}$TeX\Xi_{\mathrm{bind}}^{(\mathrm{chain})}, and $\Xi_{\mathrm{bind}}^{(\mathrm{cont})}$TeX\Xi_{\mathrm{bind}}^{(\mathrm{cont})} are realization-specific binding measures and do not replace the proxy-based quantity $\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}$TeX\Xi_{\mathrm{bind}}^{(\mathrm{proxy})} used in the main criterion.

Toy Model I: Single phase-coupled harmonic oscillator

We begin with the simplest possible bound system: a single harmonic oscillator whose kinetic term is renormalized by the phase geometry.

Effective Hamiltonian

Consider a one-dimensional degree of freedom $q$TeXq with conjugate momentum $p$TeXp and Hamiltonian

\hat{H}
=
\frac{1}{2m_{\mathrm{eff}}(\Xi)}\,\hat{p}^2
+
\frac{1}{2}m\omega_0^2 \hat{q}^2,

where the effective mass is

m_{\mathrm{eff}}(\Xi)
=
\frac{m}{1-\Xi}.

This choice reflects the idea that phase curvature modifies the effective kinetic term via $g^{\mu\nu}_{\mathrm{eff}} \sim (1-\Xi)g^{\mu\nu}$TeXg^{\mu\nu}_{\mathrm{eff}} \sim (1-\Xi)g^{\mu\nu}, within the same effective-metric scaling.

We assume $\Xi\ll 1$TeX\Xi\ll 1 in the regime of interest, so the oscillator frequency becomes

\omega_{\mathrm{eff}}
=
\sqrt{\frac{k}{m_{\mathrm{eff}}}}
=
\sqrt{\frac{k}{m}(1-\Xi)}
=
\omega_0\sqrt{1-\Xi},

with $k=m\omega_0^2$TeXk=m\omega_0^2.

The energy levels are then

E_n(\Xi)
=
\hbar\omega_{\mathrm{eff}}\left(n+\frac{1}{2}\right)
=
\hbar\omega_0\sqrt{1-\Xi}
\left(n+\frac{1}{2}\right).

Phase rigidity from a uniform phase twist

To connect with the phase rigidity definition reference, we imagine that the cosmological phase field induces a small, uniform ``phase twist'' parameter $\theta$TeX\theta across the region occupied by the bound system. At leading order, we parametrize this by a small change in $\Xi$TeX\Xi:

\Xi \to \Xi + \delta\Xi(\theta),
\quad
\delta\Xi(\theta)\propto\theta.

The ground-state energy becomes

E_0(\theta)
=
\frac{1}{2}\hbar\omega_0\sqrt{1-\Xi-\delta\Xi(\theta)}.

Expanding for small $\Xi$TeX\Xi and $\delta\Xi$TeX\delta\Xi:

E_0(\theta)
\simeq
\frac{1}{2}\hbar\omega_0
\left[
1
-\frac{1}{2}(\Xi+\delta\Xi(\theta))
+ \mathcal{O}(\Xi^2)
\right].

If we write

\delta\Xi(\theta) = \alpha \theta + \frac{1}{2}\beta\theta^2 + \cdots,

then

\left.\frac{\partial^2 E_0}{\partial\theta^2}\right|_{\theta=0}
=
-\frac{1}{4}\hbar\omega_0
\left.\frac{\partial^2\delta\Xi(\theta)}{\partial\theta^2}\right|_{\theta=0}
=
-\frac{1}{4}\hbar\omega_0\,\beta.

We identify the phase rigidity (up to a positive-definite sign convention) as

\mathcal{R}_{\mathrm{ph}}^{(\mathrm{HO})}
\sim
\hbar\omega_0\,|\beta|.

Thus, in this simplest example, the phase rigidity is proportional to the characteristic binding frequency $\omega_0$TeX\omega_0. The precise coefficient depends on how the phase twist enters $\Xi$TeX\Xi, which in turn depends on the microscopic completion. At leading order it suffices to retain the scaling:

\mathcal{R}_{\mathrm{ph}}^{(\mathrm{HO})}
\propto
\hbar\omega_0.

Dimensionless binding measure

Following reference, we define a dimensionless binding-induced phase measure

\Xi_{\mathrm{bind}}^{(\mathrm{HO})}
\equiv
\frac{\mathcal{R}_{\mathrm{ph}}^{(\mathrm{HO})}}
{\Lambda_\Xi^4/M_{\mathrm{eff}}^2}
\sim
\frac{\hbar\omega_0}{\Lambda_\Xi^4/M_{\mathrm{eff}}^2}.

For characteristic frequencies $\omega_0$TeX\omega_0 in the atomic or solid-state range and for cosmological-scale $\Lambda_\Xi$TeX\Lambda_\Xi, this measure is expected to be extremely large compared to the cosmological $\Xi_{\mathrm{cos}}(L,\omega_{\mathrm{drv}})$TeX\Xi_{\mathrm{cos}}(L,\omega_{\mathrm{drv}}) on the present-epoch slow-drive window, consistent with the main-window inequality $\Xi_{\mathrm{cos}}\ll\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}$TeX\Xi_{\mathrm{cos}}\ll\Xi_{\mathrm{bind}}^{(\mathrm{proxy})}.

Linear response to a slow phase drive

We now add a time-dependent perturbation that represents a slow cosmological phase drive. We model it as a small modulation of $\Xi$TeX\Xi:

\Xi(t) = \Xi_0 + \delta\Xi(t),

with $\delta\Xi(t)\propto e^{-i\omega t}$TeX\delta\Xi(t)\propto e^{-i\omega t} and $\omega\ll\omega_0$TeX\omega\ll\omega_0.

To linear order in $\delta\Xi$TeX\delta\Xi, the Hamiltonian becomes

\hat{H}(t)
=
\hat{H}_0
+
\hat{V}(t),

with

\hat{V}(t)
\simeq
\frac{\partial\hat{H}}{\partial\Xi}\bigg|_{\Xi_0}
\delta\Xi(t).

The induced change in an observable $\hat{O}$TeX\hat{O} is, in linear response,

\delta\langle \hat{O}(\omega)\rangle
=
\chi_{O\Xi}(\omega)\,\delta\Xi(\omega),

with susceptibility

\chi_{O\Xi}(\omega)
=
\sum_{m\neq 0}
\left(
\frac{\matrixel{0}{\hat{O}}{m}\matrixel{m}{\partial\hat{H}/\partial\Xi}{0}}
{E_0-E_m+\hbar\omega+i0^+}
+
\text{c.c.}
\right).

For the harmonic oscillator, the dominant contribution comes from the first excited state. The result has the schematic form

\chi_{O\Xi}^{(\mathrm{HO})}(\omega)
\sim
\frac{\omega_0}{\omega_0^2-\omega^2-i\gamma\omega},

where $\gamma$TeX\gamma represents possible damping. In the adiabatic limit $\omega\ll\omega_0$TeX\omega\ll\omega_0,

\chi_{O\Xi}^{(\mathrm{HO})}(\omega\ll\omega_0)
\sim
\frac{1}{\omega_0}.

Thus the susceptibility of the bound system to slow phase modulation is suppressed by the binding frequency: the larger $\omega_0$TeX\omega_0, the smaller the response to a given $\delta\Xi(\omega\ll\omega_0)$TeX\delta\Xi(\omega\ll\omega_0). This is the microscopic counterpart of the statement that $\Xi_{\mathrm{bind}}^{(\mathrm{HO})}$TeX\Xi_{\mathrm{bind}}^{(\mathrm{HO})} grows with $\omega_0$TeX\omega_0 and dominates over $\Xi_{\mathrm{cos}}$TeX\Xi_{\mathrm{cos}} for typical condensed-matter systems on the fixed slow-drive window.

Toy Model II: One-dimensional phase rotor chain

To capture spatially extended rigidity, we next consider a simple lattice model: a one-dimensional chain of phase rotors with nearest neighbour coupling.

Hamiltonian and continuum limit

Let $\phi_j$TeX\phi_j be a phase variable on site $j$TeXj (e.g.\ a coarse-grained phase of a condensate or a lattice displacement phase). We consider the Hamiltonian

H
=
\sum_{j}
\left[
\frac{1}{2I}\pi_j^2
+
\frac{K}{2}(\phi_{j+1}-\phi_j)^2
+
U(\phi_j)
\right],

where:

- $I$TeXI is an effective moment of inertia, - $K$TeXK is a stiffness controlling nearest-neighbour phase differences, - $U(\phi_j)$TeXU(\phi_j) encodes local binding (e.g.\ pinning to equilibrium).

In the small-fluctuation regime around a uniform equilibrium $\phi_j=\phi_0$TeX\phi_j=\phi_0, we may expand $U(\phi)$TeXU(\phi) as

U(\phi)
\simeq
\frac{1}{2}M^2(\phi-\phi_0)^2,

with $M$TeXM an effective mass scale. Then reference becomes a harmonic chain. In the continuum limit $x=ja$TeXx=ja with lattice spacing $a$TeXa, we write $\phi_j\to\phi(x)$TeX\phi_j\to\phi(x) and obtain

H
\simeq
\int dx
\left[
\frac{1}{2I}\pi^2(x)
+
\frac{K}{2}(\partial_x\phi)^2
+
\frac{M^2}{2}\phi^2(x)
\right].

The normal modes have dispersion

\omega^2(k)
=
\omega_0^2 + v^2 k^2,
\quad
\omega_0^2 = \frac{M^2}{I},\quad
v^2 = \frac{K}{I}.

Energy cost of a long-wavelength phase twist

We now consider a slowly varying phase twist across a system of size $L$TeXL, represented by a static configuration

\phi(x)
=
\phi_0 + \theta\frac{x}{L},
\quad
x\in[0,L],

where $\theta$TeX\theta is the total phase difference between the two ends. The gradient is

\partial_x\phi
=
\frac{\theta}{L}.

The energy cost of this twist (relative to the uniform state) is

\Delta E(\theta)
=
\int_0^L dx\,
\frac{K}{2}(\partial_x\phi)^2
=
\int_0^L dx\,
\frac{K}{2}\left(\frac{\theta}{L}\right)^2
=
\frac{K}{2L}\theta^2.

From the definition in reference, the phase rigidity at scale $L$TeXL is proportional to the second derivative of $\Delta E$TeX\Delta E with respect to $\theta$TeX\theta:

\mathcal{R}_{\mathrm{ph}}^{(\mathrm{chain})}(L)
\sim
\frac{\partial^2 \Delta E}{\partial\theta^2}
=
\frac{K}{L}.

In a more refined treatment, one may include the effect of $\omega_0$TeX\omega_0 and other microscopic parameters, but at the level used here it suffices that

\mathcal{R}_{\mathrm{ph}}^{(\mathrm{chain})}(L)
\propto
\frac{K}{L},

i.e.\ the phase rigidity increases with stiffness $K$TeXK and decreases with system size $L$TeXL.

Dimensionless binding measure and scaling

We define the corresponding binding-induced phase measure

\Xi_{\mathrm{bind}}^{(\mathrm{chain})}(L)
\equiv
\frac{\mathcal{R}_{\mathrm{ph}}^{(\mathrm{chain})}(L)}
{\Lambda_\Xi^4/M_{\mathrm{eff}}^2}
\sim
\frac{K/L}{\Lambda_\Xi^4/M_{\mathrm{eff}}^2}.

If we relate $K$TeXK to microscopic frequencies via

K
\sim
I \omega_{\mathrm{lat}}^2,

Phase decoupling must be stated sector by sector when the scalar target is compact. On a spatial loop of length $\ell$TeX\ell with winding $n$TeXn, the canonical gradient energy obeys

E_\gamma\ge \frac{n^2L_\chi^2}{2\ell}.

A smooth bound configuration with $n\ne0$TeXn\ne0 cannot relax to a globally constant phase while remaining in the same target space. Complete relocking is therefore available only in the zero-winding sector or after a defect-mediated topology change.

This lower bound supplies a quantitative addition to the energetic decoupling criterion. Any proposed bound-state branch must compare its binding and interface energies with the irreducible winding cost and must specify whether the relevant loops are contractible inside the physical domain. The bound does not by itself prove the existence or stability of a soliton; it excludes constant-phase endpoints in nonzero sectors and identifies the additional defect or ultraviolet degrees of freedom required to connect them.

The closure test for the phase decoupling of bound structures 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 internal scales, drift rates, boundary fluxes, and environmental contrasts. Let $a$TeXa range over the independent constitutive inputs comprising binding potential, environmental phase profile, boundary matching, and relaxation kernel.