Noise, Dissipation, and Null Tests of Phase-Decoupling

On a fixed measurement band, the problem reduces to a null test and channel-wise upper bounds on residual phase response. Bound media suppress large-scale phase driving when internal rigidity dominates the low-energy phase sector. The issue is whether that suppression remains stable once damping, stochastic forcing, disorder, and readout back-action are included at the level of effective collective modes. The analysis stays at the level of effective phase dynamics and their mapping to standard observables; no microscopic bath identification is required, and a null residual is not converted into a positive estimate of cosmological driving.

Let $\varphi(x,t)$TeX\varphi(x,t) denote a coarse-grained collective phase mode. All noisy and dissipative effects considered below act on such modes and do not introduce an additional fundamental field. Disorder and weak measurement back-action enter through an effective damping/self-energy structure and a corresponding noise spectrum, following standard open-system reductions for low-energy degrees of freedom [citation]. Only the structural link between damping and noise is used below; no detailed bath model is imposed.

Representative platform classes include high-$Q$TeXQ mechanics, cold-atom coherence probes, and superconducting circuits. Current experiments already access ultrahigh-$Q$TeXQ levitated oscillators, long-lived bulk-acoustic or cubic-SiC electromechanical modes, controlled phase-coherence and phase-fluctuation observables in cold-atom simulators, and near-millisecond to millisecond superconducting-qubit coherence together with correlated-noise diagnostics [citation]. These platforms enter only as channel anchors for the null-test and upper-bound logic, not as independent microphysical derivation targets. In each case the measurement band, filtering and de-embedding rule, baseline model, and residual statistic are fixed before upper-bound extraction.

Any cosmological contribution enters measured response or noise statistics only as a residual after the measurement context has been fixed. The closure conditions are written in terms of the triplet

\Xicos(L,\omega) \ll \Xibind(L,\omega),
\qquad
\gph(L,\omega) \ll \wbind(L),
\qquad
\Scos(L,\omega) \ll \Sint(L,\omega),

where the first inequality enforces rigidity dominance, the second requires an underdamped phase mode, and the third bounds any cosmological contribution below the internal-noise floor on the chosen band. The rigidity, damping, and noise quantities used below are platform-dependent response proxies. No microscopic carrier identification beyond the chosen effective mode is assumed. Once the measurement band, filtering and de-embedding rule, baseline model, and residual statistic have been fixed, changing any of them defines a different analysis convention.

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

Effective phase dynamics

Consider a continuum phase mode with effective action

S[\varphi] = \int d^4x\, \left[
\frac{Z}{2}(\partial_t\varphi)^2
- \frac{Y}{2}(\nabla\varphi)^2
- \frac{M^2}{2}\varphi^2
+ J_{\mathrm{cos}}(x,t)\,\varphi
\right],

where $Z$TeXZ, $Y$TeXY, and $M$TeXM encode the local phase rigidity. Dissipation and noise are introduced at the effective equation-of-motion level as

Z\,\partial_t^2\varphi + \eta\,\partial_t\varphi - Y\nabla^2\varphi + M^2\varphi
= J_{\mathrm{cos}}(x,t) + \xi(x,t).

Here $\eta$TeX\eta is an effective damping coefficient and $\xi(x,t)$TeX\xi(x,t) is a stochastic forcing term with correlator

\langle \xi(x,t)\,\xi(x',t')\rangle = \mathcal{N}(x-x',t-t').

Equation reference is an effective representation of environmental couplings for the phase mode, not a modification of the underlying field content. Standard open-system reductions motivate this form [citation].

Response kernel and phase spectrum

Fourier transformation of reference gives

\left[-Z\omega^2 - i\eta\omega + Y\bm{k}^2 + M^2\right]\varphi(\omega,\bm{k})
= J_{\mathrm{cos}}(\omega,\bm{k}) + \xi(\omega,\bm{k}).

The retarded phase-response kernel is therefore

\chi_{\mathrm{ph}}(\omega,\bm{k})
= \frac{\delta\langle\varphi(\omega,\bm{k})\rangle}{\delta J_{\mathrm{cos}}(\omega,\bm{k})}
= \frac{1}{-Z\omega^2 - i\eta\omega + Y\bm{k}^2 + M^2}.

The operational phase spectrum is defined by

S_{\mathrm{ph}}(\omega,\bm{k})
\equiv
\int dt\int d^3x\,e^{i\omega t-i\bm{k}\cdot\bm{x}}
\langle \varphi(\bm{x},t)\,\varphi(0,0)\rangle_{\mathrm{noise}}.

With the symmetrized convention,

S_{\mathrm{ph}}^{(\mathrm{sym})}(\omega,\bm{k})
\equiv
\int dt\int d^3x\,e^{i\omega t-i\bm{k}\cdot\bm{x}}
\frac{1}{2}\left\langle\{\varphi(\bm{x},t),\varphi(0,0)\}\right\rangle_{\mathrm{noise}},

Gaussian forcing with spectral density $S_{\xi}(\omega,\bm{k})$TeXS_{\xi}(\omega,\bm{k}) yields

S_{\mathrm{ph}}(\omega,\bm{k}) = |\chi_{\mathrm{ph}}(\omega,\bm{k})|^2\,S_{\xi}(\omega,\bm{k}).

Fluctuation--dissipation structure

At thermal equilibrium, the symmetrized spectrum reference and the retarded kernel reference satisfy the standard fluctuation--dissipation relation [citation]

S_{\mathrm{ph}}^{(\mathrm{sym})}(\omega,\bm{k})
=
\hbar\,\coth\!\left(\frac{\hbar\omega}{2k_B T}\right)\Im\chi_{\mathrm{ph}}(\omega,\bm{k}),

which reduces in the classical regime $\hbar\omega\ll k_B T$TeX\hbar\omega\ll k_B T to

S_{\mathrm{ph}}^{(\mathrm{sym})}(\omega,\bm{k})
\simeq
\frac{2k_B T}{\omega}\,\Im\chi_{\mathrm{ph}}(\omega,\bm{k}).

Only two structural consequences are used below: the same mechanisms that generate damping also generate noise, and equilibrium noise levels cannot be varied independently of dissipation without changing the response structure.

The fluctuation--dissipation relation is used here only as an equilibrium anchor for the chosen thermalized sector. Nonequilibrium channels such as mechanical vibration, pulsed analogue drives, cosmic-ray bursts, or readout back-action are not inferred from reference and reference; they are bounded only through fixed baseline spectra, control runs, and residual estimators on the same analysis convention.

Internal and cosmological contributions

The effective noise acting on the phase mode is separated into an internal component $\Sint(\omega,\bm{k})$TeX\Sint(\omega,\bm{k}) and a possible residual component $\Scos(\omega,\bm{k})$TeX\Scos(\omega,\bm{k}) associated with large-scale phase driving. The latter is treated only as an upper-bound term. Operationally, any cosmological contribution must appear as a subdominant residual in measured spectra after the measurement bandwidth, time window, filtering, and baseline subtraction have been fixed. Accordingly, the closure requires

\Scos(\omega,\bm{k}) \ll \Sint(\omega,\bm{k})

on the chosen band. A later microscopic model may relate $\Scos$TeX\Scos to $\Xicos$TeX\Xicos through a platform-dependent map,

\Scos(\omega,\bm{k}) \lesssim f\!\left(\Xicos(L,\omega)\right),

but no such specification is required for the upper-bound logic used here.

The three models below serve only as proxy realizations for the rigidity--damping--noise triplet on a fixed measurement band: a damped harmonic oscillator, a pinned rotor chain, and a phase mode with Ohmic damping. They are not offered as a microphysical derivation of constitutive matter response. Their role is only to display how the same declared-band closure appears across representative dissipative channels.

Damped harmonic oscillator

Consider a harmonic coordinate $q$TeXq with conservative Hamiltonian

\hat H = \frac{\hat p^2}{2m} + \frac{1}{2}m\omega_0^2\hat q^2.

At the effective level,

m\ddot q + \gamma\dot q + m\omega_0^2 q = F_{\mathrm{cos}}(t) + \xi(t),

with response kernel

\chi_{qF}(\omega)=\frac{1}{-m\omega^2-i\gamma\omega+m\omega_0^2}.

The characteristic damping rate and binding frequency are

\gph^{(\mathrm{HO})}=\frac{\gamma}{2m},
\qquad
\wbind^{(\mathrm{HO})}=\omega_0.

Robust phase decoupling requires

\Xicos\ll\Xibind,
\qquad
\gph^{(\mathrm{HO})}\ll\wbind^{(\mathrm{HO})},
\qquad
\Scos(\omega)\ll\Sint(\omega).

The second inequality is the standard high-$Q$TeXQ condition with $Q\equiv \omega_0/(2\gph^{(\mathrm{HO})})\gg 1$TeXQ\equiv \omega_0/(2\gph^{(\mathrm{HO})})\gg 1.

Pinned rotor chain

For a phase chain with variables $\phi_j$TeX\phi_j, moment of inertia $I$TeXI, stiffness $K$TeXK, and pinning scale $M$TeXM, consider

I\ddot{\phi}_j + \eta_{\phi}\dot{\phi}_j + K(\phi_{j+1}-2\phi_j+\phi_{j-1}) + M^2(\phi_j-\phi_0)
= F_{\mathrm{cos},j}(t) + \xi_j(t).

In the continuum limit this reduces to a special case of reference. The static energy cost of a phase twist $\theta$TeX\theta across a length $L$TeXL is

\Delta E(\theta)=\frac{K}{2L}\theta^2,
\qquad
\mathcal R_{\mathrm{ph}}^{(\mathrm{chain})}(L)\sim\frac{K}{L}.

Dissipation and noise introduce a coherence length $L_{\phi}$TeXL_{\phi} beyond which correlations are suppressed. A representative order-of-magnitude estimate is

L_{\phi}\sim\sqrt{\frac{Y}{\eta_{\phi}\gamma_{\phi}}},

where $\gamma_{\phi}$TeX\gamma_{\phi} is a local decoherence rate. The estimate reference is used only as a coherence proxy. The relevant statement is the fixed-band closure

L\ll L_{\phi},
\qquad
\Xicos(L,\omega)\ll\Xibind^{(\mathrm{chain})}(L,\omega),
\qquad
\Scos(\omega,\bm{k})\ll\Sint(\omega,\bm{k}),

on the same $(k,\omega)$TeX(k,\omega) window.

Phase mode with Ohmic damping

For an Ohmic environment, the retarded phase kernel remains

\chi_{\mathrm{ph}}(\omega,\bm{k})
= \frac{1}{-Z\omega^2-i\eta\omega+Y\bm{k}^2+M^2}.

At fixed $\bm{k}$TeX\bm{k} on the chosen band, define

\omega_0^2(\bm{k})\equiv \frac{Y\bm{k}^2+M^2}{Z}.

For weak damping, $\eta\ll 2Z\omega_0$TeX\eta\ll 2Z\omega_0, the poles are

\tilde\omega_{\pm}(\bm{k})\simeq \pm\omega_0(\bm{k})-i\frac{\eta}{2Z}.

Accordingly,

\wbind(\bm{k})\equiv \omega_0(\bm{k}),
\qquad
\gph\simeq\frac{\eta}{2Z},
\qquad
Q\equiv\frac{\wbind}{2\gph}\simeq \frac{Z\wbind}{\eta}\gg 1.

The robustness closure is then written directly as

\Xicos(L,\omega)\ll\Xibind(L,\omega),
\qquad
Q(L,\omega)\gg 1,
\qquad
\Scos(L,\omega)\ll\Sint(L,\omega).

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

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

A numerical upper bound on $\Scos$TeX\Scos is meaningful only after the measurement context has been fixed: bandwidth, time window, filtering, calibration, baseline subtraction, and internal-noise characterization. Once those ingredients are chosen, they are held fixed during bound extraction; changing any of them defines a different analysis convention rather than a refinement within the same one. A reportable upper bound is assigned only in readout channels whose measured observable admits, on that convention, a baseline-plus-excess decomposition. Channels that do not admit such a decomposition remain admissible as residual diagnostics, but no cosmological upper bound is extracted from them. Only the structural mapping from the robustness inequalities to representative observables is used below. A fully explicit bound-extraction protocol is worked out only for a single mechanical resonance. The cold-atom and superconducting sections provide admissible channel mappings; no platform-specific numerical bound is extracted there without a separately fixed convention tuple analogous to $\mathcal C_m$TeX\mathcal C_m. Detailed operational inputs are collected in Appendix reference.

High-Q mechanical resonators

High-$Q$TeXQ mechanical resonators are naturally modeled by the damped harmonic mode. Let $\omega_m$TeX\omega_m be the resonance frequency, $Q_m$TeXQ_m the quality factor, and $S_x(\omega)$TeXS_x(\omega) the displacement-noise spectrum. The identifications

\wbind^{(\mathrm{HO})}\leftrightarrow \omega_m,
\qquad
\gph^{(\mathrm{HO})}\leftrightarrow \frac{\omega_m}{2Q_m},
\qquad
\Sint(\omega)\leftrightarrow S_x(\omega)

convert reference into

\Xicos(L,\omega_m)\ll\Xibind^{(\mathrm{HO})}(L,\omega_m),
\qquad
\frac{\omega_m}{2Q_m}\ll\omega_m,
\qquad
\Scos(L,\omega_m)\ll S_x(\omega_m).

The second inequality is the standard high-$Q_m$TeXQ_m condition realized in modern levitated, bulk-acoustic, and multimode cubic-SiC electromechanical platforms [citation]. The third states only that any residual large-scale contribution must remain below the measured noise floor on the chosen readout band.

To convert reference into a reportable upper bound, fix one resonance and one analysis convention

\mathcal C_m \equiv
\bigl(\Omega_m,\mathcal F_m,T_{\mathrm{acq}},\mathcal T_x,\mathcal B_x,\Delta_\Gamma,\Delta_S\bigr),
\qquad
\Omega_m=[\omega_m-B_m,\omega_m+B_m],

where $\Omega_m$TeX\Omega_m is the analysis band, $\mathcal F_m$TeX\mathcal F_m the filtering and de-embedding rule, $T_{\mathrm{acq}}$TeXT_{\mathrm{acq}} the measurement window, $\mathcal T_x$TeX\mathcal T_x the transfer calibration from raw readout to displacement, $\mathcal B_x$TeX\mathcal B_x the baseline model extracted from control runs, and $\Delta_\Gamma,\Delta_S$TeX\Delta_\Gamma,\Delta_S the residual estimators.

Fixed analysis convention..

For the declared mechanical channel, the fixed convention $\mathcal C_m$TeX\mathcal C_m is the analysis convention: it fixes the measurement band, filtering and de-embedding rule, measurement window, transfer calibration, baseline model, and residual statistics through which calibrated data are converted into any reportable upper-bound statement.

Let $\Gamma_{\mathrm{meas}}$TeX\Gamma_{\mathrm{meas}} and $S_{x,\mathrm{meas}}(\omega)$TeXS_{x,\mathrm{meas}}(\omega) denote the calibrated linewidth and displacement spectrum, with baseline values $\Gamma_{\mathrm{int}}$TeX\Gamma_{\mathrm{int}} and $S_{x,\mathrm{int}}(\omega)$TeXS_{x,\mathrm{int}}(\omega).

Fixed reporting rule..

A reportable cosmological upper bound is assigned only for channels that are analyzed on one fixed convention $\mathcal C_m$TeX\mathcal C_m and admit a baseline-plus-excess decomposition of the measured linewidth and spectrum on that same convention. Channels without that decomposition remain residual diagnostics only. All upper-bound statements below are read only within this fixed reporting framework.

Define

\Delta_\Gamma \equiv \frac{|\Gamma_{\mathrm{meas}}-\Gamma_{\mathrm{int}}|}{\omega_m},
\qquad
\Delta_S \equiv \sup_{\omega\in\Omega_m}
\frac{|S_{x,\mathrm{meas}}(\omega)-S_{x,\mathrm{int}}(\omega)|}{S_{x,\mathrm{int}}(\omega)}.

Residual summary pair..

On the fixed convention $\mathcal C_m$TeX\mathcal C_m, the ordered pair $(\Delta_\Gamma,\Delta_S)$TeX(\Delta_\Gamma,\Delta_S) summarizes the residual comparison for the fixed reporting analysis. All downstream upper-bound claims are read only relative to this residual pair and its stated tolerances on the same convention.

Proposition (Channel-wise upper-bound extraction within the fixed reporting framework)..

Within this fixed reporting framework, assume that, after the chosen calibration and de-embedding, the measured channel admits the additive decomposition

\Gamma_{\mathrm{meas}}=\Gamma_{\mathrm{int}}+\delta\Gamma_{\mathrm{cos}},
\qquad
S_{x,\mathrm{meas}}(\omega)=S_{x,\mathrm{int}}(\omega)+S_{x,\mathrm{cos}}(\omega)
\quad (\omega\in\Omega_m),

with any calibration and control-run uncertainty already absorbed into the stated tolerances $\varepsilon_\Gamma$TeX\varepsilon_\Gamma and $\varepsilon_S$TeX\varepsilon_S. If

\Delta_\Gamma\le \varepsilon_\Gamma,
\qquad
\Delta_S\le \varepsilon_S,

The comparison of the residual pair $(\Delta_\Gamma,\Delta_S)$TeX(\Delta_\Gamma,\Delta_S) with the tolerance pair $(\varepsilon_\Gamma,\varepsilon_S)$TeX(\varepsilon_\Gamma,\varepsilon_S) is the final consistency test for the declared mechanical channel. Under that test, the reportable upper bounds on the same convention are

\frac{|\delta\Gamma_{\mathrm{cos}}|}{\omega_m}\le \varepsilon_\Gamma,
\qquad
\frac{|S_{x,\mathrm{cos}}(\omega)|}{S_{x,\mathrm{int}}(\omega)}\le \varepsilon_S
\quad (\omega\in\Omega_m).

If the chosen readout channel does not admit reference on the fixed convention, reference is not used for that channel. Changing $\Omega_m$TeX\Omega_m, $\mathcal F_m$TeX\mathcal F_m, $T_{\mathrm{acq}}$TeXT_{\mathrm{acq}}, $\mathcal T_x$TeX\mathcal T_x, $\mathcal B_x$TeX\mathcal B_x, or the residual statistic defines a new analysis convention.

Resonance protocol..

Choose one isolated resonance and keep the full convention $\mathcal C_m$TeX\mathcal C_m fixed across baseline and test runs. First determine $\Gamma_{\mathrm{int}}$TeX\Gamma_{\mathrm{int}} and $S_{x,\mathrm{int}}$TeXS_{x,\mathrm{int}} from control runs on the same device and readout chain. Next acquire the test data without changing the analysis band, filtering/de-embedding rule, calibration map, baseline model, or residual statistic. Finally report only the pair $(\Delta_\Gamma,\Delta_S)$TeX(\Delta_\Gamma,\Delta_S) together with the upper bounds in reference. If either residual exceeds its stated tolerance, the robustness conditions fail for that resonance on that convention.

Cold-atom condensates

For a weakly interacting neutral condensate, the low-energy phase field $\theta(x,t)$TeX\theta(x,t) is described by

S_{\mathrm{sf}}[\theta]
= \int d^4x\,\left[
\frac{\rho_s}{2c_s^2}(\partial_t\theta)^2
- \frac{\rho_s}{2}(\nabla\theta)^2
\right],

with superfluid stiffness $\rho_s$TeX\rho_s and sound speed $c_s$TeXc_s. This action supplies a laboratory proxy for $\Xibind$TeX\Xibind. The relevant observables are a coherence-length proxy, phonon linewidths or damping rates, and phase/density fluctuation spectra on a chosen $(k,\omega)$TeX(k,\omega) window. Recent excitation-spectrum measurements, phase-fluctuation extraction, noise-correlation methods, and curved-spacetime simulator protocols give direct access to those quantities [citation].

The mapping is then

\Xibind^{(\mathrm{sf})}(L) \ \text{from stiffness and geometry},

\gph(k) \ \text{from phonon damping},

\Sint(k,\omega) \ \text{from measured phase or density spectra}.

This subsection records only an admissible channel template for the cold-atom class. Whenever a readout channel admits a baseline-plus-excess decomposition on a fixed $(k,\omega)$TeX(k,\omega) convention, a null residual constrains only the corresponding excess linewidth or spectral contribution in that same channel. No channel-specific numerical upper bound is extracted here without a separately fixed convention tuple, baseline model, and residual statistic analogous to $\mathcal C_m$TeX\mathcal C_m. Controlled expansion or simulator protocols may be used as laboratory drive surrogates, but any induced excess contribution must still be interpreted through the same response map, baseline accounting, and residual estimator used for ordinary laboratory drives on that same convention [citation].

Superconducting circuits

Superconducting circuits provide phase-like coordinates with tunable dissipation and long monitoring baselines. This subsection records only an admissible channel template for the superconducting-circuit class. Let $\phi(t)$TeX\phi(t) denote a generalized phase coordinate with effective Ohmic damping. The operational observables are coherence times $(T_1,T_2)$TeX(T_1,T_2), mode linewidths, and flux/charge/phase-noise spectra. Recent measurements report systematic coherence improvements from surface engineering, disorder-controlled coherence systematics, near-millisecond to millisecond transmon coherence, mechanically induced correlated errors, cosmic-ray-induced correlated errors, synchronous cosmic-ray coincidences, and correlated charge-noise dynamics in regimes directly relevant to long-baseline residual tests [citation].

For a chosen circuit and readout chain, any contribution attributed to $\Xicos$TeX\Xicos or $\Scos$TeX\Scos must remain below the intrinsic drift, internal noise, and systematic uncertainty on the chosen band. The mechanically induced, cosmic-ray-induced, and correlated charge-noise channels measured in current devices are nonequilibrium by construction; they are bounded through fixed baseline spectra, control runs, and residual estimators, not through fluctuation--dissipation inference. A null residual is converted into an upper bound only for readout channels whose baseline-plus-excess decomposition is fixed on that same convention. No channel-specific numerical upper bound is extracted here without a separately fixed circuit-level convention tuple, baseline model, and residual statistic.

Across the damped models and platform classes considered here, the same closure reappears:

\Xicos(L,\omega)\ll\Xibind(L,\omega),
\qquad
\gph(L,\omega)\ll\wbind(L),
\qquad
\Scos(L,\omega)\ll\Sint(L,\omega).

The first inequality enforces rigidity dominance, the second requires a well-defined underdamped phase mode, and the third requires internal spectra to dominate any residual cosmological contribution on the chosen band. When all three hold, damping and stochastic forcing do not overturn the phase-decoupling regime. For any chosen platform setting, the measurement band, filtering convention, transfer or de-embedding rule, baseline model, and residual statistic are fixed before bound extraction and then held fixed throughout the analysis.

For a chosen readout channel, any cosmological contribution must remain below the internal-noise and systematic floor on the chosen band. A reproducible residual that persists after environmental controls and exceeds the stated tolerance rules out the corresponding robustness conditions on that same channel and convention. When the channel admits a baseline-plus-excess decomposition on a fixed convention, a null residual yields only the corresponding upper limit on excess damping or spectral contribution in that same channel. Improved sensitivity tightens those channel-dependent upper limits once a quantitative map from $\Xicos$TeX\Xicos or $\Scos$TeX\Scos to the measured observable is available. Without such a map, a null result remains a bound on admissible residual size rather than a numerical estimate of cosmological driving.

Causality and dissipation impose analytic restrictions before a noise excess is assigned to phase decoupling. A passive scalar response has a positive-real, equivalently frequency-rotated Herglotz, representation, so its dispersive and absorptive parts cannot be tuned independently. In the stronger finite completely monotone class, the moment Hankel matrices must be positive semidefinite and their stabilized rank is the minimal number of positive relaxation modes.

These statements add two fail-closed null tests. A negative principal minor excludes the positive-pole memory model, and $\mu_0\mu_2-\mu_1^2>0$TeX\mu_0\mu_2-\mu_1^2>0 excludes the one-mode branch. Only a response passing those tests may use one common relaxation time in its fluctuation, dissipation, switching, and recovery predictions. Passivity alone does not imply complete monotonicity, so failure of the latter calls for a broader causal kernel rather than a violation of the former.

The closure test for the noise--dissipation null test 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 noise spectra, response functions, residual floors, and control-channel nulls. Let $a$TeXa range over the independent constitutive inputs comprising dissipative kernel, fluctuation spectrum, environmental controls, and acquisition filter.

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

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

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

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

If the rank is constant locally, these restrictions are tangent to a compatibility manifold of codimension $m-r$TeXm-r. For this sector, the finite closure is admissible only if the response and fluctuation data are generated by one passive causal model with a predeclared memory order and analysis band.

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 independently fitted noise and response spectra need not satisfy a common dissipative realization. Failure of that condition therefore rejects the proposed microscopic closure before parameter estimation can be counted as evidence for it.

Noise and dissipation do not by themselves destroy phase-decoupling. At the level of effective collective modes, the fixed-band closure is encoded in the triplet

\Xicos\ll\Xibind,
\qquad
\gph\ll\wbind,
\qquad
\Scos\ll\Sint.

These inequalities enforce rigidity dominance, an underdamped phase mode, and internal-noise dominance on the chosen band. $\Scos$TeX\Scos therefore enters only as an upper-bound term. If a readout channel admits a baseline-plus-excess decomposition on a fixed convention, a null residual yields only the corresponding upper limit on excess damping or spectral weight in that channel. A reproducible excess contribution that survives control subtraction and exceeds the stated tolerance rules out the corresponding robustness conditions on that same channel and convention. Without a quantitative map from $\Xicos$TeX\Xicos or $\Scos$TeX\Scos to a measured observable, these results remain channel-wise upper bounds rather than numerical estimates of cosmological driving. Accordingly, the paper closes at the level of declared-channel robustness and channel-wise upper bounds. For the explicit mechanical channel, all upper-bound claims are read only within the fixed reporting framework determined by the fixed convention $\mathcal C_m$TeX\mathcal C_m, the residual pair $(\Delta_\Gamma,\Delta_S)$TeX(\Delta_\Gamma,\Delta_S), and the final comparison test reference. Outside that framework, a channel remains diagnostic only, and no cosmological upper bound is assigned there. Only the mechanical channel is worked out as a fully explicit bound-extraction example; the cold-atom and superconducting sections remain admissible mapping templates until a platform-specific convention tuple, baseline model, and residual statistic are fixed. It does not by itself furnish a positive estimate of cosmological driving or a constitutive closure of matter response.

Throughout the platform analysis, $\Scos(L,\omega)$TeX\Scos(L,\omega) enters only as an upper-bound contribution. Controlled drive surrogates, when used, serve only to calibrate residual response on the fixed convention and are not identified with direct cosmological forcing. The analysis convention is fixed in advance and is not retuned after residual inspection. Without both a model for $\Scos$TeX\Scos and a measurement context, no numerical estimate of $\Xicos$TeX\Xicos is extracted.

Any numerical upper bound on $\Scos(L,\omega)$TeX\Scos(L,\omega) or on an induced laboratory drive therefore requires the following operational inputs.

Common context..

- the measurement bandwidth and filtering convention, the acquisition time window, and any averaging used to form the reported spectra; - the readout transfer function and calibration map used to convert raw readout into the reported observable; - a baseline internal-noise spectrum measured with the drive surrogate off, or with an equivalent control configuration, together with the drift model used to remove slow environmental variation.

Platform-specific minimum inputs..

- High-$Q$TeXQ mechanics: $(\omega_m,Q_m)$TeX(\omega_m,Q_m) and a displacement or equivalent noise spectrum on a chosen band around $\omega_m$TeX\omega_m. - Cold atoms: a coherence-length proxy, a phonon linewidth or damping proxy, and a phase or density fluctuation spectrum on a chosen $(k,\omega)$TeX(k,\omega) window. - Superconducting circuits: coherence times $(T_1,T_2)$TeX(T_1,T_2) or linewidths of the relevant mode together with a chosen flux, charge, or phase-noise spectrum on the readout band.

If these inputs are not supplied, no numerical upper bound on $\Scos(L,\omega)$TeX\Scos(L,\omega) is extracted.

Funding and competing interests..

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