Boundary Response and Analogue Tests of Phase-Decoupling

Low-energy phase sectors of bound media carry an intrinsic rigidity that suppresses phase response under weak driving. The regime of interest here is the competition boundary where an effective large-scale phase drive becomes comparable to that rigidity. Let $\Xicos(L,\Omega)$TeX\Xicos(L,\Omega) denote a coarse-grained drive variable and $\Xibind(L,\Omega)$TeX\Xibind(L,\Omega) a normalized phase-rigidity variable on a fixed scale $L$TeXL and hierarchy-frequency window $\Omega$TeX\Omega. Deep decoupling corresponds to

\Xicos(L,\Omega)\ll \Xibind(L,\Omega),

whereas the boundary-response analysis below is restricted to

\Xicos(L,\Omega)\sim \Xibind(L,\Omega).

Ultracold Bose gases and superfluids provide a particularly clean laboratory realization of this regime. Rapidly expanding ring condensates, curved-spacetime simulators, curved light-cone propagation, particle-production protocols in time-dependent effective metrics, and entanglement-sensitive cold-atom analogues of preheating have already been demonstrated or quantitatively benchmarked on such platforms [citation]. Response extraction on fixed momentum and frequency channels is also mature in these systems, through Bragg spectroscopy and dynamic-structure-factor measurements ranging from the original condensate measurements to recent pulse-shaped Bragg control and wide-band excitation spectroscopy in dipolar gases [citation]. Recent nondestructive optomechanical schemes further extend readout toward unequal-time density correlations in Bose--Einstein-condensate platforms [citation]. Taken together, these results justify treating the boundary regime directly as a response problem on analogue platforms.

The boundary-response problem is organized by the underlying competition asymmetry $\varepsilon$TeX\varepsilon, the reciprocal scan coordinate $\epshat$TeX\epshat, the pair-normalized kernel $\chi_{\mathrm{ph}}(k,\omega;\epshat)$TeX\chi_{\mathrm{ph}}(k,\omega;\epshat) reported from midpoint scans on that coordinate, and the midpoint slope $\alpha(k,\omega)$TeX\alpha(k,\omega). The admitted leading law is derived only on finite windows around $\epshat=0$TeX\epshat=0. Unless a fixed convention identifies them on the admitted window, the midpoint law is stated in the reported coordinate $\epshat$TeX\epshat and not directly as a constitutive law in the underlying asymmetry $\varepsilon$TeX\varepsilon. The present result is a laboratory response law on one declared reciprocal scan class near the competition regime. It does not redefine the underlying bridge criterion, and it does not provide cosmological estimation, robustness or null-test sealing, or matter-side constitutive closure. Dissipative robustness, noise budgets, and full null-test closure are not treated here.

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Hierarchy variables and asymmetry parameter

Let $\LXi$TeX\LXi and $\Meff$TeX\Meff denote the fixed scales used to render the competition variables dimensionless. Only the combination $\LXi^4/\Meff^2$TeX\LXi^4/\Meff^2 enters below. The normalized rigidity variable is

\Xibind(L,\Omega)
\equiv
\frac{\mathcal R_{\mathrm{ph}}(L,\Omega)}{\LXi^4/\Meff^2},

where $\mathcal R_{\mathrm{ph}}$TeX\mathcal R_{\mathrm{ph}} is the phase-rigidity functional evaluated on the reported hierarchy layer $(L,\Omega)$TeX(L,\Omega). The asymmetry parameter around the competition boundary is then defined by

\varepsilon(L,\Omega)
\equiv
\frac{\Xibind(L,\Omega)-\Xicos(L,\Omega)}{\Xibind(L,\Omega)+\Xicos(L,\Omega)}.

Hence

\varepsilon\to +1 \Longleftrightarrow \Xicos\ll\Xibind,
\qquad
\varepsilon\to -1 \Longleftrightarrow \Xicos\gg\Xibind,
\qquad
\varepsilon=0 \Longleftrightarrow \Xicos=\Xibind.

remark: Variable layers. The pair $(L,\Omega)$TeX(L,\Omega) labels the coarse-graining layer used to define competition between drive and rigidity. The pair $(k,\omega)$TeX(k,\omega) labels the measured response channel. On a fixed reciprocal scan family these labels are held separately by the fixed protocol.

A useful regime organizer is the midpoint interpolation

\chi_{\mathrm{ph}}^{\mathrm{heur}}(k,\omega)
\sim
\frac{\Xicos(L,\Omega)}{\Xicos(L,\Omega)+\Xibind(L,\Omega)}
=
\frac{1-\varepsilon(L,\Omega)}{2}.

Equation reference is not used as an independent constitutive law. It serves only to organize the scan and to identify the midpoint near $\varepsilon=0$TeX\varepsilon=0.

Reciprocal scan class

A reciprocal scan class consists of one apparatus family, one admitted mode family, one finite response band $B\subset\mathbb R_{+}$TeXB\subset\mathbb R_{+}, one reported de-embedding convention, and paired runs through $+\epshat$TeX+\epshat and $-\epshat$TeX-\epshat about the same midpoint on the reported scan coordinate. For each $(k,\omega)\in B$TeX(k,\omega)\in B, let

\Aresp(k,\omega;\epshat)>0

denote the de-embedded response amplitude extracted under those fixed conventions and reported on that scan coordinate. The reported boundary kernel is the pair-normalized quantity

\chi_{\mathrm{ph}}(k,\omega;\epshat)
\equiv
\frac{\Aresp(k,\omega;\epshat)}
{\Aresp(k,\omega;\epshat)+\Aresp(k,\omega;-\epshat)}.

The midpoint identity

\chi_{\mathrm{ph}}(k,\omega;\epshat)+\chi_{\mathrm{ph}}(k,\omega;-\epshat)=1

is therefore a property of the reciprocal reporting convention. Equations reference and reference define the declared reciprocal normalization identity on that reported reciprocal convention, and all midpoint and normalization statements below are read only relative to this declared identity.

To prevent convention drift from being re-read as boundary physics, fix a finite set $\mathfrak D$TeX\mathfrak D of admissible de-embedding variants sharing the same apparatus family, mode family, reference channel, baseline subtraction, windowing rule, and normalization target. One member of $\mathfrak D$TeX\mathfrak D is the reported convention; the others are stability witnesses only. For each $d\in\mathfrak D$TeXd\in\mathfrak D, let $\chi_{\mathrm{ph}}^{(d)}$TeX\chi_{\mathrm{ph}}^{(d)} and $\alpha^{(d)}$TeX\alpha^{(d)} be extracted from the same reciprocal scan. The scan is admitted only if

\delta_{\mathrm{de}}
\equiv
\sup_{(k,\omega)\in B}
\max_{d,d'\in\mathfrak D}
\left(
\left|\chi_{\mathrm{ph}}^{(d)}-\chi_{\mathrm{ph}}^{(d')}\right|
+
\left|\alpha^{(d)}-\alpha^{(d')}\right|
\right)
\le \delta_{\mathrm{de},*},

for one family-fixed tolerance $\delta_{\mathrm{de},*}>0$TeX\delta_{\mathrm{de},*}>0. The reciprocal scan class together with the pair-normalized kernel reference, the midpoint identity reference, and the de-embedding stability gate reference is the declared reciprocal response map for the present paper. All closure claims below are read only on this declared response map.

Scan coordinate and reciprocal pairing

For one apparatus family, fix one monotone laboratory control or proxy family $u$TeXu on an admitted window $\mathcal W_{\varepsilon}$TeX\mathcal W_{\varepsilon} together with one reported map

\hat\varepsilon:u\mapsto \hat\varepsilon(u)

used to order the competition variable on that window. The control data are

\Ueps=(u,\hat\varepsilon,\mathcal W_{\varepsilon},\delta_{\mathrm{pair}}),

where $\delta_{\mathrm{pair}}>0$TeX\delta_{\mathrm{pair}}>0 is a family-fixed reciprocal pairing tolerance. When a laboratory proxy rather than a direct microscopic estimate is used, $\hat\varepsilon$TeX\hat\varepsilon is the only scan variable admitted into the reciprocal reporting convention.

definition: Reciprocal midpoint pair. A pair of runs $(u_{+},u_{-})$TeX(u_{+},u_{-}) is admitted as one reciprocal midpoint pair only if

\left|\hat\varepsilon(u_{+})+\hat\varepsilon(u_{-})\right|\le \delta_{\mathrm{pair}},
\qquad
\left|\left|\hat\varepsilon(u_{+})\right|-\left|\hat\varepsilon(u_{-})\right|\right|\le \delta_{\mathrm{pair}}.

If reference fails, the runs are not admitted into the reciprocal scan class. All downstream midpoint claims below are read only relative to this declared reciprocal midpoint-pair class on the admitted window.

Within one reciprocal scan family, the map $\epshat$TeX\epshat, the tolerance $\delta_{\mathrm{pair}}$TeX\delta_{\mathrm{pair}}, the response observable, the baseline subtraction, and the estimator are fixed before the paired runs are compared. Post hoc replacement of the control proxy, midpoint, or reciprocal partner is inadmissible.

Equation reference defines the underlying competition asymmetry. The load-bearing reciprocal law below is written in the reported scan coordinate $\epshat$TeX\epshat supplied by reference. When $\epshat$TeX\epshat is a laboratory proxy rather than a direct microscopic estimate, all midpoint fits, residual gates, and reported susceptibilities are understood with respect to $\epshat$TeX\epshat on the admitted window. If a direct estimate is available, the fixed convention may set $\epshat=\varepsilon$TeX\epshat=\varepsilon on that window.

proposition: Local midpoint expansion of the pair-normalized response kernel on a reciprocal scan class. This stage fixes an intermediate midpoint law on the declared reciprocal scan class and does not yet assert final closure. Assume, on that declared class and with respect to the fixed reported scan coordinate $\epshat$TeX\epshat on the admitted window, that $\Aresp(k,\omega;\epshat)$TeX\Aresp(k,\omega;\epshat) is locally analytic in $\epshat$TeX\epshat near $\epshat=0$TeX\epshat=0 for each $(k,\omega)\in B$TeX(k,\omega)\in B and that $\Aresp(k,\omega;0)\neq 0$TeX\Aresp(k,\omega;0)\neq 0. Then the pair-normalized kernel reference admits the expansion

\chi_{\mathrm{ph}}(k,\omega;\epshat)
=
\frac12-\alpha(k,\omega)\,\epshat+\mathcal O(\epshat^3),

with

\alpha(k,\omega)
\equiv
-\left.\frac{\partial \chi_{\mathrm{ph}}(k,\omega;\epshat)}{\partial \epshat}\right|_{\epshat=0}
=
-\frac12\left.\frac{\partial}{\partial \epshat}\ln \Aresp(k,\omega;\epshat)\right|_{\epshat=0}.

proof. Write

\Aresp(\epshat)=A_0+A_1\epshat+A_2\epshat^2+A_3\epshat^3+\cdots

with $A_0\neq 0$TeXA_0\neq 0. Then

\Aresp(\epshat)+\Aresp(-\epshat)=2A_0+2A_2\epshat^2+\mathcal O(\epshat^4),

so reference gives

\chi_{\mathrm{ph}}(\epshat)=\frac12+\frac{A_1}{2A_0}\epshat+\mathcal O(\epshat^3).

Defining $\alpha=-A_1/(2A_0)$TeX\alpha=-A_1/(2A_0) yields reference, and the logarithmic form in reference follows from $A_1/A_0=\partial_{\epshat}\ln\Aresp\vert_{\epshat=0}$TeXA_1/A_0=\partial_{\epshat}\ln\Aresp\vert_{\epshat=0}.

As a proposition-level statement, reference is a local midpoint expansion on the declared reciprocal scan class. Its use as an admitted leading law for the stated scan family remains conditional on the finite-window residual gate reference below and does not widen the result beyond that declared class.

Finite-window residual gate

For one midpoint window $|\epshat|\le \varepsilon_{\max}$TeX|\epshat|\le \varepsilon_{\max} and admitted band $B$TeXB, define the finite-window residual

r_{\mathrm{fit}}(k,\omega)
\equiv
\sup_{|\epshat|\le \varepsilon_{\max}}
\frac{
\left|
\chi_{\mathrm{ph}}(k,\omega;\epshat)-\left(\frac12-\alpha(k,\omega)\,\epshat\right)
\right|
}{|\epshat|^{3}+\varepsilon_{\mathrm{reg}}^{3}},

with one fixed regularizer $\varepsilon_{\mathrm{reg}}>0$TeX\varepsilon_{\mathrm{reg}}>0 used only to avoid a vanishing denominator at the midpoint. The leading law reference is admitted on the reciprocal class only if

\sup_{(k,\omega)\in B} r_{\mathrm{fit}}(k,\omega)\le \delta_{\mathrm{fit}}

for one family-fixed tolerance $\delta_{\mathrm{fit}}>0$TeX\delta_{\mathrm{fit}}>0. If reference fails, higher-order structure is resolved on the stated window and the linear midpoint law is not admitted there.

Equation reference is a regime organizer only. Equation reference is the reporting definition on the reciprocal scan. Equation reference is the admitted leading law only on reciprocal pairs satisfying reference and on finite windows satisfying reference, and reference defines the measured reciprocal boundary susceptibility. The physical content of the scan lies in $\alpha(k,\omega)$TeX\alpha(k,\omega) together with reciprocal pairing, de-embedding stability, and finite-window residual control rather than in the midpoint normalization itself.

Single-mode realization and purity gate

A restricted microscopic anchor is obtained by declaring one dominant low-energy phase mode $Q_{*}(t)$TeXQ_{*}(t) over the scan band and by keeping the apparatus family, de-embedding map, and reporting convention fixed throughout the reciprocal scan. This restricted single-mode anchor is a restricted subclass used only to exhibit one concrete realization of the admitted reciprocal law. At that level the mode obeys the causal linear-response equation

M_{*}\,\ddot Q_{*}(t)+\Gamma_{*}(\epshat)\,\dot Q_{*}(t)+K_{*}(\epshat)\,Q_{*}(t)=J_{*}(\epshat)\,D_{*}(t),

where $D_{*}(t)$TeXD_{*}(t) is the reported drive channel. In frequency space,

Q_{*}(\omega;\epshat)
=
\frac{J_{*}(\epshat)}
{K_{*}(\epshat)-M_{*}\omega^2+i\omega\Gamma_{*}(\epshat)}
\,D_{*}(\omega).

The corresponding response amplitude is

\Aresp^{(*)}(k,\omega;\epshat)
\equiv
\left|
\frac{Q_{*}(\omega;\epshat)}{D_{*}(\omega)}
\right|
=
\left|
\frac{J_{*}(\epshat)}
{K_{*}(\epshat)-M_{*}\omega^2+i\omega\Gamma_{*}(\epshat)}
\right|.

The general reciprocal law reference--reference applies directly to this subclass.

The single-mode anchor is admitted only when one admitted mode family dominates the same reciprocal scan band. Let $\Qadm$TeX\Qadm denote that mode family, and define the purity witness

\eta_{\mathrm{sm}}(k,\omega;\epshat)
\equiv
\frac{\left|\Aresp^{(*)}(k,\omega;\epshat)\right|^2}
{\sum_{q\in \Qadm}\left|\Aresp^{(q)}(k,\omega;\epshat)\right|^2}.

The quantity $\eta_{\mathrm{sm}}$TeX\eta_{\mathrm{sm}} is an analysis-level witness defined on the admitted mode decomposition $\Qadm$TeX\Qadm and on the fixed de-embedding class; it is not a stand-alone observable independent of that decomposition. The single-mode anchor is used only if

\inf_{(k,\omega)\in B,\;|\epshat|\le \varepsilon_{\max}}
\eta_{\mathrm{sm}}(k,\omega;\epshat)\ge 1-\delta_{\mathrm{sm}}

for one family-fixed tolerance $0<\delta_{\mathrm{sm}}\ll 1$TeX0<\delta_{\mathrm{sm}}\ll 1. If reference fails, the reciprocal scan remains a multimode response problem. The admitted mode family $\Qadm$TeX\Qadm together with the purity gate reference is the declared admissible residual carrier class for the restricted single-mode anchor, and all anchor-level claims below are read only relative to this declared class.

With midpoint expansions

K_{*}(\epshat)=K_{0}+K_{1}\epshat+\mathcal O(\epshat^2),\qquad
\Gamma_{*}(\epshat)=\Gamma_{0}+\Gamma_{1}\epshat+\mathcal O(\epshat^2),\qquad
J_{*}(\epshat)=J_{0}+J_{1}\epshat+\mathcal O(\epshat^2),

the reciprocal boundary susceptibility becomes

\alpha(\omega)
=
\frac{\Delta_{0}K_{1}+\Gamma_{0}\Gamma_{1}\omega^2}
{2\left[\Delta_{0}^{2}+(\Gamma_{0}\omega)^2\right]}
-\frac{J_{1}}{2J_{0}},
\qquad
\Delta_{0}\equiv K_{0}-M_{*}\omega^2.

Operational meaning and causal-embedding gate

The coefficient $\alpha(k,\omega)$TeX\alpha(k,\omega) is the midpoint logarithmic slope of the de-embedded response amplitude on the reciprocal scan. It is therefore a measured susceptibility of the scan itself rather than a free fit parameter.

The extracted kernel is admitted only if it belongs to one causal complex transfer function $H_{\mathrm{resp}}(k,\omega)$TeXH_{\mathrm{resp}}(k,\omega) on the scan band $B=[\omega_{\min},\omega_{\max}]$TeXB=[\omega_{\min},\omega_{\max}], with $\Aresp=\lvert H_{\mathrm{resp}}\rvert$TeX\Aresp=\lvert H_{\mathrm{resp}}\rvert. Fix one finite-band extrapolation and Hilbert-transform convention $\mathcal K_{B}$TeX\mathcal K_{B} on that band. Define the residual

\varepsilon_{\mathrm{KK}}(k)
\equiv
\frac{
\left\|
\Re H_{\mathrm{resp}}-\mathcal K_{B}[\Im H_{\mathrm{resp}}]
\right\|_{L^{2}(B)}
+
\left\|
\Im H_{\mathrm{resp}}+\mathcal K_{B}[\Re H_{\mathrm{resp}}]
\right\|_{L^{2}(B)}
}
{\left\|H_{\mathrm{resp}}\right\|_{L^{2}(B)}}.

The scan is causality-admissible only if

\sup_{k}\varepsilon_{\mathrm{KK}}(k)\le \delta_{\mathrm{KK}}

for one family-fixed tolerance $\delta_{\mathrm{KK}}>0$TeX\delta_{\mathrm{KK}}>0. The transfer family specified by $H_{\mathrm{resp}}$TeXH_{\mathrm{resp}}, the fixed finite-band convention $\mathcal K_{B}$TeX\mathcal K_{B}, and the gate reference is the declared reference transfer witness for the present paper, and all downstream transfer-admissibility statements are read only relative to this declared witness on the stated scan band. Equations reference--reference provide an existence-level admissibility requirement on the finite scan band under the stated extrapolation and phase-reconstruction convention. They do not assert uniqueness of the full phase retrieval or of the global continuation of the response. Finite-band causal macromodeling and stable transfer-function reconstruction provide a practical template for this gate [citation].

Phase-only action and rigidity scale

For a neutral superfluid or weakly interacting Bose gas, the low-energy phase field $\theta(x,t)$TeX\theta(x,t) is described by the effective action

S_{\mathrm{sf}}[\theta]
=
\int dt\,d^{d}x\,
\left[
\frac{\rho_{s}}{2c_{s}^{2}}(\partial_{t}\theta)^2
-
\frac{\rho_{s}}{2}(\nabla\theta)^2
\right],

where $\rho_{s}$TeX\rho_{s} is the superfluid stiffness and $c_{s}$TeXc_{s} the sound speed [citation]. The corresponding rigidity functional scales as

\mathcal R_{\mathrm{ph}}^{(\mathrm{sf})}(L)\sim \rho_{s}L^{d-2},

so that

\Xibind^{(\mathrm{sf})}(L)
\sim
\frac{\rho_{s}L^{d-2}}{\LXi^{4}/\Meff^{2}}.

For quasi-one-dimensional condensates such as ring traps, this reduces to the expected inverse-length scaling of phase rigidity.

Platform realization of the boundary regime

On a laboratory platform the large-scale drive is replaced by an engineered quantity $\Xilab(L,\Omega)$TeX\Xilab(L,\Omega) generated by trap modulation, interaction tuning, or synthetic-gauge control. The platform-level reciprocal embedding is

\chi_{\mathrm{ph}}^{(\mathrm{sf})}(k,\omega;\epshat)
\equiv
\frac{\Aresp^{(\mathrm{sf})}(k,\omega;\epshat)}
{\Aresp^{(\mathrm{sf})}(k,\omega;\epshat)+\Aresp^{(\mathrm{sf})}(k,\omega;-\epshat)},

with the reported scan variable assigned by one fixed monotone control/proxy map $\epshat(u)$TeX\epshat(u) that orders $\Xilab/\Xibind^{(\mathrm{sf})}$TeX\Xilab/\Xibind^{(\mathrm{sf})} on the admitted laboratory window. Equation reference only specifies how the reciprocal scan is reported on the platform; it does not add content beyond the general law reference.

Current Bose-gas experiments supply the relevant control and readout ingredients. Rapid expansion in a ring condensate realizes redshifting and reheating-like dynamics [citation]; two-dimensional condensates have been used as quantum field simulators in curved spacetime [citation]; curved light-cone propagation has been measured directly in coupled one-dimensional quantum gases [citation]; particle production has been mapped to a scattering problem in time-dependent effective spacetimes [citation]; entangled pair production has been quantified and then observed in cold-atom analogues of expanding-universe or preheating dynamics [citation]; and driven-superfluid experiments together with backreaction analyses delimit the onset of nonlinear corrections [citation]. These results make Bose--Einstein-condensate and superfluid platforms a natural analogue class for testing the boundary-response law. They motivate feasibility of the reciprocal boundary-response scan only; they do not by themselves verify the admitted boundary law, yield a cosmological estimate, or establish a matter-side closure.

The analogue task is to realize a family of runs in which the ratio $\Xilab/\Xibind$TeX\Xilab/\Xibind is moved through order unity while the apparatus family, the admitted mode family, the reported response observable, the de-embedding convention family, the control/proxy map $\epshat$TeX\epshat, the reciprocal pairing tolerance $\delta_{\mathrm{pair}}$TeX\delta_{\mathrm{pair}}, and the midpoint fit rule remain fixed. A minimal implementation is a condensate in a time-dependent ring or expanding trap, where the imposed drive modifies long-wavelength phase response and the rigidity scale is controlled independently by density, interaction strength, and geometry [citation].

On such platforms the response observable can be taken from phase quadratures, density response, or dynamic structure factors, provided that the same readout channel is used across the reciprocal pair. Bragg spectroscopy provides momentum- and frequency-resolved access to collective excitations in condensates, from the original determination of the condensate response to multibranch Bogoliubov spectra, pulse-shaped momentum-selective control, and recent wide-band excitation measurements in dipolar gases [citation]. In situ dynamic-structure-factor measurements provide a complementary route when a continuous response monitor is available, and nondestructive optomechanical protocols now extend access toward unequal-time density correlations on Bose--Einstein-condensate platforms [citation].

The reported object is the reciprocal scan pair admitted by reference,

\Aresp(k,\omega;\pm\epshat)
\equiv
\left.
\frac{\text{response amplitude}}{\text{drive amplitude}}
\right|_{\pm\epshat},

from which the pair-normalized kernel reference, the susceptibility reference, and the finite-window residual reference are extracted. The reciprocal scan specification is summarized in reference.

The proposal fails if any of the following occurs:

- the inferred response family admits no finite-band causal embedding satisfying reference on the scan band under the fixed extrapolation and phase-reconstruction convention; - the midpoint scan cannot be organized as a reciprocal class because the pairing gate reference fails on the stated window, or the de-embedding stability witness reference fails under the admitted convention family; - the restricted single-mode anchor is invoked but the purity gate reference fails on the scan band; or - the extracted boundary slope $\alpha(k,\omega)$TeX\alpha(k,\omega) is not stable under the fixed analysis conventions or the admitted leading law fails the finite-window residual gate reference on the reciprocal class.

remark: Local-dominance expectation. Phenomenologically admissible cosmological driving is expected to keep ordinary local structures in the rigidity-dominated regime. No quantitative cosmological estimate is made here. The result is restricted to the boundary-response law and to its analogue extraction.

On reciprocal scan families whose de-embedded response amplitude is locally analytic in the reported scan coordinate and nonzero at the midpoint, the pair-normalized kernel obeys the admitted finite-window midpoint law

\chi_{\mathrm{ph}}(k,\omega;\epshat)=\frac12-\alpha(k,\omega)\,\epshat+\mathcal O(\epshat^3),

on reciprocal pairs and midpoint windows that satisfy the stated admission gates. Here $\alpha(k,\omega)$TeX\alpha(k,\omega) is the midpoint logarithmic slope of the de-embedded response amplitude with respect to the reported scan coordinate. The physical content of the result lies in the extracted susceptibility $\alpha(k,\omega)$TeX\alpha(k,\omega) together with reciprocal pairing, de-embedding stability, finite-window residual control, single-mode purity when invoked, and finite-band causal embedding; the midpoint organizer and reciprocal normalization do not by themselves constitute a material law. A restricted single-mode realization and a worked Bogoliubov anchor exhibit one subclass of this law on Bose--Einstein-condensate and superfluid platforms.

The reported kernel must remain stable under fixed de-embedding variants, the reciprocal scan must satisfy the pairing gate, the admitted leading law must satisfy the finite-window residual gate on the stated midpoint window, the single-mode anchor must satisfy the purity gate whenever it is invoked, and the full response family must admit one finite-band causal embedding on the scan band under the fixed extrapolation and phase-reconstruction convention. Failure of any of these conditions excludes the proposed boundary response on that scan family. This stage remains intermediate and gate-conditioned: it does not redefine the analogue bridge criterion, and it does not yet assert final closure, noise-budget sealing, dissipation-side closure, cosmological estimation, or matter-side constitutive closure beyond the declared reciprocal response map and its stated gates.

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Consider a quasi-one-dimensional condensate with one phonon mode $q_{*}$TeXq_{*} in a time-dependent ring or effectively expanding trap. Linearization of the Gross--Pitaevskii/Bogoliubov dynamics about the operating point and projection onto that one mode yields

\ddot{\vartheta}_{q_{*}}(t)
+\Gamma_{q_{*}}(\epshat)\,\dot{\vartheta}_{q_{*}}(t)
+\omega_{q_{*}}^{2}(\epshat)\,\vartheta_{q_{*}}(t)
=
J_{q_{*}}(\epshat)\,D_{q_{*}}(t),

where $D_{q_{*}}(t)$TeXD_{q_{*}}(t) is the reported drive channel and $\vartheta_{q_{*}}$TeX\vartheta_{q_{*}} the reported phase-mode response. In frequency space,

\vartheta_{q_{*}}(\omega;\epshat)
=
\frac{J_{q_{*}}(\epshat)}
{\omega_{q_{*}}^{2}(\epshat)-\omega^{2}-i\omega\Gamma_{q_{*}}(\epshat)}
\,D_{q_{*}}(\omega).

The corresponding response amplitude is

\Aresp^{(q_{*})}(k,\omega;\epshat)
=
\left|
\frac{\vartheta_{q_{*}}(\omega;\epshat)}{D_{q_{*}}(\omega)}
\right|
=
\left|
\frac{J_{q_{*}}(\epshat)}
{\omega_{q_{*}}^{2}(\epshat)-\omega^{2}-i\omega\Gamma_{q_{*}}(\epshat)}
\right|.

The pair-normalized kernel follows from reference with $\Aresp\to \Aresp^{(q_{*})}$TeX\Aresp\to \Aresp^{(q_{*})}.

Expanding about the midpoint,

\omega_{q_{*}}^{2}(\epshat)=\omega_{0}^{2}+\omega_{1}^{2}\epshat+\mathcal O(\epshat^{2}),\qquad
\Gamma_{q_{*}}(\epshat)=\Gamma_{0}+\Gamma_{1}\epshat+\mathcal O(\epshat^{2}),\qquad
J_{q_{*}}(\epshat)=J_{0}+J_{1}\epshat+\mathcal O(\epshat^{2}),

one obtains

\chi_{\mathrm{ph}}^{(q_{*})}(k,\omega;\epshat)
=
\frac12-\alpha_{q_{*}}(\omega)\,\epshat+\mathcal O(\epshat^{3}),

with

\alpha_{q_{*}}(\omega)
=
\frac{\Delta_{0}\omega_{1}^{2}+\Gamma_{0}\Gamma_{1}\omega^{2}}
{2\left[\Delta_{0}^{2}+(\Gamma_{0}\omega)^2\right]}
-\frac{J_{1}}{2J_{0}},
\qquad
\Delta_{0}\equiv \omega_{0}^{2}-\omega^{2}.

Equations reference and reference provide an explicit Bogoliubov realization of the admitted reciprocal law.

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Funding and competing interests..

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