Dynamic Boundary Insertion, Detector Memory, and Pre-Latch Trigger Sinks in Wave--Detector Systems

Detector-side dynamics are now directly resolvable in several platform classes. Spatially resolved detector tomography reconstructs local response maps in single-atom detectors [citation], time-resolved single-shot parity measurements are implemented in interferometric semiconductor devices [citation], single-electron interferometers resolve time-dependent fields through phase and contrast on picosecond scales [citation], and superconducting thermal or microwave detectors make readout duration, thresholding, and internal loss channels experimentally visible [citation]. On the formal side, absorbing-boundary and detector-surface laws treat the time and place of detection as outputs of a boundary rule rather than as externally appended statistics [citation], while process-tensor tomography, hidden-model reconstruction from multiple time measurements, Lindblad-like tomography of non-Markovian dynamical maps, and multi-time tomography resolve detector-relevant memory directly on controlled hardware [citation]. Related detector-side distinctions between arrival times and detection times are analyzed in recent absorbing-boundary work [citation].

On activated detector-family windows with visible switching, scanning, or finite detector bandwidth, admissible pre-latch closure must preserve bulk propagation away from the detector tube, represent reversible near-boundary reorganization before irreversible counting, store finite short-window memory in the detector tube, produce a nonnegative trigger current, reduce correctly to instantaneous local opening when detector memory is removed, and remain causally compatible with any later durable-record stage. Within the declared first-order local pre-latch class, these constraints select the minimal admissible detector-tube closure: a real insertion potential, one detector-memory field, and a positive trigger sink on the declared pre-latch window. On a certified one-pole detector-family window, the same three-sector law admits a compressed response reading in which the conservative insertion and positive trigger sectors are the dispersive and absorptive faces of one retarded detector-boundary susceptibility, while the detector-memory coordinate is its dominant local mode [citation]. That reduced reading is admitted only after residual control against the reconstructed detector-tube response on the declared operating band. Recent detector-characterization studies and reporting standards now quantify mid-circuit measurement error rates, precision bounds for detector estimation, and family-fixed benchmarking requirements directly on hardware or declared protocol windows [citation]. Fixed-surface absorbing-boundary laws and static detector tomography enter only through the recovery limits of reference, and any later durable-record description may depend on the present layer only through $\Jtr$TeX\Jtr or an admitted causal coarse-graining of it. No latching, reset, or objectivity claim is made here.

Let $\Gam(t)$TeX\Gam(t) be the active detector interface. The detector enters the wave law through a real tube-localized insertion potential $\Ug$TeX\Ug, a detector-memory field $\Mmem$TeX\Mmem, and a positive pre-latch trigger sink $\Kg$TeX\Kg. The real insertion term reorganizes near-boundary propagation without by itself generating counts. The memory field stores the short-lived persistence of that detector-conditioned state. The sink converts a controlled fraction of the boundary-conditioned wave structure into a nonnegative trigger current. Within the declared detector tube window, the three sectors are selected at baseline order by the simultaneous requirements of reversible reorganization, finite lag, trigger production, instantaneous-opening recovery, and later durable-record admissibility. On certified one-pole windows, the memory field is the dominant local coordinate of the detector-tube response.

The source of the detector-memory field is fixed before fitting observables. Under locality on the detector tube, phase-offset invariance, first-order truncation, and isotropic scalar reduction, the pullback of the propagation density-current pair to the moving detector tube yields a four-channel baseline source basis: incoming normal flux, gate-density, insertion-rate density, and scan-gradient coupling. This pullback closes the declared lowest-order source baseline within the declared isotropic first-order scalar class.

Three recovery limits are fixed. The static-memoryless limit yields an instantaneous local-opening law. The thin-layer limit joins the absorbing-boundary detector class. The weak-coupling null limit returns to the unperturbed propagation sector. Within the declared one-pole window, the same $\tau_M$TeX\tau_M controls feedthrough-subtracted count--drive hysteresis, short-window excess build-up, threshold lag beyond the static reference, scan-speed excess visibility, and post-switch recovery. The detector-side output of the wave/tube layer is the pre-latch trigger current; durable amplification, persistent records, entropy export, and reset require additional nonequilibrium variables.

On the declared pre-latch window, the baseline detector-tube closure is the minimal class-bound triad $(\Ug,\Mmem,\Kg)$TeX(\Ug,\Mmem,\Kg) of reference. The one-pole susceptibility form and the reduced coordinate $m$TeXm introduced in reference are secondary certified reductions admitted only on detector-family windows with fixed direct-feedthrough subtraction and residual control. Process-tensor, reaction-coordinate, pseudomode, and durable-record constructions enter only as characterization, reduced-window comparison, or admissible trigger-to-record coarse-graining statements; they do not supply the baseline law, a latching theory, or a reset model.

definition: Dynamic boundary insertion. Let $\Gam(t)$TeX\Gam(t) denote the active detector interface and let $d_{\Gam}(\mathbf{x},t)$TeXd_{\Gam}(\mathbf{x},t) be its signed distance function. On the declared pre-latch detector-tube window, the detector is treated as a dynamic boundary insertion when its physical role in the theory is to alter the admissible near-surface organization of an already propagating field inside a thin boundary tube

\Tube(t)=\{\mathbf{x}: |d_{\Gam}(\mathbf{x},t)|<\epsilon\},

rather than to be represented only as a final classical readout map. The interface may move geometrically, activate through a gate profile, or do both.

definition: Detector-memory field. A field $\Mmem(\mathbf{x},t)$TeX\Mmem(\mathbf{x},t) supported on the detector tube is called a detector-memory field if it measures how much of a previously boundary-adapted detector-response state persists locally after the instantaneous boundary forcing has changed. It is not a second wavefunction and not a hidden classical record. On the declared one-pole working window it is the minimal local state-space coordinate of the detector-boundary susceptibility.

definition: Local commit. A local commit on the declared pre-latch measurement window is the event-local transfer of boundary-accessible propagation structure into a declared detector-side outcome channel. A local commit is therefore stronger than a reversible near-boundary distortion and weaker than a durable macroscopic record. In the baseline law it is represented by the nonnegative trigger current of reference.

definition: Trigger current. The detector-side trigger current on the declared pre-latch detector-tube window is

\Jtr(t):=\int \Kg(\mathbf{x},t)|\psib(\mathbf{x},t)|^2\,d^dx.

It is the nonnegative current of event-local commit. It is weaker than a durable-record current because it does not by itself encode latching, amplification, or reset.

definition: Detector-boundary susceptibility. Let $\PiGam(t)$TeX\PiGam(t) denote multiplication by the localization kernel $\epsloc(d_{\Gam}(\mathbf{x},t))$TeX\epsloc(d_{\Gam}(\mathbf{x},t)). A detector-boundary susceptibility on the declared pre-latch detector-tube window is a retarded kernel $\SigR(t,t')$TeX\SigR(t,t') supported on the detector tube such that, on the declared pre-latch window, the detector-corrected propagation law can be written as

i\hbar\partial_t\psib
=
\hat H_0\psib+
\int_{-\infty}^{t}dt'\,\SigR(t,t')\,\PiGam(t')\psib(t').

Its dispersive part reorganizes near-boundary propagation, its dominant pole fixes detector memory, and its absorptive part fixes the trigger loss on the active detector band.

These declared objects are the fixed primitive layer for the detector-boundary balance and reduction statements below; all such statements are read only on the declared pre-latch detector-tube window and within the declared first-order local class fixed here.

remark: Four-term detector response. The closure below is written in the response form

\partial_t\Phi=\mathcal F_{\mathrm{bulk}}[\Phi]+\mathcal B_{\Gam}[\Phi]+\mathcal M[\Phi]-\mathcal C[\Phi;\Gam].

Here the bulk term is the imported propagation law, the boundary term is the conservative insertion potential, the memory term is the detector-memory equation, and the commit term is the positive trigger sink.

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

Once activated, the detector inserts a localized boundary condition into the wave field. That insertion reorganizes the nearby field, the detector retains a finite memory of the reorganized state, and only then does part of the resulting structure convert into a trigger current of local commit. A common gate schedule may synchronize detector activation without thereby fixing identical pre-latch detector history, because $\Mmem$TeX\Mmem and $\Jtr$TeX\Jtr depend on the causal boundary trajectory on the detector tube.

Let $\psib(\mathbf{x},t)$TeX\psib(\mathbf{x},t) be a declared propagation descriptor satisfying away from the detector tube the baseline law

i\hbar\partial_t\psib=\hat H_0\psib,

where $\hat H_0$TeX\hat H_0 is self-adjoint on the declared propagation sector. For Schr\"odinger-type benchmarks one has

\rhob=|\psib|^2,
\qquad
\jb=\frac{\hbar}{m}\operatorname{Im}(\psib^*\nabla\psib),

with continuity equation $\partial_t\rhob+\nabla\!\cdot\!\jb=0$TeX\partial_t\rhob+\nabla\!\cdot\!\jb=0. In other propagation sectors, $\rhob$TeX\rhob and $\jb$TeX\jb are replaced by the corresponding declared density-current pair.

Let $\epsloc(d)$TeX\epsloc(d) be a smooth nonnegative localization kernel supported in $|d|<\epsilon$TeX|d|<\epsilon. The baseline detector-boundary closure is

i\hbar\partial_t\psib
 =
 \hat H_0\psib+\Ug\psib-\frac{i\hbar}{2}\Kg\psib,
 
 

 \Ug(\mathbf{x},t)
 =
 \epsloc(d_{\Gam})\,\bigl(u_g g_{\Gam}(t)+u_M\Mmem(\mathbf{x},t)\bigr),
 
 

 \partial_t\Mmem+\vb\!\cdot\!\nabla \Mmem
 =
 -\tau_M^{-1}\Mmem+\Sg[\psib,\Gam,g_{\Gam}],
 
 

 \Kg(\mathbf{x},t)
 =
 \epsloc(d_{\Gam})\,\gamma_{\Gam}\,G(\Mmem)
 \,S\!\left(\Lam-\Lambda_*\right),
 
 

 \Lam
 =
 a_J\frac{\Jnp}{\rhob+\rho_*}+a_g g_{\Gam}(t)+a_M\Mmem.

Here $\rho_*>0$TeX\rho_*>0 is a family-fixed low-density regularizer used only to keep the normalized incoming-flux gate finite on the declared weak-signal working window, and $\Lambda_*$TeX\Lambda_* is the family-fixed trigger threshold on that same declared window. The constant $\gamma_{\Gam}\ge 0$TeX\gamma_{\Gam}\ge 0 is a family-fixed coupling rate, $G(\Mmem)\ge 0$TeXG(\Mmem)\ge 0 a nonnegative response factor, and $S(z)$TeXS(z) a smooth monotone gate with

S'(z)\ge 0,
\qquad
\lim_{z\to-\infty}S(z)=0,
\qquad
\lim_{z\to+\infty}S(z)=1.

The incoming normal flux is defined by

\Jnp=\max\!\bigl(\jb\cdot\nb,0\bigr),
\qquad
\nb=\nabla d_{\Gam}/|\nabla d_{\Gam}|,

and the tangential projector by $\Ppar=\mathbf I-\nb\otimes\nb$TeX\Ppar=\mathbf I-\nb\otimes\nb, so that $\gradpar=\Ppar\nabla$TeX\gradpar=\Ppar\nabla.

remark: Interpretation of the added terms. The potential $\Ug$TeX\Ug is a conservative boundary-reorganization term. By itself it changes phases and near-boundary mode organization but does not produce counts. The field $\Mmem$TeX\Mmem is a detector-memory layer that stores the short-lived persistence of the boundary-adapted state. The operator $\Kg$TeX\Kg is a positive local-commit sink: it drains norm from the propagation descriptor and converts that drain into event counts. None of these terms is interpreted as a complete detector micro-Hamiltonian. On a certified one-pole causal window they may be read as the dispersive, memory, and absorptive faces of one causal detector-boundary response.

proposition: Norm--trigger balance. Within the declared detector-boundary closure class, assume $\hat H_0$TeX\hat H_0 is self-adjoint on the declared propagation sector, $\Ug$TeX\Ug is real-valued, and $\Kg\ge0$TeX\Kg\ge0 pointwise. Then the propagation norm obeys

\frac{d}{dt}\|\psib(t)\|_2^2
=
-\int \Kg(\mathbf{x},t)\,|\psib(\mathbf{x},t)|^2\,d^dx
\equiv -\Jtr(t)\le 0.

Consequently,

\Ntr(t)=\int_0^t \Jtr(s)\,ds
=\|\psib(0)\|_2^2-\|\psib(t)\|_2^2

measures the accumulated event-local commits in the declared detector channel.

proof. Differentiate $\|\psib\|_2^2=\langle\psib,\psib\rangle$TeX\|\psib\|_2^2=\langle\psib,\psib\rangle and use reference. The contribution of $\hat H_0+\Ug$TeX\hat H_0+\Ug is purely imaginary because $\hat H_0$TeX\hat H_0 is self-adjoint and $\Ug$TeX\Ug is real-valued. The non-Hermitian term contributes $-\langle\psib,\Kg\psib\rangle$TeX-\langle\psib,\Kg\psib\rangle, which is nonpositive because $\Kg\ge0$TeX\Kg\ge0.

theorem: Minimal pre-latch closure decomposition in the declared local class. Consider detector laws on a declared detector-tube window that (i) coincide with the imported bulk propagation law outside $\Tube$TeX\Tube, (ii) admit reversible near-boundary reorganization, (iii) generate a nonnegative trigger current $\Jtr$TeX\Jtr, (iv) display finite lag or oriented count--drive hysteresis with one dominant response time, (v) reduce in the static-memoryless limit to an instantaneous local-opening law, and (vi) admit a later durable-record extension whose state variables are insensitive to reversible post-switch recovery. Then, within that declared class, any admissible lowest-order local closure is represented, up to coefficient renaming and higher-order gradient, tensor, or nonlocal corrections, by the triad $(\Ug,\Mmem,\Kg)$TeX(\Ug,\Mmem,\Kg): a real insertion sector, one detector-memory coordinate, and a positive pre-latch trigger sink.

proof. The exclusions are separated. Appendix B records the corresponding baseline arguments. A purely absorptive localized correction supplies loss but no independent conservative tube-local reorganization sector. A state-free instantaneous reduced law has zero oriented loop area, in agreement with reference. Without a positive sink, reference collapses to norm conservation and no admissible nonnegative trigger input is available for any later record layer. If durable-record variables were inserted directly into the wave-side law, the post-switch separation between reversible boundary recovery and monotone trigger accumulation would fail. The only surviving minimal pre-latch sectors are therefore a real insertion term, one detector-memory coordinate, and a positive trigger sink, up to higher-order, tensor, or nonlocal corrections.

remark: Class-bound minimality. reference is a minimality statement for the declared detector-tube local class. It does not claim microscopic uniqueness across all detector architectures or across explicitly enlarged detector-side state spaces.

proposition: Certified one-pole causal reading on the declared window. Assume the activated detector tube couples locally to detector-side coordinates and residual detector degrees of freedom that begin in a declared ready state, and linearize the pre-latch dynamics on the declared working window. For any declared residual detector sector whose linearized elimination yields a retarded self-energy on the certified pre-latch working window, the three-sector closure reference--reference admits the reduced retarded reading reference on that window. On the local frequency-domain window one may write

\SigR(\omega)
=
\Ug(\omega)-\frac{i\hbar}{2}\Kg(\omega),

with $\Ug$TeX\Ug the dispersive boundary-response part and $\Kg\ge0$TeX\Kg\ge0 the absorptive boundary-response part on the active pre-latch band. Consequently $\Ug$TeX\Ug and $\Kg$TeX\Kg need not be treated as independently tuned detector insertions on that window; they are two projections of one causal detector-boundary susceptibility on that window.

proof. Tracing out a linearly coupled detector remainder on the certified pre-latch window produces a retarded self-energy on the localized boundary sector. Retardation gives reference, while the Hermitian and anti-Hermitian parts of the reduced self-energy yield the decomposition reference. Positivity of the pre-latch trigger channel fixes the absorptive contribution to be nonnegative on the active band. The proposition therefore states an admissible reduced reading on the certified pre-latch window, not a unique microscopic detector Hamiltonian.

remark: Dispersion--absorption consistency. Because $\SigR$TeX\SigR is causal, the frequency dependences of $\Ug$TeX\Ug and $\Kg$TeX\Kg are linked on any declared smooth operating window by Hilbert-transform consistency. Detector families that require separately retuned phase-shift and trigger-loss kernels on the same certified one-pole window already lie outside the baseline closure.

Specificity also lies in restricting which local boundary data may source the detector-memory field. The source may not be an arbitrary regression functional of everything measurable near the detector. Within the declared isotropic first-order scalar detector class, the admissible source is fixed by locality, phase-offset invariance, first-order kinematics, and boundary geometry.

proposition: Declared isotropic first-order scalar source basis. Let $(\rhob,\jb)$TeX(\rhob,\jb) be the density-current pair exported by the declared propagation law, and let $(g_{\Gam},d_{\Gam},\vb)$TeX(g_{\Gam},d_{\Gam},\vb) specify the moving detector tube. Assume that the detector-memory source is

- localized to the detector tube, - invariant under global phase offsets $\psib\mapsto e^{i\phi_0}\psib$TeX\psib\mapsto e^{i\phi_0}\psib, - built only from first-order pullbacks of propagation and boundary data in the declared coarse-graining window, and - reduced in the isotropic baseline to scalar couplings.

Then every admissible first-order scalar source on that declared baseline is a linear combination of

\Sg
=
\epsloc(d_{\Gam})\left(
\chi_J\Jnp
+\chi_g g_{\Gam}\rhob
+\chi_{\dot g}\tau_M\dot g_{\Gam}\rhob
+\chi_v\ell_{\Gam}\,\vbpar\!\cdot\!\gradpar\rhob
\right),

and any additional source channel belongs to an explicitly enlarged class outside the declared isotropic first-order scalar baseline. Higher derivatives, unrestricted nonlocal kernels, polarization tensors, and object-wise spectral filters lie outside the declared baseline law.

proof. Locality and tube support require a factor localized by $\epsloc(d_{\Gam})$TeX\epsloc(d_{\Gam}). Phase-offset invariance excludes the absolute phase itself, so the detector tube can depend only on $\rhob$TeX\rhob, currents, boundary motion, and gate variables exported by the propagation law. Appendix B classifies the first-order isotropic pullback scalars: modulo the continuity equation and higher-order curvature, tensor, or nonlocal terms, the moving tube yields four baseline scalar channels and no further independent scalar channel within the declared baseline. Within the declared baseline, the boundary-normal channel is represented by the incoming normal flux $\Jnp$TeX\Jnp. A static activated detector contributes the gate-density scalar $g_{\Gam}\rhob$TeXg_{\Gam}\rhob. Temporal insertion contributes $\dot g_{\Gam}\rhob$TeX\dot g_{\Gam}\rhob. Tangential motion contributes only through contraction with a tangential gradient, giving $\vbpar\cdot\gradpar\rhob$TeX\vbpar\cdot\gradpar\rhob. Any additional source channel would lie outside the declared isotropic first-order scalar baseline.

remark: Baseline completeness of the source class. The source proposition is complete only relative to the declared isotropic first-order scalar pullback baseline. Anisotropy, polarization tensors, additional detector-side coordinates, and unrestricted nonlocal kernels are not excluded in general; they belong to explicitly enlarged classes.

remark: Roles of the four channels. The term $\chi_J\Jnp$TeX\chi_J\Jnp is the direct influx channel. The term $\chi_g g_{\Gam}\rhob$TeX\chi_g g_{\Gam}\rhob measures detector activation in the presence of local wave density. The term $\chi_{\dot g}\tau_M\dot g_{\Gam}\rhob$TeX\chi_{\dot g}\tau_M\dot g_{\Gam}\rhob is the insertion-rate channel: rapid switching modifies the detector-side state even at fixed instantaneous gate amplitude. The term $\chi_v\ell_{\Gam}\,\vbpar\cdot\gradpar\rhob$TeX\chi_v\ell_{\Gam}\,\vbpar\cdot\gradpar\rhob is the scan-gradient channel: a tangentially moving detector builds memory according to the signed local structure it traverses. These four channels close only the declared isotropic first-order scalar baseline seen by the moving detector tube; anisotropy, polarization tensors, additional detector-side coordinates, and unrestricted nonlocal kernels belong to explicitly enlarged classes.

proposition: Lowest-order conservative insertion sector. Assume the detector-side reorganization term is (i) localized to the detector tube, (ii) conservative, (iii) invariant under global phase offsets, (iv) scalar in the isotropic baseline, and (v) at most first order in instantaneous activation and detector conditioning. Then the lowest-order admissible conservative sector on that baseline is represented by reference, up to coefficient renaming and higher-order gradient, tensor, or nonlocal corrections. In particular, reversible near-boundary reorganization must enter before trigger formation through a real insertion term.

proof. Locality enforces the factor $\epsloc(d_{\Gam})$TeX\epsloc(d_{\Gam}). Conservativity and phase-offset invariance require a real scalar multiplying $\psib$TeX\psib. At first order the only isotropic detector-side scalars are the gate amplitude and one detector-conditioning coordinate. Their linear combination yields reference. Any additional gradient, tensor, or history dependence is higher order or belongs to the memory source rather than to the conservative insertion sector.

proposition: Baseline admissible pre-latch trigger sink. Assume the trigger sink is (i) localized to the detector tube, (ii) nonnegative, (iii) null in the weak-coupling limit, (iv) monotone in an instantaneous trigger variable, (v) causal with respect to the local detector state, and (vi) first order in the isotropic baseline. Then the lowest-order admissible sink is

\Kg(\mathbf{x},t)=
\epsloc(d_{\Gam})\,\gamma_{\Gam}\,G(\Mmem)
\,S\!\left(\Lam-\Lambda_*\right),

with $\gamma_{\Gam}\ge0$TeX\gamma_{\Gam}\ge0, $G(\Mmem)\ge0$TeXG(\Mmem)\ge0, and $S$TeXS obeying reference. Moreover the insertion-rate and scan-asymmetry channels may enter the sink only through $\Mmem$TeX\Mmem: direct dependence on $\dot g_{\Gam}$TeX\dot g_{\Gam} or $\vbpar\!\cdot\!\gradpar\rho_{\psi}$TeX\vbpar\!\cdot\!\gradpar\rho_{\psi} in $\Lam$TeX\Lam introduces an extra detector-side state and violates the one-memory closure.

proof. Locality yields the tube factor $\epsloc(d_{\Gam})$TeX\epsloc(d_{\Gam}). Nonnegativity and the null-coupling limit require an overall nonnegative prefactor $\gamma_{\Gam}G(\Mmem)$TeX\gamma_{\Gam}G(\Mmem) that vanishes when detector coupling is turned off. Monotone threshold response requires a gate functional depending on a single instantaneous trigger variable $\Lam$TeX\Lam through a monotone switch $S$TeXS. Causality restricts detector-side history dependence to the present local state, which within the one-memory baseline is represented solely by $\Mmem$TeX\Mmem. The dynamic channels $\dot g_{\Gam}$TeX\dot g_{\Gam} and $\vbpar\!\cdot\!\gradpar\rho_{\psi}$TeX\vbpar\!\cdot\!\gradpar\rho_{\psi} already enter the theory through reference and reference. If they are inserted directly into $\Lam$TeX\Lam, then after linearization on a certified one-pole window they contribute frequency-odd absorptive terms proportional to $i\omega g_{\Gam}(\omega)$TeXi\omega g_{\Gam}(\omega) or to the Fourier image of the scan-gradient channel. Those terms cannot be represented on a finite operating band by a family-fixed scalar multiple of the one-pole coordinate $m(\omega)=\chi_{\Gamma}u(\omega)/(1-i\omega\tau_M)$TeXm(\omega)=\chi_{\Gamma}u(\omega)/(1-i\omega\tau_M) without either frequency-dependent retuning or an additional pole. Direct dynamic sink channels therefore lie outside the one-memory certified baseline. Thus the lowest-order admissible trigger sink is reference, equivalent to reference.

Within a declared detector family, the coefficients $u_g,u_M,\gamma_{\Gam},a_J,a_g,a_M,\chi_J,\chi_g,\chi_{\dot g},\chi_v,\ell_{\Gam},\tau_M$TeXu_g,u_M,\gamma_{\Gam},a_J,a_g,a_M,\chi_J,\chi_g,\chi_{\dot g},\chi_v,\ell_{\Gam},\tau_M must be fixed once the family and operating window are declared. They may not be re-fitted independently for each observable. A new detector family may carry a different parameter set, but one family may not hide object-wise retuning inside the source or sink.

Neighboring detector formalisms fix recovery and admissibility only; they do not alter the three-sector pre-latch detector-tube closure.

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

On quasistatic fixed-surface windows, reference and reference contract the closure to static local-opening or absorbing-boundary classes. When switching, scanning, or finite detector bandwidth is resolved on the same declared detector-family window, the triad $(\Ug,\Mmem,\Kg)$TeX(\Ug,\Mmem,\Kg) is the minimal admissible pre-latch closure. The recovery statements below are compatibility limits. They do not append detector-side state variables to fixed-surface absorbing-boundary laws or to static detector maps. Recent boundary-quadruple results make the fixed-surface contraction-semigroup side explicit [citation]. Within that detector-boundary closure, the baseline pre-latch memory variable is the field $\Mmem$TeX\Mmem; the coordinate $m$TeXm appears only after the certified one-pole compression of reference.

Three recovery limits are especially important.

corollary: Static-memoryless local-opening reduction. In the static-memoryless limit of the declared detector-boundary closure, assume $\vb=0$TeX\vb=0, $\dot g_{\Gam}=0$TeX\dot g_{\Gam}=0, $\tau_M\to0$TeX\tau_M\to0, and $u_M\Mmem=o(1)$TeXu_M\Mmem=o(1) in the declared working window. Then $\Mmem=\tau_M\Sg+o(\tau_M)$TeX\Mmem=\tau_M\Sg+o(\tau_M) and the closure reduces to the instantaneous opening law

\Kg(\mathbf{x},t)=
\epsloc(d_{\Gam})\,\gamma_{\Gam}\,G(0)\,
S\!\left(
 a_J\frac{\Jnp}{\rhob+\rho_*}+a_g g_{\Gam}-\Lambda_*
\right)+o(1).

No oriented count--drive loop area, no ramp-lag shift, and no scan-memory length survive this limit.

corollary: Static open-gate limit. If the detector boundary is fixed, $\dot g_{\Gam}=0$TeX\dot g_{\Gam}=0, $\vb=0$TeX\vb=0, the memory field is negligible, $\Mmem\approx0$TeX\Mmem\approx0, and the gate is homogeneously open on the active detector area, then

\Kg(\mathbf{x},t)=\gamma_0\epsloc(d_{\Gam})+o(1),
\qquad
\Jtr(t)\propto\int_{\Tube(t)}|\psib(\mathbf{x},t)|^2\,d^dx.

Thus the baseline law recovers an operational local Born-type rate law in the static homogeneous limit.

remark: Relation to detector tomography. The static limit reference is exactly where detector tomography is most powerful. Spatially resolved detector tomography now determines nonuniform local efficiency maps in single-atom detectors [citation], and recent self-characterization and semidefinite measurement-tomography schemes constrain how detector maps may be reconstructed from count data [citation]. Once detector activation, switching, or scan history enters the same working window, a dynamic boundary term and a finite detector-memory layer are required before the count law closes.

proposition: Zero-memory immediate-trigger absorbing-boundary recovery. Assume a fixed detector surface, a negligible memory field, a vanishing memory time $\tau_M\to0$TeX\tau_M\to0, and a thin-layer limit in which

\Kg(\mathbf{x},t)\to 2\kappa(\mathbf{s})\,\delta_{\Gam}(\mathbf{x}),
\qquad
\Ug\to0,

with $\delta_{\Gam}$TeX\delta_{\Gam} the surface delta distribution and with the local trigger gate effectively open on the active boundary band. Then reference reduces to the usual complex-absorbing-potential class, and boundary-layer elimination yields the absorbing-boundary rule class discussed in [citation]. Thus absorbing-boundary detection is the zero-memory immediate-trigger limit of the dynamic detector-boundary closure.

remark: Dynamic extension of absorbing boundaries. Absorbing boundaries and imaginary potentials already capture detection-time laws at idealized fixed surfaces [citation]. By themselves, they do not encode finite detector memory under switching, scanning, or bandwidth-limited activation. The baseline law adds that pre-latch layer while preserving the absorbing-boundary limit of reference.

corollary: Weak-coupling null limit. If $u_g,u_M,\gamma_{\Gam},\chi_J,\chi_g,\chi_{\dot g},\chi_v\to0$TeXu_g,u_M,\gamma_{\Gam},\chi_J,\chi_g,\chi_{\dot g},\chi_v\to0 with fixed $\hat H_0$TeX\hat H_0, fixed detector geometry, and fixed initial data, then $\Ug,\Kg,\Sg\to0$TeX\Ug,\Kg,\Sg\to0 in the detector tube and the solution of reference converges to the solution of the unperturbed bulk law reference. No detector-induced asymptotic channel survives this limit.

remark: Scattering consistency. The weak-coupling null limit is not optional. A detector-boundary law must disappear smoothly when detector couplings are turned off; otherwise the detector contaminates the asymptotic propagation sector. The same requirement is consistent with absorbing-boundary recovery at fixed surfaces and with boundary-quadruple parameterizations of fixed-surface contraction semigroups [citation].

Boundary sensitivity can also be more than a mathematical convenience. Recent work on Unruh--deWitt detectors shows that detector response may depend on boundary conditions at an interface even when the detector is classically isolated from the region where those conditions are imposed [citation]. The nonrelativistic or effectively one-mode setting used here has the same structural feature: boundary data can modify detector response already at the detector layer.

The full tube equations reference--reference are the baseline theory. For practical identification the theory must also yield a reduced law with directly testable cross-observable consequences. The reduced coordinate introduced below is admitted only on windows certified by reference; it is not part of the baseline identity of reference. On a certified one-pole window, write the linearized trigger current relative to a static or quasistatic reference as

\delta\Jtr(t)=\alpha_{\Gamma}u(t)+\Jexc(t),

where $\alpha_{\Gamma}$TeX\alpha_{\Gamma} is the family-fixed direct feedthrough fixed by that reference on the same family/window and $\Jexc$TeX\Jexc is the memory-conditioned excess trigger current. The memory coordinate cannot be omitted, because the loop criterion below would otherwise force zero oriented area in the feedthrough-subtracted channel.

proposition: No-memory no-loop criterion. Let a reduced detector law in a declared detector family be instantaneous in the sense that $\Jexc(t)=F(u(t))$TeX\Jexc(t)=F(u(t)) with no internal detector-side state variable. Then for any periodic drive $u(t)$TeXu(t) of period $T$TeXT, the oriented loop area satisfies

A_{R\text{--}u}=\oint \Jexc\,du = 0.

Consequently, any experimentally resolved nonzero count--drive loop area in the feedthrough-subtracted channel requires at least one detector-side state variable.

proof. If $\Jexc$TeX\Jexc is a single-valued function of $u$TeXu, then the parametric curve $(u,\Jexc)$TeX(u,\Jexc) is traversed on the graph of $F$TeXF, and the line integral around one closed period is exactly $\oint F(u)\,du=0$TeX\oint F(u)\,du=0.

Assume therefore a declared detector family and operating window in which the detector-tube response is dominated by one pole.

proposition: One-pole detector-tube compression. Assume the linearized pre-latch detector response on the declared working window is causal and time-translation invariant to leading order, and that its detector-tube susceptibility has one dominant simple pole separated from the rest of the spectrum by a gap larger than the working frequencies. Let the raw trigger current be decomposed as in reference, with the direct feedthrough coefficient $\alpha_{\Gamma}$TeX\alpha_{\Gamma} fixed from the same family/window in a static or quasistatic reference. Then every minimal local state-space realization on that certified window is equivalent, up to coefficient renaming, to one real coordinate $m(t)$TeXm(t) satisfying

\tau_M\dot m + m = \chi_{\Gamma} u(t),

\Jexc(t) = \beta_{\Gamma} m(t),

where $u(t)$TeXu(t) is the reduced boundary drive assembled from the admissible channels of reference. The coefficient $\chi_{\Gamma}$TeX\chi_{\Gamma} is family-fixed, $\beta_{\Gamma}$TeX\beta_{\Gamma} is the linear excess-trigger sensitivity at the chosen working point, and $\alpha_{\Gamma}$TeX\alpha_{\Gamma} is not refit after certification on that family/window.

proof. A causal time-translation-invariant linear response law is determined by its susceptibility. If the susceptibility has one dominant simple pole and no comparable secondary pole on the declared window, the minimal local state-space realization is one first-order real coordinate. The direct feedthrough allowed by the static local-opening sector is fixed separately by the decomposition reference. The remaining memory-conditioned dynamic channel therefore obeys reference, and the linearized excess trigger response takes the form reference.

remark: One-pole susceptibility representation. Equation reference is equivalent to the causal convolution

m(t)=\int_{-\infty}^{t}\frac{\chi_{\Gamma}}{\tau_M}e^{-(t-s)/\tau_M}u(s)\,ds,
\qquad
\chi_M(\omega)=\frac{\chi_{\Gamma}}{1-i\omega\tau_M}.

On the declared window, $m$TeXm is the lowest-order sufficient coordinate of the detector-tube process rather than an arbitrary added variable.

remark: Finite-mode support reading. On certified one-pole windows, the reduced coordinate $m$TeXm can be matched to the real working-point amplitude of a corresponding finite-mode embedding, such as a reaction coordinate or pseudomode [citation]. That comparison records only a possible reduced reading of the certified window and does not define the baseline three-sector closure.

proposition: Residual-controlled one-pole admissibility. Fix before calibration a detector family, an operating band $\Omwork$TeX\Omwork, a protocol norm $\normw{\cdot}$TeX\normw{\cdot}, and an admissibility tolerance $\epstol$TeX\epstol; these declarations are held fixed throughout certification and all cross-observable tests on that same family/window. Let $\Tresp$TeX\Tresp denote the reconstructed linear detector-tube response on $\Omwork$TeX\Omwork, and let $\Tone$TeX\Tone be the one-pole realization that matches the static gain and dominant pole of reference. Define

\epsmem
=
\frac{\normw{\Tresp-\Tone}}{\normw{\Tresp}}.

The reduction reference--reference for the memory-conditioned excess channel is admissible only when $\epsmem\le\epstol$TeX\epsmem\le\epstol on that declared band. If $\epsmem>\epstol$TeX\epsmem>\epstol, then higher poles or explicit multi-time kernels belong to the baseline detector response and the one-pole closure fails on that family/window.

proof. A one-pole realization is admissible only if it approximates the reconstructed detector-tube response on the predeclared band within the predeclared protocol tolerance. If the residual exceeds that tolerance, then no single local memory coordinate with one time scale reproduces the detector-tube response on that certified window, so the pole relations and no-retuning predictions cease to be family-stable. The missing response must then be carried by additional poles or by an explicit multi-time kernel.

remark: Operational memory witness. For frequency-sweep data one may take $\normw{\cdot}$TeX\normw{\cdot} to be a weighted $L^{\infty}$TeXL^{\infty} or $L^2$TeXL^2 norm of the reconstructed boundary response on $\Omwork$TeX\Omwork; for multi-time reconstruction one may use the corresponding norm of a detector-tube multi-time response reconstructed by process-tensor tomography [citation]. Changing $\Omwork$TeX\Omwork, $\normw{\cdot}$TeX\normw{\cdot}, or $\epstol$TeX\epstol after calibration constitutes a family/window change rather than a new fit of the same closure.

Harmonic gate or flux loops Let $u(t)=u_1\cos(\omegad t)$TeXu(t)=u_1\cos(\omegad t). Then the periodic solution of reference is

m(t)=\chi_{\Gamma}u_1\frac{\cos(\omegad t-\varphi_M)}{\sqrt{1+(\omegad\tau_M)^2}},
\qquad
\tan\varphi_M=\omegad\tau_M.

The resulting hysteresis area in the $(u,\Jexc)$TeX(u,\Jexc)-plane is

A_{R\text{--}u}
=
\oint \Jexc\,du
=
\pi\beta_{\Gamma}\chi_{\Gamma}u_1^2\frac{\omegad\tau_M}{1+(\omegad\tau_M)^2}.

The loop area peaks at $\omegad\tau_M=1$TeX\omegad\tau_M=1. Thus harmonic data identify $\tau_M$TeX\tau_M directly.

Finite-window capture If the detector is switched on at $t=0$TeXt=0 for a short capture window of duration $\Tc$TeX\Tc while the reduced drive is approximately constant, $u(t)=u_0\Theta(t)$TeXu(t)=u_0\Theta(t), then

m(t)=\chi_{\Gamma}u_0\bigl(1-e^{-t/\tau_M}\bigr).

Accordingly, any short-window linearized excess-capture observable inherits the factor

\etacap(\Tc)=1-e^{-\Tc/\tau_M}.

Equation reference does not assert a universal efficiency law for all detectors. It states the baseline build-up factor of the detector-memory channel in the declared short-window regime. When $\Tc\ll\tau_M$TeX\Tc\ll\tau_M, the detector is memory-limited; when $\Tc\gg\tau_M$TeX\Tc\gg\tau_M, the dynamic boundary has fully conditioned the detector tube.

Ramp thresholds and onset lag For a ramped drive $u(t)=\dot u\,t$TeXu(t)=\dot u\,t with constant rate $\dot u$TeX\dot u, reference gives

m(t)=\chi_{\Gamma}\dot u\left(t-\tau_M+\tau_M e^{-t/\tau_M}\right).

In the regime $t\gtrsim \tau_M$TeXt\gtrsim \tau_M, the memory-conditioned excess response behaves as if the drive were delayed by one memory time. If $u_*$TeXu_* is the static threshold extracted from the same detector family after fixing the direct feedthrough baseline, the dynamic onset of the excess channel satisfies to leading order

u_{\mathrm{on}}^{\uparrow}\approx u_*+\nu_{\Gamma}\dot u\,\tau_M,
\qquad
u_{\mathrm{on}}^{\downarrow}\approx u_*-\nu_{\Gamma}|\dot u|\,\tau_M,

with a family-fixed coefficient $\nu_{\Gamma}>0$TeX\nu_{\Gamma}>0. The up--down onset split therefore scales linearly with ramp rate and memory time.

Scan-speed visibility and memory length Now let a detector scan tangentially across a one-dimensional interference pattern of pitch $\ellint$TeX\ellint, so that the reduced drive is

u(t)=u_0+u_1\cos(\kgate v_s t),
\qquad
\kgate=\frac{2\pi}{\ellint},

with scan speed $v_s$TeXv_s. Applying reference yields the observed excess fringe visibility

\Vex(v_s)=\frac{\Vbarex}{\sqrt{1+(\kgate v_s\tau_M)^2}}.

The same law defines a scan-memory length

\Lmem\equiv v_s\tau_M.

Thus the detector samples a memory-filtered pattern extending over a distance $\Lmem$TeX\Lmem along the scan path.

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

corollary: Parameter-free pole relations on a certified one-pole detector-family window. Within the certified one-pole detector-family window, the hysteresis-peak frequency, half-build time, and half-power scan speed are

\omega_{\mathrm{peak}}=\tau_M^{-1},
\qquad
T_{1/2}=\tau_M\ln 2,
\qquad
v_{1/2}=\frac{\ellint}{2\pi\tau_M},

where $\omega_{\mathrm{peak}}$TeX\omega_{\mathrm{peak}} maximizes reference, $T_{1/2}$TeXT_{1/2} satisfies $\etacap(T_{1/2})=1/2$TeX\etacap(T_{1/2})=1/2, and $v_{1/2}$TeXv_{1/2} satisfies $\Vex(v_{1/2})=\Vbarex/\sqrt{2}$TeX\Vex(v_{1/2})=\Vbarex/\sqrt{2}. Hence

\omega_{\mathrm{peak}}T_{1/2}=\ln 2,
\qquad
\omega_{\mathrm{peak}}=\frac{2\pi v_{1/2}}{\ellint},

with no extra fit parameter beyond $\tau_M$TeX\tau_M.

proof. The identities follow directly from reference, reference, and reference.

This section fixes only the admissible variable-passing contract from the pre-latch layer to later detector-architecture variables. It does not introduce a metastable cycle, hazard law, record-bearing current account, or reset dynamics.

proposition: Post-switch recovery/trigger separation. Suppose detector activation and detector motion are switched off at time $t_0$TeXt_0, so that $g_{\Gam}=0$TeXg_{\Gam}=0, $\vb=0$TeX\vb=0, and no new incoming flux enters the tube for $t>t_0$TeXt>t_0. Then the detector-memory field obeys

\Mmem(\mathbf{x},t)=\Mmem(\mathbf{x},t_0)e^{-(t-t_0)/\tau_M},
\qquad t>t_0,

and the conservative detector perturbation decays with the same factor,

\Ug(\mathbf{x},t)=\epsloc(d_{\Gam})u_M\Mmem(\mathbf{x},t).

The trigger current remains nonnegative and tends to zero,

\Jtr(t)\ge 0,
\qquad
\Jtr(t)\to 0\quad (t\to\infty),

whereas the accumulated trigger count remains monotone,

\Ntr(t_0+T)-\Ntr(t_0)=\int_{t_0}^{t_0+T}\Jtr(s)\,ds\ge0.

Therefore the post-switch detector response splits into an exponentially recoverable boundary-distortion tail and a nonrecoverable trigger tail.

proof. With source and advection terms turned off, reference reduces to $\partial_t\Mmem=-\tau_M^{-1}\Mmem$TeX\partial_t\Mmem=-\tau_M^{-1}\Mmem, giving reference. Equation reference follows from reference. Since $\Kg\ge0$TeX\Kg\ge0, reference implies reference. As $\Mmem\to0$TeX\Mmem\to0 and the direct gate closes, $\Lam$TeX\Lam falls below threshold and the trigger current vanishes. The integrated trigger count remains monotone by construction.

definition: Operational local-commit boundary. Within a declared switch-off protocol, the local-commit sector is exactly the part of the detector response that contributes to the monotone trigger accumulation $\Ntr(t)$TeX\Ntr(t) after switch-off. Purely reversible boundary distortion is the complementary detector response that relaxes with the post-switch factor $e^{-(t-t_0)/\tau_M}$TeXe^{-(t-t_0)/\tau_M} and produces no additional trigger count.

proposition: Admissible trigger-to-record factorization on the declared pre-latch trigger-to-record coarse-graining window. Let $Q(t)$TeXQ(t) be any later durable-record variable whose dynamics is causal and insensitive to reversible detector distortions after switch-off. Then the wave/detector layer can enter the evolution law for $Q$TeXQ only through the trigger current $\Jtr$TeX\Jtr or a causal coarse-graining of it. In the linearized baseline,

\Jlat(t)=\int_0^{\infty}\ell(s)\,\Jtr(t-s)\,ds,
\qquad \ell(s)\ge0,

defines the admissible linearized coarse-grained record-input current on the declared pre-latch window. Record-bearing latching, metastable hazard structure, entropy export, and reset are not part of this proposition. Direct dependence on $\Ug$TeX\Ug or $\Mmem$TeX\Mmem is inadmissible.

proof. After switch-off, $\Mmem$TeX\Mmem and $\Ug$TeX\Ug decay exponentially by reference--reference, whereas the accumulated trigger count $\Ntr(t)$TeX\Ntr(t) remains monotone by reference. A later record variable that depended directly on $\Ug$TeX\Ug or $\Mmem$TeX\Mmem would therefore inherit sensitivity to reversible boundary recovery. Any admissible durable-record variable must depend only on the nonnegative trigger current or a causal coarse-graining of it. In the linearized baseline, causal coarse-graining is represented by reference.

theorem: Compatibility of static reduction with declared durable-record coarse-graining. Let $\mathcal R_0$TeX\mathcal R_0 denote the static-memoryless reduction $\tau_M\to0$TeX\tau_M\to0, $\dot g_{\Gam}=0$TeX\dot g_{\Gam}=0, $\vb=0$TeX\vb=0, and let $\mathcal L$TeX\mathcal L be the causal coarse-graining map in reference with fixed kernel $\ell\in L^1(\mathbb R_+)$TeX\ell\in L^1(\mathbb R_+). If $\Jtr^{(\tau_M)}\to \Jtr^{(0)}$TeX\Jtr^{(\tau_M)}\to \Jtr^{(0)} locally in $L^1$TeXL^1 under $\mathcal R_0$TeX\mathcal R_0, then

\mathcal L\!\left[\mathcal R_0\Jtr\right]
=
\mathcal R_0\!\left[\mathcal L\Jtr\right].

Thus reducing first to an instantaneous local-opening law and then forming durable records yields the same durable-record current as forming the pre-latch trigger current first and then taking the static-memoryless limit. Equivalently, the diagram

{ccc}
\Jtr^{(\tau_M)}  \xrightarrow{\ \mathcal L\ }  \Jlat^{(\tau_M)} 

\downarrow{\mathcal R_0}    \downarrow{\mathcal R_0} 

\Jtr^{(0)}  \xrightarrow{\ \mathcal L\ }  \Jlat^{(0)}

commutes on the declared static-memoryless window.

proof. By reference, $\mathcal L$TeX\mathcal L is convolution with the fixed integrable kernel $\ell$TeX\ell. Local $L^1$TeXL^1 convergence of $\Jtr^{(\tau_M)}$TeX\Jtr^{(\tau_M)} together with $\ell\in L^1$TeX\ell\in L^1 implies convergence of the convolutions by dominated convergence. Therefore the static-memoryless limit commutes with durable-record coarse-graining.

Several experimentally active detector families expose the detector-side structure encoded by the closure: a propagating field, an activated interface, finite response time or bandwidth, and a count or readout channel. These platforms are used only to identify observables that can certify or reject the declared pre-latch closure.

Spatially resolved atom and matter-wave detectors Spatially resolved single-atom detector tomography already shows that a detector surface carries local response structure rather than a single global efficiency [citation]. Wave-packet imaging in continuous space now resolves the projected spatial distribution of a single-atom wave packet before fluorescence imaging [citation]. Those systems test two separate parts of the law: reference in the static projection limit and reference for the excess visibility channel whenever the detector is scanned, activated in time, or reconstructed from finite-duration pinning windows.

Superconducting and microwave detectors Single-shot qubit readout with a bolometer explicitly exposes a finite readout duration and a tradeoff between fidelity and measurement time [citation]. Cyclically operated microwave single-photon counters make the finite conversion window and detector duty cycle explicit [citation]. Thermal spectroscopy of superconducting circuits now offers a direct-current route to frequency-resolved detector characterization over a broad band, again emphasizing detector-side response time and conditioning rather than a single static efficiency number [citation]. These detector-side observables are summarized in $\tau_M$TeX\tau_M, reference, and reference, with static direct feedthrough fixed separately on the same family. Threshold microwave detection by underdamped Josephson junctions provides a direct threshold-type platform for the commit sink reference [citation].

Time-resolved interferometric readout Single-electron interferometry now detects time-dependent electromagnetic fields by measuring both phase and contrast of the interference pattern, with temporal resolution limited by the electronic wave-packet width [citation]. Time-resolved parity measurement in interferometric semiconductor devices likewise shows that single-shot readout can depend critically on the finite measurement window and the dynamics of the readout loop [citation]. These platforms are especially natural because reference and reference directly couple excess counts, excess visibility, and window duration.

Detector characterization beyond static POVMs Detector self-characterization and semidefinite measurement tomography remain indispensable in the static or quasistatic limit [citation]. The baseline law adds the structures that become visible once the detector is dynamically inserted into the wave field: a memory coordinate, an insertion-rate channel, and a commit sink. Process-tensor methods now characterize fixed multi-time memory in terms of experimentally accessible quantities [citation], and full multi-time quantum process tomography on superconducting qubits shows that such structure can be reconstructed in practice [citation]. Those developments make it realistic to test whether the single-time closure reference is sufficient or whether a broader kernel is required. Recent photodetector reporting standards likewise emphasize family-fixed calibration, bandwidth declaration, and protocol consistency in detector benchmarking [citation]. The present layer fixes only the pre-latch detector-boundary memory closure between local opening and later durable readout thermodynamics; it neither revises the local opening grammar nor closes the later latching/reset architecture.

Family-fixed one-pole protocol

proposition: Family-fixed no-retuning protocol. Fix before calibration a detector family, a working point, an operating band $\Omwork$TeX\Omwork, a protocol norm $\normw{\cdot}$TeX\normw{\cdot}, and a tolerance $\epstol$TeX\epstol, fix the direct feedthrough coefficient $\alpha_{\Gamma}$TeX\alpha_{\Gamma} from a static or quasistatic reference on that same family/window, and certify $\epsmem\le\epstol$TeX\epsmem\le\epstol on the declared dynamic window. Determine $\tau_M$TeX\tau_M from one primary calibration observable on the certified one-pole window - for example phase lag, gain roll-off, weak-window build-up, a loop-area peak, or a step-on capture law - while keeping the family/window declarations fixed. Then reference, reference, reference, and reference predict the feedthrough-subtracted trigger-side observables of that same detector family/window without retuning $\tau_M$TeX\tau_M, $\alpha_{\Gamma}$TeX\alpha_{\Gamma}, changing $\Omwork$TeX\Omwork, changing $\normw{\cdot}$TeX\normw{\cdot}, enlarging the source basis, or separating the dispersive and absorptive sectors. Any such change rejects the one-pole baseline on that family/window.

proof. Once $\epsmem\le\epstol$TeX\epsmem\le\epstol, the reduced dynamic law is fixed by reference--reference on the declared band. The static reference fixes $\alpha_{\Gamma}$TeX\alpha_{\Gamma}, and the chosen dynamic calibration observable fixes $\tau_M$TeX\tau_M together with the corresponding working-point gain combination before any cross-observable test is attempted. Equations reference, reference, reference, and reference then depend on that same $\tau_M$TeX\tau_M on the same declared family/window after the direct feedthrough subtraction is held fixed. A second $\tau_M$TeX\tau_M, a changed $\alpha_{\Gamma}$TeX\alpha_{\Gamma}, a changed protocol band or norm, an enlarged source class, or an independently tuned absorptive sector therefore signals failure of the one-pole baseline.

A minimal same-family certification protocol has five steps.

- Static reference. In a fixed-boundary or quasistatic regime, determine the local response map or threshold curve, fix the direct feedthrough coefficient $\alpha_{\Gamma}$TeX\alpha_{\Gamma}, and verify the static reductions of reference and reference. - Window certification. Declare $\Omwork$TeX\Omwork, $\normw{\cdot}$TeX\normw{\cdot}, and $\epstol$TeX\epstol before fitting, reconstruct the linear detector-tube response on $\Omwork$TeX\Omwork, and verify $\epsmem\le\epstol$TeX\epsmem\le\epstol using reference. - Calibration observable. On the certified one-pole window, use one primary calibration observable only - for example phase lag, gain roll-off, weak-window build-up, loop-area spectrum, or the step-on capture law - to infer $\tau_M$TeX\tau_M and the working-point gain combination $\beta_{\Gamma}\chi_{\Gamma}$TeX\beta_{\Gamma}\chi_{\Gamma}, with $\alpha_{\Gamma}$TeX\alpha_{\Gamma} already fixed by the static reference. - Cross-observable prediction without retuning. With $\alpha_{\Gamma}$TeX\alpha_{\Gamma}, $\tau_M$TeX\tau_M, $\Omwork$TeX\Omwork, $\normw{\cdot}$TeX\normw{\cdot}, $\epstol$TeX\epstol, and the family-fixed coefficients frozen, predict the remaining observables not used in the calibration step - including threshold lag reference, scan-speed excess visibility reference, finite-window excess capture reference, and post-switch recovery - from the same detector family and operating window. - Failure rule. If independent observables require different $\tau_M$TeX\tau_M values, altered direct-feedthrough subtraction, altered protocol declarations, additional source channels, residual memory above $\epstol$TeX\epstol, or separately retuned dispersive and absorptive sectors before a declared family/window change, reject the one-pole baseline closure.

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

Calibrated benchmark: thermal-detector qubit readout Thermal-detector qubit readout already exposes a finite detector-side time constant on the same platform that carries the trigger signal [citation]. Ref. [citation] reports an effective thermal time constant $\tau_b=9.4\,\mu\mathrm{s}$TeX\tau_b=9.4\,\mu\mathrm{s} on the calibrated ground-state branch and a reported readout pulse length $t_{\mathrm{RO}}=13.9\,\mu\mathrm{s}$TeXt_{\mathrm{RO}}=13.9\,\mu\mathrm{s}. On that declared family/window we take $\tau_M=\tau_b$TeX\tau_M=\tau_b as a same-family calibration ansatz rather than a universal detector identity. Taking $\tau_M=\tau_b$TeX\tau_M=\tau_b on that declared family/window gives the same-family one-pole consequences

T_{1/2}=\tau_M\ln 2=6.52\,\mu\mathrm{s},
\qquad
\eta_{\mathrm{cap}}(13.9\,\mu\mathrm{s})=1-e^{-13.9/9.4}=0.772,

and

\omega_{\mathrm{peak}}=\tau_M^{-1}=0.106\,\mu\mathrm{s}^{-1},
\qquad
f_{\mathrm{peak}}=\frac{\omega_{\mathrm{peak}}}{2\pi}=16.9\,\mathrm{kHz}.

These are same-family cross-observable consequences of one declared $\tau_M$TeX\tau_M, not new fit parameters.

A calibrated detector-family benchmark on that platform is therefore: declare the operating point, fix the static direct feedthrough on that family, and certify $\epsmem\le\epstol$TeX\epsmem\le\epstol on one pre-latch window; extract $\tau_M$TeX\tau_M once from a weak-modulation lag or short-window build-up observable, and then test reference--reference, reference, and reference without changing $\Omwork$TeX\Omwork, $\normw{\cdot}$TeX\normw{\cdot}, or $\epstol$TeX\epstol. Pulse-duration sweeps probe $\eta_{\mathrm{cap}}$TeX\eta_{\mathrm{cap}}, weak periodic modulation probes $f_{\mathrm{peak}}$TeXf_{\mathrm{peak}}, ramped activation probes the onset shift, and post-switch relaxation probes the same $\tau_M$TeX\tau_M in the recovery tail.

Ref. [citation] also shows that the effective thermal time constant depends on probe conditions. Accordingly, changing probe power, bias, or operating point defines a new family/window declaration rather than a refit inside one certified window. If build-up, weak-modulation, threshold-lag, or post-switch observables on one fixed declaration require a distinct $\tau_M$TeX\tau_M, the one-pole detector-family reduction is rejected on that window.

The baseline law is rejected whenever any of the following occurs within a declared detector family.

- One-pole window is not certified: the reconstructed detector-tube response yields $\epsmem>\epstol$TeX\epsmem>\epstol on the declared operating band, or certification requires changing the response norm or tolerance within one detector family. - No family-fixed memory time: with $\epsmem\le\epstol$TeX\epsmem\le\epstol and the direct feedthrough subtraction fixed on the declared window, a single $\tau_M$TeX\tau_M extracted from one declared calibration observable does not predict threshold lag, excess-visibility roll-off, post-switch recovery, or short-window excess capture. - Pole relations fail: the measured hysteresis-peak frequency, half-build time, and half-power scan speed do not satisfy the parameter-free relations reference--reference. - Dispersion--absorption mismatch: phase or contrast shifts measured on the declared calibration window and trigger losses attributed to $\Ug$TeX\Ug and $\Kg$TeX\Kg cannot be fitted by one causal boundary susceptibility on the declared operating window. - Object-wise retuning is required: different observables or nominally identical devices demand different $\tau_M$TeX\tau_M or different source channels without a declared family change. - No ramp asymmetry: threshold onset fails to show the linear $|\dot u|\tau_M$TeX|\dot u|\tau_M split of reference in a regime where the detector is known to be dynamically bandwidth-limited. - Visibility scaling is wrong: scanned or time-swept interference data do not follow the $v_s\tau_M/\ellint$TeXv_s\tau_M/\ellint scaling of reference in the excess channel even when static tomography confirms the baseline static response. - No post-switch separation: after detector deactivation the near-boundary recovery tail does not relax with the same $\tau_M$TeX\tau_M that governs loops and capture, or trigger accumulation fails to separate monotonically from the recovery tail. - Memory needs higher structure: the data require explicit multi-time kernels, polarization tensors, or several competing memory times already in the baseline operating window. - No boundary insertion sensitivity: activating, moving, or bandwidth-limiting the detector changes neither trigger counts nor contrast beyond what is already explained by static detector tomography. - Trigger sink is too strong: the model predicts trigger current while destroying propagation in a regime where the observed detector is demonstrably weakly invasive, signalling that the sink must be weakened or that the working point lies outside the baseline window. - Static/null limits are wrong: in a regime independently known to be static and memoryless, the data do not reduce to reference, or turning off detector couplings fails to recover the unperturbed propagation sector of reference. - Trigger-to-record factorization fails: after conditioning on $\Jtr$TeX\Jtr or its declared coarse-graining, later durable-record observables still depend directly on reversible boundary variables. - Reduction and record limits do not commute: the durable-record current obtained after static-memoryless reduction does not agree with the reduced limit of the durable-record current in the sense of reference.

The detector tube carries three pre-latch objects with distinct functions. The pair $(\Ug,\Mmem)$TeX(\Ug,\Mmem) describes reversible boundary adaptation: it reorganizes phase and contrast near the detector, lags under switching or scanning, and relaxes after switch-off. The trigger current $\Jtr$TeX\Jtr is different: it is nonnegative and accumulates monotonically. The post-switch identities therefore separate reversible boundary distortion from event-local trigger generation without inserting any persistent record variable into the wave-side equations. Only on a certified one-pole detector-family window may $\Ug$TeX\Ug, $\Mmem$TeX\Mmem, and $\Kg$TeX\Kg be read, in a secondary response description, as the dispersive, dominant-pole, and absorptive faces of one causal detector-boundary susceptibility.

The pullback source theorem fixes the detector-memory sector at first order. The source is not a regression over arbitrary local features; within the declared isotropic first-order scalar class it is restricted to incoming normal flux, gate-density, insertion-rate density, and scan-gradient coupling. The trigger sink is constrained in the same spirit: locality, positivity, null-coupling recovery, monotone thresholding, and one-memory causality restrict the baseline sink to reference. If the data require extra tensor channels, nonlocal kernels, independently retuned dispersive and absorptive sectors, or several comparable poles in the same operating window, the baseline closure fails rather than absorbing the discrepancy into ad hoc detector structure.

The one-pole compression is a certified reduction of that baseline law on declared detector-family windows. Within the declared detector-tube window, $m$TeXm is the minimal local realization of the dominant causal susceptibility, and admissibility of that reduction is fixed by the residual witness $\epsmem\le\epstol$TeX\epsmem\le\epstol. The resulting pole relations tie together feedthrough-subtracted hysteresis, excess capture, excess visibility, threshold lag beyond the static reference, and post-switch recovery through one family-fixed $\tau_M$TeX\tau_M, which is fixed once on the certified one-pole window and then tested on independent trigger-side observables without retuning. Success of that reduction shows that one measurable detector-memory coordinate suffices on the declared window. Failure identifies either a broader memory kernel or a later nonequilibrium amplification sector already active inside the working window.

The admissible trigger-to-record coarse-graining is fixed by post-switch separation. A later record variable may depend on the pre-latch layer only through $\Jtr$TeX\Jtr or a causal coarse-graining of it; the architectural readout cycle itself is not closed here. With that factorization, the static-memoryless reduction commutes with durable-record coarse-graining. The same pre-latch layer joins instantaneous local opening to later record formation without mixing reversible boundary recovery into the durable-record channel. Process-tensor, tomography, reaction-coordinate, and pseudomode descriptions therefore remain characterization tools or certified reduced-window descriptions; none replaces the three-sector detector-tube law on the declared pre-latch window.

The present dynamic detector-boundary closure closes the activated finite-bandwidth pre-latch layer between bulk propagation and later durable-record dynamics. Within the declared first-order local class, reversible near-boundary reorganization requires $\Ug$TeX\Ug, one dominant pre-latch memory time requires $\Mmem$TeX\Mmem, and event-local trigger generation requires $\Kg$TeX\Kg. Within the declared isotropic first-order scalar baseline, the admissible memory source is fixed by the moving detector tube and the exported propagation density-current pair, while the admissible pre-latch trigger sink is fixed by locality, positivity, null-coupling recovery, monotone thresholding, and one-memory causality. Enlarged detector-coordinate sets, tensor-valued source classes, or explicitly nonlocal kernels define enlarged models rather than counterexamples within that baseline.

The same family-fixed time scale $\tau_M$TeX\tau_M governs feedthrough-subtracted count--drive hysteresis, short-window excess capture, threshold lag beyond the static reference, scan-speed excess visibility, and post-switch recovery. The one-pole reduction is secondary to the baseline closure and is admitted only on detector-family windows for which $\Omwork$TeX\Omwork, $\normw{\cdot}$TeX\normw{\cdot}, and $\epstol$TeX\epstol are declared before calibration and the residual witness satisfies $\epsmem\le\epstol$TeX\epsmem\le\epstol. On that certified window the same-family protocol first fixes the static direct feedthrough and then fixes $\tau_M$TeX\tau_M once before testing the remaining trigger-side observables without retuning. Only on that window may the three-sector closure be read, in a secondary response description, as the dispersive, dominant-pole, and absorptive faces of one detector-boundary susceptibility. Process-tensor, tomography, reaction-coordinate, and pseudomode descriptions remain characterization tools or certified reduced-window descriptions on that window; they do not replace the detector-tube law itself.

The trigger current $\Jtr$TeX\Jtr is the detector-side output of the wave/tube layer retained as an admissible causal input to any later durable-record description. The static-memoryless reduction and the durable-record coarse-graining commute, so the same pre-latch closure joins instantaneous local opening to later record formation without inserting latching or reset into the wave-side equations. The instantaneous local-opening law appears here only as a recovery limit of the present closure. In the fixed-surface thin-layer zero-memory limit the closure returns to the absorbing-boundary detector class; in the quasistatic map limit it returns to detector tomography. Within the declared first-order local pre-latch class, the triad $(\Ug,\Mmem,\Kg)$TeX(\Ug,\Mmem,\Kg) is therefore the minimal detector-tube closure whenever switching, scanning, or finite detector bandwidth is already part of the observable dynamics. No durable-record criterion, amplification law, reset dynamics, or objectivity theorem is derived here. No metastable ready--activated--latched architecture, durable-record persistence criterion, or reset rule is part of the present result. If no declared detector-family window supports the same-family one-pole relations without retuning, or if later durable-record variables remain directly sensitive to reversible boundary variables rather than to $\Jtr$TeX\Jtr or its causal coarse-graining, the closure is rejected on that window rather than implicitly enlarged.

The reduced formulas in reference follow from the single-pole response reference. We record the short derivations.

Harmonic loops..

For $u(t)=u_1\cos(\omegad t)$TeXu(t)=u_1\cos(\omegad t), solve reference in steady state to obtain reference. Since $\Jexc=\beta_{\Gamma}m$TeX\Jexc=\beta_{\Gamma}m, the parametric loop $(u(t),\Jexc(t))$TeX(u(t),\Jexc(t)) is an ellipse whose oriented area is exactly reference; the direct feedthrough term $\alpha_{\Gamma}u$TeX\alpha_{\Gamma}u contributes zero area because $\oint u\,du=0$TeX\oint u\,du=0.

Step activation..

For $u(t)=u_0\Theta(t)$TeXu(t)=u_0\Theta(t), the solution of reference is reference. Any linearized excess-capture observable proportional to the memory-conditioned opening factor therefore acquires the build-up law reference.

Scanning across a sinusoidal pattern..

Let the scanned signal be $u(t)=u_0+u_1\cos(\kgate v_s t)$TeXu(t)=u_0+u_1\cos(\kgate v_s t). The memory coordinate responds with the same frequency but reduced amplitude $u_1\chi_{\Gamma}/\sqrt{1+(\kgate v_s\tau_M)^2}$TeXu_1\chi_{\Gamma}/\sqrt{1+(\kgate v_s\tau_M)^2}. The observed excess contrast therefore inherits the suppression factor in reference once the static visibility floor has been subtracted. $\square$TeX\square

The modal statements in reference, reference, and reference are restricted to the declared isotropic first-order local baseline. This appendix records the corresponding exclusions.

B.1. Pure absorption does not supply reversible tube reorganization..

Consider a localized detector correction with no real insertion sector,

i\hbar\partial_t\psi=(\hat H_0-\tfrac{i\hbar}{2}K)\psi,
\qquad K\ge 0.

Such a law supplies only tube-local norm loss. There is then no independent conservative tube-local generator that can persist as a reversible phase or contrast distortion while the trigger channel is held fixed. Pure absorption can therefore represent local loss, but not the reversible near-boundary reorganization required in reference.

B.2. Classification of first-order isotropic pullback scalars..

On the moving detector tube the basic local data are $(\rhob,\jb,g_{\Gam},d_{\Gam},\vb)$TeX(\rhob,\jb,g_{\Gam},d_{\Gam},\vb). Global phase invariance removes the absolute phase. At zeroth order the only scalar activation channel is $g_{\Gam}\rhob$TeXg_{\Gam}\rhob. At first order, the only isotropic scalars built from one derivative or one velocity insertion are $\Jnp$TeX\Jnp, $\dot g_{\Gam}\rhob$TeX\dot g_{\Gam}\rhob, and $\vbpar\!\cdot\!\gradpar\rhob$TeX\vbpar\!\cdot\!\gradpar\rhob. Terms involving $\partial_t\rhob$TeX\partial_t\rhob reduce by the continuity equation to $-\nabla\!\cdot\!\jb$TeX-\nabla\!\cdot\!\jb and hence to the normal-flux channel plus curvature or higher-order tangential corrections. Curvature tensors, polarization tensors, and unrestricted nonlocal averages therefore lie outside the declared isotropic first-order scalar baseline. This yields reference.

B.3. Direct dynamic sink channels do not fit one certified pole..

Linearize the trigger variable on a certified one-pole window and suppose, contrary to reference, that

\delta\Lam(\omega)=c_M m(\omega)+c_{\dot g}\,i\omega g_{\Gam}(\omega)+c_v q_v(\omega)+\cdots,

where $m(\omega)=\chi_{\Gamma}u(\omega)/(1-i\omega\tau_M)$TeXm(\omega)=\chi_{\Gamma}u(\omega)/(1-i\omega\tau_M) is the certified one-pole coordinate and $q_v$TeXq_v is the Fourier image of the scan-gradient channel. The factor $i\omega$TeXi\omega is frequency odd and cannot equal a family-fixed scalar multiple of $m(\omega)$TeXm(\omega) on a finite band $\Omwork$TeX\Omwork: equality would require

c_{\dot g}\,i\omega
=
c_M\frac{\chi_{\Gamma}}{1-i\omega\tau_M}
\qquad
\text{for all }\omega\in\Omwork,

which is impossible unless the band collapses to a point or coefficients are retuned frequency by frequency. The scan-gradient term is analogous. Direct absorptive dependence on $\dot g_{\Gam}$TeX\dot g_{\Gam} or scan asymmetry therefore requires either an additional local detector state or an independently retuned absorptive kernel. Both lie outside the one-memory certified baseline, so those channels must enter through $\Mmem$TeX\Mmem alone.

B.4. Durable-record variables do not belong to the wave-side baseline..

If a durable-record variable were inserted directly into the pre-latch wave-side law, then after switch-off the same variable would respond both to reversible tube recovery and to monotone trigger accumulation. That contradicts the separation of reference and reference. Durable-record coordinates therefore belong only to the later coarse-grained sector.