Detector Thermodynamics, Metastable Amplification, and Irreversible Readout

Outcome statistics can be represented abstractly by POVMs, response kernels, or readout channels. Recent work has sharpened detector characterization through spatially resolved detector tomography, mid-circuit measurement benchmarking, and detector-estimation bounds [citation]. A physical detector with irreversible readout requires an additional layer: a driven nonequilibrium cycle that maintains a metastable ready state, converts a microscopic trigger into a resolved output class, exports entropy to uncontrolled degrees of freedom, and resets before the next shot.

That architecture is explicit across established detector families. Geiger-mode avalanche photodiodes combine a bias-maintained metastable state with quenching and recharge circuitry; recent gated calibrations and realistic detector-response analyses resolve dark counts, afterpulsing, dead time, and pulse-to-pulse recovery directly [citation]. Transition-edge sensors operate near a dissipative electrothermal instability, and transition physics, dark-count characterization, geometry-dependent parameter/noise studies, and thermal-detector single-shot readout make the same ready--activation--readout--recovery structure explicit [citation]. Superconducting nanowire detectors rely on hotspot-triggered electrothermal switching and recovery, with latching, reset, and recovery control resolved experimentally and in reduced circuit models [citation].

Irreversible readout and apparatus resetting carry finite energetic cost, and finite-time protocols raise that cost through entropy production [citation]. At the detector-platform level the same coupling appears through explicit tradeoffs among afterpulsing, recovery, count rate, latching, and dead-time distortion [citation]. A threshold may initiate activation, but irreversible detector readout requires one coupled cycle containing a ready state, a switching channel, a record-bearing output class, a sink, and reset.

The detector-facing event variable is the local-commit rate $\Rc$TeX\Rc, defined operationally as the coarse-grained current into the record-bearing resolved output class of the detector cycle. It is not a relabeling of raw switching flux. A device may transduce a signal or cross a threshold without entering such an output class; only devices with irreversible readout are considered here. The detector class is therefore fixed by the minimum thermodynamic structure required for durable record formation: metastability on the declared readout window, signal-assisted escape on the same window, dissipative latching with finite gain, and reset compatible with the declared repetition rate. Born-rule derivations, intrinsic non-unitary collapse models, and intersubjective objectivity are not considered.

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Metastable ready state

A detector here is not defined by a single threshold parameter. It is defined by a nonequilibrium cycle whose ready state is metastable. Let $R$TeXR denote the ready state, $A$TeXA an activated state after signal-assisted escape, and $L$TeXL the record-bearing resolved output class on the declared readout channel. Throughout, $L$TeXL need not be a permanently self-trapped material state; it denotes the output class that the declared readout resolves as an event. The ready state is maintained by bias work $\Wbias$TeX\Wbias and separated from spontaneous activation by a free-energy barrier $\Fbar$TeX\Fbar.

definition: Metastable ready state. A detector is said to possess a metastable ready state if, on the declared readout window $\tauread$TeX\tauread, the spontaneous switching probability is small:

\Gdark \,\tauread \ll 1,

where $\Gdark$TeX\Gdark is the spontaneous escape rate from $R$TeXR in the absence of a signal. All downstream metastability, switching, latching, and benchmark-facing claims are read only relative to this declared readout window.

For activated switching, the natural minimal form is Arrhenius/Kramers-like:

\Gdark = \Gamma_0\,e^{-\betath \Fbar},

with $\Gamma_0$TeX\Gamma_0 an attempt rate and $\betath=(\kB \Teff)^{-1}$TeX\betath=(\kB \Teff)^{-1} the inverse effective temperature of the relevant bath or noise environment. Equation reference is used as the baseline effective metastability law on the declared platform window; no universal microscopic escape law for every detector family is assumed.

A barrier-separated ready state is operationally necessary because otherwise the device cannot simultaneously maintain low dark switching and appreciable signal responsiveness on the same declared readout window. Detector design is therefore a metastability problem rather than a bare threshold problem.

Signal-assisted escape

Let $\Esig$TeX\Esig denote the signal energy delivered into the detector mode that participates in triggering. To leading order, the signal lowers or bypasses the ready-state barrier and produces a signal-assisted switching rate

\Gsw(\Esig)
=
\Gamma_0 \exp\!\left[-\betath\!\left(\Fbar-\etaabs \Esig\right)\right],

where $0<\etaabs\le 1$TeX0<\etaabs\le 1 is an effective absorption/coupling factor. Equation reference is a minimal effective activated-sector law on the declared platform window. It is not a claim that identical microscopic barrier-lowering physics governs all detector technologies; in TES, SNSPD, and thermal-detector platforms it serves only as a coarse-grained surrogate for signal-assisted escape into the activated sector. The resulting raw activation probability on the declared readout window is

\Ptrig(\tauread)
=
1-\exp\!\left[-\int_0^{\tauread}\left(\Gsw(t)+\Gdark\right)dt\right].

This quantity refers only to signal-assisted activation out of the ready sector on the declared readout window; it is not yet the detector-side single-hit readout probability for a durable record on that window, which is governed below by the integrated local-commit current.

definition: Signal-admissible switching. The switching layer is called signal-admissible on $\tauread$TeX\tauread if

\int_0^{\tauread}\Gsw(t)\,dt \gtrsim 1
\qquad\text{and}\qquad
\int_0^{\tauread}\Gsw(t)\,dt \gg \Gdark \tauread.

All downstream activation-gate and detector-facing counting claims are read only relative to this declared readout window and its signal-dominance convention.

The first inequality ensures an $O(1)$TeXO(1) chance of signal-triggered escape on the declared readout window. The second ensures that activation is signal-dominated rather than dark-count dominated.

Threshold crossing versus irreversible readout A threshold description is often sufficient to describe whether a microscopic perturbation can trigger a switching excursion. It is not sufficient to describe a detector with irreversible readout. Threshold crossing alone does not explain why the ready state is stable against dark switching on the declared window, does not explain how a transient activation is converted into a stable output class, and does not explain why reset and dead time are tied to dissipation. In particular, a threshold model does not distinguish a short-lived activation that relaxes back to the ready state from a transition that enters a record-bearing latched sector. Threshold crossing is therefore treated as, at most, a partial description of the $R\to A$TeXR\to A step. Microscopic trigger formation upstream of the activated sector is taken as declared input data for the present cycle. Once a signal-responsive activation channel into $A$TeXA has been fixed, the detector problem is the thermodynamic conversion of that activation into a durable output class and a reusable reset cycle. A detector theory therefore requires the full metastable cycle $R\to A\to L\to R$TeXR\to A\to L\to R, together with the associated dark-count, entropy-export, and reset constraints.

Latching, gain, and eventwise entropy export

Switching by itself is not yet a measurement record. The activated state $A$TeXA must be converted into a stable output, and that conversion is the latching/amplification step. Let $\Glat$TeX\Glat be the rate for $A\to L$TeXA\to L, and let $\Gain$TeX\Gain denote the output distinguishability of the two readout classes (no event versus event) in the chosen measurement channel. We keep $\Gain$TeX\Gain deliberately operational; it may be a pulse-height separation, a current contrast, a calorimetric energy separation, or another platform-defined output distinction.

definition: Latched readout. A detector has latched readout on the output timescale $\tau_{\mathrm{out}}$TeX\tau_{\mathrm{out}} if

\Glat\,\tau_{\mathrm{out}} \gtrsim 1,

and the output classes remain distinguishable with declared gain $\Gain>1$TeX\Gain>1 on the same interval. All downstream record-formation and event-count claims are read only relative to this declared output timescale, gain convention, and resolved output class.

To make that output stable, the detector must export entropy and heat to uncontrolled degrees of freedom. We therefore define the eventwise entropy production of a click by

\Sigclick \equiv \betath \Qclick + \Delta s_{\mathrm{det}},

where $\Qclick$TeX\Qclick is the heat dumped to sink channels during the event-bearing cycle and $\Delta s_{\mathrm{det}}$TeX\Delta s_{\mathrm{det}} is the detector-state entropy change. The minimal one-bit record-formation bound motivates the compatibility condition

\Qclick \gtrsim \kB \Teff \ln 2,

with the right-hand side replaced by $\kB\Teff \ln M$TeX\kB\Teff \ln M for an $M$TeXM-way distinguishable record alphabet [citation].

Equation reference is used only as a lower compatibility floor for stable record creation on the declared output alphabet. It is neither sufficient for detector admissibility nor a platform-independent estimator of the total dissipated heat of a concrete device; physical detectors can dissipate far above that floor, and apparatus-level thermodynamic models likewise treat readout and reset as finite-cost operations [citation].

Reset, dead time, and the nonequilibrium cycle

A detector is only useful as a repeated measurement device if it resets. Let $\Greset$TeX\Greset denote the rate $L\to R$TeXL\to R under the bias-maintained reset channel, and define the dead time $\taudead$TeX\taudead as the characteristic recovery time to return the ready-state occupation to its declared operating level. At the cycle level, the detector free-energy balance is

\Delta F_{\mathrm{det}}
=
\Wbias + \Esig - \Qclick - \Qreset,

where $\Qreset$TeX\Qreset is the heat dumped during recovery. Sustained repeated operation therefore requires net positive entropy export to the environment:

\Sigdot = \Jout - \Jin > 0,

with $\Jin$TeX\Jin and $\Jout$TeX\Jout the entropy currents entering and leaving the detector subsystem. Here the sign convention is chosen so that $\Sigdot>0$TeX\Sigdot>0 denotes net entropy export from the detector subsystem to uncontrolled sink channels on the declared cycle. Equation reference is a cycle-level entropy-bookkeeping identity for the declared detector operation. Once a Markovian state model and local detailed balance are specified, the resolved stochastic entropy-production rate is given more explicitly by Eq. reference below.

The compatibility condition for the declared repetition rate $\Prep$TeX\Prep is

\Prep \,\taudead \ll 1.

Equation reference is operational rather than universal. It says only that the reset cycle must be fast compared to the use-rate claimed for the detector. Detector-platform studies likewise resolve reset and recovery as independent dynamical layers rather than as passive afterthoughts of threshold crossing: gated SPAD work tracks dead-time-limited recovery and afterpulsing directly, while recent SNSPD recovery-control experiments and electrothermal models expose the same reset--latching tradeoff in a different material language [citation]. Autonomous-reset experiments in superconducting qubit platforms provide a complementary thermodynamic example outside the detector benchmarks used here [citation].

The thermodynamic detector layer can be represented by a minimal three-state master equation. Let $p_R,p_A,p_L$TeXp_R,p_A,p_L be the ready, activated, and latched occupations, normalized by $p_R+p_A+p_L=1$TeXp_R+p_A+p_L=1. This occupation normalization, together with the declared identification of detector events as window integrals of the local-commit current, is fixed throughout the paper; all downstream tomography-facing identities and benchmark comparisons are read only relative to this normalization convention. Then

\dot p_R = -(\Gsw+\Gdark)p_R + \Grel p_A + \Greset p_L, 

\dot p_A = (\Gsw+\Gdark)p_R - (\Glat+\Grel) p_A, 

\dot p_L = \Glat p_A - \Greset p_L.

Here $\Grel$TeX\Grel is the non-recording relaxation rate for $A\to R$TeXA\to R. With Eqs. reference--reference, the minimal cycle is population conserving, $\dot p_R+\dot p_A+\dot p_L=0$TeX\dot p_R+\dot p_A+\dot p_L=0, while the detector-facing event count still receives contributions only from the latching current into $L$TeXL.

Why switching rate is not yet detector event rate

The minimal cycle distinguishes three physically different processes:

R\to A : \text{internal activation only},\nonumber

A\to R : \text{failed transient without record formation},\nonumber

A\to L : \text{record-bearing latching transition}.

Only the last transition creates a stable detector event on the declared readout pipeline. The generic switching current out of the ready state,

J_{R\to A}(t)=(\Gsw+\Gdark)p_R(t),

therefore differs in principle from the detector-facing event current. Whenever the non-recording relaxation channel $A\to R$TeXA\to R is present, one has internal activations that do not become stable readout events. The detector-facing observable must therefore count entry into the latched record-bearing sector rather than generic threshold crossing. A shared switching or readout window therefore does not by itself fix identical durable-record accumulation, because only the $A\to L$TeXA\to L flux contributes to the record-bearing output class. This is the reason for introducing the local-commit rate $\Rc$TeX\Rc.

Coarse-graining to a local-commit rate

Let $\Rc(t)$TeX\Rc(t) denote the local-commit rate, meaning the coarse-grained probability current into the latched record-bearing sector. This definition is operational: the readout channel distinguishes the presence or absence of a latched event on the declared observation window, but it need not resolve the internal microstructure of the activated state. In that sense $\Rc$TeX\Rc is not a relabeling of a switching rate. It is the minimal measurement-facing variable that discards internal activations and counts only those trajectories that enter the stable output class. The analytical payoff is immediate whenever failed transients are substantial: event accounting based on the generic switching current reference overcounts recorded events and conflates dark-triggered activation with stable readout, whereas $\Rc$TeX\Rc separates switching statistics from recorded-event statistics at the point where the detector actually forms a durable output class.

Upstream trigger formation is treated as declared input data. If a pre-latch trigger observable is imported from a detector-boundary model, it fixes only the activation channel into $A$TeXA; it is not identified with the detector-facing event current unless an additional latching coarse graining into the resolved output class is declared.

proposition: Local-commit rate from the unique latching current on a declared single-hit observation window. Fix a declared observation window $\Omega$TeX\Omega on which the readout resolves entry into the latched sector but not the internal activated microstructure. Assume that the declared observation window begins from the ready preparation, with no pre-existing latched occupancy counted on $\Omega$TeX\Omega. Assume either that retriggering, afterpulsing memory, and unresolved activated substructure are negligible on $\Omega$TeX\Omega, or that they are absorbed into an extended state space that still carries a unique record-forming transition into the resolved output class. If the only record-forming transition on the declared detector pipeline is $A\to L$TeXA\to L, then the local-commit rate is realized by

\Rc(t)=\Glat\,p_A(t).

Consequently, on a declared single-hit observation window $\Omega$TeX\Omega that resolves at most one durable entry into $L$TeXL, the detector-side single-hit readout probability is the window integral of the local-commit current,

P_{\mathrm{det}}(\Omega)
=
\int_{\Omega}\Rc(t)\,dt.

proof. Within the detector cycle, a recorded event occurs if and only if the activated state enters the latched sector during the declared observation window. The rate of such events is therefore the probability current into $L$TeXL, namely $\Glat p_A$TeX\Glat p_A. By contrast, the total switching current out of $R$TeXR equals reference and includes activations that subsequently relax through $A\to R$TeXA\to R without ever forming a stable record. Thus, whenever $\Grel>0$TeX\Grel>0, one generically has

\Rc(t)=\Glat p_A(t) \neq (\Gsw+\Gdark)p_R(t),

so the detector event rate cannot be identified with threshold crossing alone. If the detector output resolves the presence or absence of a latched event but not additional microstructure inside $A$TeXA, then coarse-graining over unresolved activated microstates yields $\Rc(t)=\Glat p_A(t)$TeX\Rc(t)=\Glat p_A(t). Because the declared observation window begins from the ready preparation and admits at most one durable entry into the resolved output class, integrating that entry current over $\Omega$TeX\Omega gives the corresponding detector-side single-hit readout probability reference.

Equation reference is the single-hit readout counting law for the local-commit current on the declared observation window. It applies when the readout coarse graining resolves at most one durable entry into the latched sector on $\Omega$TeX\Omega. It is not a derivation of accessible-event probability semantics or of the Born rule. Correlated multi-click, explicit afterpulsing-memory, or retriggering statistics require an extended counting model. All downstream single-hit readout-probability, quasi-steady, and benchmark-facing comparisons are read only relative to this declared single-hit observation window and the readout coarse graining fixed above.

The identities reference and reference are the exact readout-layer structural identities used in this paper; benchmark-family inversion overlays, platform tolerances, and numeric estimator checks are kept separate from this exact structural claim.

A compact analytical payoff appears already in the quasi-steady activated sector. If the activated occupation changes slowly on the readout window, one has

p_A \approx \frac{\Gsw+\Gdark}{\Glat+\Grel}\,p_R,

and therefore

\Rc
\approx
\underbrace{\frac{\Glat}{\Glat+\Grel}}_{\displaystyle \eta_{\mathrm{rec}}}
(\Gsw+\Gdark)p_R.

The local-commit rate is thus the generic switching current weighted by a record-formation efficiency

\eta_{\mathrm{rec}}\equiv \frac{\Glat}{\Glat+\Grel},

which quantifies how much of the activated flux actually enters the durable output class. This makes the analytical payoff explicit: on platforms where failed transients are substantial, $\eta_{\mathrm{rec}}<1$TeX\eta_{\mathrm{rec}}<1 and switching-based event accounting systematically overestimates recorded events. Operationally, generic trigger flux can overcount detector events whenever activated trajectories decay before entering the latched sector.

Entropy production rate

For a Markovian detector cycle with transition rates $W_{ij}$TeXW_{ij} between states $i,j\in\{R,A,L\}$TeXi,j\in\{R,A,L\}, the nonequilibrium entropy production rate is

\Sigdot
=
\sum_{i<j} J_{ij}\,
\ln\!\frac{W_{ij}p_i}{W_{ji}p_j},
\qquad
J_{ij}\equiv W_{ij}p_i-W_{ji}p_j,

which is non-negative under the standard stochastic-thermodynamic assumptions of local detailed balance [citation]. The content relevant for this paper is structural:

- maintaining the ready state out of equilibrium requires nonzero entropy production; - readout quality is therefore not independent of dissipation; - and reset cannot be regarded as thermodynamically free.

At the level needed here, readout quality, recovery speed, and dissipation are coupled rather than independently tunable. Apparatus-level thermodynamic analyses treat irreversible readout and apparatus resetting as finite-cost processes [citation], while detector benchmarks and realistic detector-response analyses exhibit the same coupling through explicit tradeoffs among afterpulsing, recovery, count rate, latching, and dead-time distortion [citation]. Those results are used only as compatibility evidence that $\Gdark$TeX\Gdark, $\taudead$TeX\taudead, and $\Sigdot$TeX\Sigdot cannot be tuned independently inside an admissible detector cycle.

The detector conditions are collected as follows.

definition: Metastable-amplification class. A detector belongs to the metastable-amplification class on a declared operating window if all of the following hold:

- Ready-state metastability: $\Gdark \tauread \ll 1$TeX\Gdark \tauread \ll 1. - Signal dominance: $\int_0^{\tauread}\Gsw(t)dt \gtrsim 1$TeX\int_0^{\tauread}\Gsw(t)dt \gtrsim 1 and dominates over $\Gdark \tauread$TeX\Gdark \tauread. - Latched amplification: $\Glat\tau_{\mathrm{out}}\gtrsim 1$TeX\Glat\tau_{\mathrm{out}}\gtrsim 1 with declared output gain $\Gain>1$TeX\Gain>1. - Entropy export: $\Sigclick>0$TeX\Sigclick>0, with at least the compatibility floor reference for a one-bit record. - Reset admissibility: $\Prep \taudead \ll 1$TeX\Prep \taudead \ll 1 for the declared repetition rate.

All downstream detector-class and benchmark-facing admissibility claims are read only relative to this declared operating window and detector-class convention.

proposition: Metastable-amplification detector admissibility on a declared operating window. A physical device functions as a detector in this framework only if it belongs to the metastable-amplification class. If any of the conditions (M1)--(M5) fail, the device may still operate as a passive sensor or threshold element, but it does not realize the irreversible detector cycle required for latched readout.

proof. The argument is classificatory. If (M1) fails, the ready state is not metastable and false positives dominate the readout window. If (M2) fails, signal-triggered switching does not dominate spontaneous activation and the device cannot be trusted as a signal detector. If (M3) fails, no stable macroscopic output is formed. If (M4) fails, no irreversible record can be maintained without contradicting the thermodynamic cost of record creation. If (M5) fails, the cycle does not reset as a reusable detector. Hence failure of any one gate removes the device from the detector class.

The criterion does not imply that all detectors have the same microscopic rates or the same barrier shape. It states only that the material devices counted as detectors here share the same thermodynamic architecture: metastable ready state, signal-assisted escape, dissipative amplification, and reset.

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

The symbol $L$TeXL is platform-agnostic throughout this section. It denotes the record-bearing resolved output class on the declared readout channel. In a TES it is the resolved electrothermal pulse delivered to the readout chain; in an SNSPD it is the nominal resolved voltage-pulse output class. Persistent self-latching that blocks reset is treated as a loss of reset admissibility rather than as the generic event state. The benchmark detector-tomography family package summarized in Table reference is fixed throughout the paper, and all downstream benchmark-facing admissibility, reduction, and numeric-conditioned comparisons are read only relative to this declared family package.

Geiger-mode avalanche devices

Geiger-mode avalanche photodiodes provide a direct example of the detector cycle. The ready state is a reverse-biased depletion region held just beyond breakdown. A signal carrier initiates avalanche multiplication, producing a large output pulse that must be actively or passively quenched before the device can recover. Dark counts, afterpulsing, and dead time are standard manifestations of the same metastable-amplification cycle rather than incidental defects [citation]. Recent gated calibrations, realistic detector-response models, and full-chip afterpulsing analyses track the same operating variables directly through tunable gate frequency, afterpulse probability, pulse-to-pulse efficiency recovery, and dead-time suppression [citation].

Operationally the device realizes all five detector conditions. In particular, the threshold picture is only the initial activation step; the quench-and-recharge cycle is what closes (M3)--(M5). Operationally:

\Fbar \leftrightarrow \text{breakdown barrier under bias},

\Gdark \leftrightarrow \text{dark-count rate},

\taudead \leftrightarrow \text{quench + recharge interval}.

A SPAD that achieves high count fidelity only by extending $\taudead$TeX\taudead or by accepting larger $\Gdark$TeX\Gdark is not anomalous; it is expressing the detector tradeoff encoded in (M1)--(M5). Active quenching can shorten $\taudead$TeX\taudead, but only by driving a tighter tradeoff with afterpulsing and dark-count control [citation].

Declared-window reduction..

Let $\Omega_n=[nT_{\mathrm g},\,nT_{\mathrm g}+\tauread]$TeX\Omega_n=[nT_{\mathrm g},\,nT_{\mathrm g}+\tauread] denote one declared gate of a gated SPAD with gate period $T_{\mathrm g}$TeXT_{\mathrm g}. When the quench/recharge schedule enforces a single-hit window and when afterpulsing or retriggering within $\Omega_n$TeX\Omega_n is negligible or encoded in an extended state space, the event count on that gate is governed by

P_{\mathrm{det}}^{(n)}
=
\int_{\Omega_n}\Rc(t)\,dt,
\qquad
\Rc^{(n)}
\approx
\eta_{\mathrm{rec}}^{(n)}(\Gsw^{(n)}+\Gdark^{(n)})p_R^{(n)}.

Time-resolved SPAD calibrations, realistic detector-response models, and full-chip afterpulsing analyses measure pulse-to-pulse detection-efficiency recovery and afterpulse probability directly as functions of gate frequency and recovery delay [citation]. On that declared window the detector event is therefore the durable quenched pulse admitted by the readout chain, not the raw microscopic threshold crossing that initiated avalanche multiplication.

Transition-edge sensors

Transition-edge sensors operate by biasing a superconducting film in the narrow region between superconducting and normal resistance, so that a tiny absorbed energy produces a large resistance change. Electrothermal feedback is therefore not an implementation detail but the gain mechanism of the detector cycle itself [citation]. Recent TES studies resolve the same structure through transition physics, dark-count characterization, and geometry-dependent parameter/noise measurements [citation]. The ready state is the bias-stabilized transition point, the activated state is the driven resistive excursion, the record-bearing output class is the resolved pulse in the readout chain, and the reset is the thermal return to the bias point.

The same validation structure appears here as well: transition biasing supplies (M1), absorbed energy secures (M2), electrothermal pulse formation gives (M3), bath coupling supplies (M4), and thermal return fixes (M5). The natural thermodynamic variables are

\Fbar \leftrightarrow \text{effective electrothermal stability margin},

\Qclick \leftrightarrow \text{heat dumped to bath per pulse},

\taudead \leftrightarrow \text{thermal recovery time}.

This is why detector performance in TES devices is inseparable from thermal conductance, bias power, and bath coupling.

Superconducting nanowire detectors

Superconducting nanowire single-photon detectors operate near a current-driven switching threshold. An absorbed photon creates a hotspot, redistributes current, and drives the nanowire into a measurable voltage state before the device cools and resets. Their dark counts, electrothermal feedback, and recovery times are naturally described by the same metastable-amplification architecture [citation]. The three-terminal electrothermal device of McCaughan and Berggren makes the switching-and-readout architecture explicit [citation]. A platform-specific tradeoff is already visible experimentally: pushing the electrical response faster to raise the count rate can stabilize electrothermal feedback and trap the device in a resistive self-latched state, destroying reusable detection [citation]. Here persistent self-latching is a loss of reset admissibility, not the nominal event state $L$TeXL. The generic detector event remains the resolved voltage-pulse output class produced by the hotspot-triggered excursion.

Borderline threshold elements and non-detectors

The criterion is meant to discriminate, not merely to redescribe successful detector families. Three borderline cases are instructive. A non-latching threshold sensor may realize the trigger step $R\to A$TeXR\to A but, if it never forms a stable output class, it fails (M3) and therefore does not define a detector event on the declared readout window; operationally, it can supply a trigger count while failing to generate a stable output-class histogram. A transient comparator that repeatedly crosses a threshold but relaxes immediately back to the ready sector without durable output fails the distinction between activation and record formation and therefore cannot identify its switching flux with $\Rc$TeX\Rc; it can report threshold crossings while failing to account for readout persistence or the dead-time envelope of a reusable detector. A one-shot analog transducer that produces a large excursion once but lacks a controlled reset path may form a signal-responsive pulse while failing (M5), so it is not an admissible repeated detector on the claimed operating rate; it may display one-shot gain while leaving no reproducible repetition window for event counting. These examples show that the criterion has discriminative bite: threshold crossing is at most a partial description of triggering, not yet a theory of irreversible readout.

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

The detector class is evaluated through explicit operational tests.

G1 (Metastability gate)..

If a claimed detector family cannot be represented by a ready state with rare spontaneous switching on the declared readout window, then it falls outside the metastable-amplification class.

G2 (Signal-over-dark gate)..

If no declared signal window exists for which $\int_0^{\tauread}\Gsw(t)dt$TeX\int_0^{\tauread}\Gsw(t)dt dominates over $\Gdark\tauread$TeX\Gdark\tauread, then the device is not functioning as a signal detector in the sense used here.

G3 (Latching gate)..

If the output pulse never forms a stable readout class on the declared output window, the device is a transient sensor but not a detector with irreversible readout.

G4 (Entropy-export gate)..

If a proposed readout architecture requires irreversible record creation while carrying parametrically vanishing $\Qclick$TeX\Qclick and $\Sigclick$TeX\Sigclick, the interpretation is rejected. This gate rules out latched readout without a thermodynamic sink.

G5 (Reset/dead-time gate)..

If repetition requires dead times incompatible with the declared operating rate, the device may still exist physically but it does not satisfy the admissible detector cycle claimed for that rate.

G6 (Boundary gate)..

If one asserts that the present detector-cycle model derives outcome probabilities, replaces unitary propagation by an intrinsic non-unitary collapse law, or certifies intersubjective classical objectivity, one has left the scope of the analysis.

These gates test only the local thermodynamic conversion of triggering into durable detector readout.

The failure conditions are therefore: Failure occurs if established detector families cannot be placed in the metastable-amplification class, if the local-commit rate cannot be realized as a latching current in an admissible detector cycle, or if a threshold-only description reproduces the same latched-output persistence and reset/dead-time architecture on the same platform without an explicit latching cycle. Failure also occurs if the coupled gain--dark-count--dead-time tradeoffs required by the detector architecture are contradicted by benchmark devices rather than merely engineered differently.

The thermodynamic amplification layer must receive a normalized event current rather than an unassigned loss term. If the pre-latch detector sink factors as $K=\sum_aL_a^\dagger L_a$TeXK=\sum_aL_a^\dagger L_a, the no-event trajectory and the jump branches form one CPTP instrument. The total jump rate $\sum_a\operatorname{Tr}(L_a^\dagger L_a\rho)$TeX\sum_a\operatorname{Tr}(L_a^\dagger L_a\rho) is then the unique probability current that may enter the metastable transition network without violating trace preservation. A resolved thermodynamic record may depend on the channel label $a$TeXa, but that dependence must be fixed by the admitted detector architecture.

This completion does not supply irreversibility for free. It only closes the probability balance at the pre-latch boundary. Persistence, entropy export, reset cost, and record stability continue to require the nonequilibrium conditions proved in this paper. A model in which the same lost norm is counted more than once, or in which amplification probabilities do not sum to the incoming jump current, fails before any thermodynamic efficiency or Landauer comparison is made.

The closure test for the detector thermodynamics and metastable amplification is applied to a dimensionless observable vector $y\in\mathbb R^m$TeXy\in\mathbb R^m formed from fixed reference scales and the declared basket of switching rates, dissipated heat, gain, error probability, and record lifetime. Let $a$TeXa range over the independent constitutive inputs comprising metastable landscape, bath spectrum, amplification channel, and reset protocol.

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

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

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

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

If the rank is constant locally, these restrictions are tangent to a compatibility manifold of codimension $m-r$TeXm-r. For this sector, the finite closure is admissible only if a single system--bath model accounts for transition rates, energy balance, amplification, and the resolved output instrument.

proof. Split surjectivity gives a bounded right inverse $R$TeXR with $D_aF\,R=I_m$TeXD_aF\,R=I_m. The Banach-space submersion theorem then makes $F$TeXF locally onto a neighborhood of the calibrated observable vector. Any smooth equality holding throughout that image must therefore vanish on an open set and contributes no model-specific local restriction. Under finite closure, the attainable first-order variations are exactly the column space of $J$TeXJ. Its orthogonal complement is $\ker J^{\mathsf T}$TeX\ker J^{\mathsf T}, whose dimension is $m-r$TeXm-r by rank--nullity, which proves reference. The constant-rank theorem supplies the stated local manifold. The sector condition is necessary because separate rate and heat fits can violate detailed energy bookkeeping and double count the incoming jump current. Failure of that condition therefore rejects the proposed microscopic closure before parameter estimation can be counted as evidence for it.

The detector cycle fixed here is $R\to A\to L\to R$TeXR\to A\to L\to R: metastable ready state, signal-assisted activation, dissipative entry into a record-bearing resolved output class, and reset. Within that cycle the detector-facing event variable is the local-commit rate $\Rc$TeX\Rc, realized in the minimal model by the latching current into the resolved output sector. Recorded-event statistics therefore need not coincide with threshold-crossing statistics when non-recording relaxation remains appreciable.

The metastable-amplification class organizes that distinction into five coupled operating requirements: metastability on the declared readout window, signal-dominated activation, dissipative latching with finite gain, positive entropy export, and reset compatible with the declared repetition rate. In the quasi-steady activated sector this yields

\Rc \approx \eta_{\mathrm{rec}}(\Gsw+\Gdark)p_R,
\qquad
\eta_{\mathrm{rec}}=\frac{\Glat}{\Glat+\Grel},

which isolates durable-record formation from internal activation. Geiger-mode avalanche photodiodes, transition-edge sensors, and superconducting nanowire detectors provide benchmark platform classes realizing that architecture on declared operating windows, whereas non-latching threshold sensors, transient comparators, and non-resetting transducers do not. The central rate laws are used as effective relations on declared platform windows rather than as universal microscopic laws across detector technologies. The analysis begins only after a declared activation channel into $A$TeXA has been fixed and is restricted to thermodynamic detector cycles and durable-record formation. Pre-latch detector-boundary realization, accessible-event probability semantics, intrinsic non-unitary collapse models, and intersubjective classical objectivity are not derived here. If no declared platform window supports signal-dominated activation, finite-gain latching, positive entropy export, and reset compatible with the claimed repetition rate, the device is excluded from the detector class rather than accommodated by enlarging the event notion.

Funding and competing interests..

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