A Fail-Closed Plateau Protocol for Residual Irreversibility Floors in Interference and Echo Observables

Experimental question and scope Interference visibility and echo amplitudes remain standard operational probes of coherence, reversibility, and control quality across electronic and photonic Mach--Zehnder interferometers, long-coherence optical platforms, neutron spin echo, and matter-wave decoherence experiments [citation]. The experimental question is whether, as isolation is increased within a declared platform family, the maximal attainable coherence continues to improve over the scanned range or instead admits a gate-stable nonzero residual-floor declaration on that scanned range.

This paper stays at the protocol level. It does not identify a microscopic completion of irreversibility, and it does not decide between environmental decoherence, intrinsic-decoherence or collapse-type dynamics, realistic-clock mechanisms, or other universal channels [citation]. Its purpose is narrower: a reproducible plateau/no-plateau protocol with gate-controlled confounder rejection, detector-envelope discrimination, revisit stability, and upper-limit reporting for interference and echo observables.

Current experimental baselines The decision problem is asymmetric. Improved local engineering may reduce the within-campaign controllable irreversibility increment, but no finite scan certifies a true zero-irreversibility boundary. The operational task is therefore to distinguish, on a declared scan window, between continued improvement of the best attainable coherence and a gate-stable plateau declaration.

Recent measurement-characterization work reinforces the need for explicit error budgets, detector-aware inference, and in situ reporting conventions rather than post-hoc curve preference [citation]. The same discipline is adopted here: detector effects enter only through the declared envelope model, its perturbation range, and the fixed family convention, while the output remains a protocol-level classification on a declared coherence observable. No imported readout-side quantity is redescribed here as a new detector microtheory, detector source law, or objectivity criterion.

Protocol elements Five ingredients are fixed explicitly:

- normalized fringe visibility and normalized echo amplitude are treated within one monotone-probe framework; - Tier 1 uses pre-registered plateau/no-plateau rules with interleaved revisits and scan-direction hysteresis checks rather than post-hoc fit preference; - detector-envelope perturbation control is mandatory, so readout ceilings are rejected before any residual-floor language is admitted; - observable definition, detector-envelope model, PSD convention, revisit rule, scan-direction interleave rule, estimator, continuation rule, and uncertainty budget are frozen before data inspection; - negative outcomes are converted into detector-aware observable-scale upper limits, with optional Bayesian support kept auxiliary.

Operational variables and notation Tier 1 fixes a campaign by declared $(L,\Tint,B)$TeX(L,\Tint,B) and varies only the remaining admissible source and operating settings $u$TeXu. Changes in $(L,\Tint,B)$TeX(L,\Tint,B) define a new campaign and enter the protocol only through Tier 2 comparisons. To avoid symbol overload, the total operational irreversibility relevant to a declared protocol setting $u$TeXu is written as

\Xiirr(u) \equiv \Xires + \Ximeas(u),

where $\Xires>0$TeX\Xires>0 is the campaign-fixed residual-floor candidate for the declared $(\Tint,L,B)$TeX(\Tint,L,B) tuple and $\Ximeas(u)$TeX\Ximeas(u) is the within-campaign increment induced by acquisition, control, detector backaction, and readout overhead.

definition: Residual irreversibility floor. A residual irreversibility floor is a strictly positive lower bound $\Xires>0$TeX\Xires>0 on the operational irreversibility budget within a declared platform class. Here $\Xires$TeX\Xires is not assumed a priori from a microscopic law; it is admitted only if a stable coherence plateau survives the declared gates, revisits, and uncertainty budget.

definition: Measurement-induced increment. For a fixed Tier 1 campaign with declared $(L,\Tint,B)$TeX(L,\Tint,B), $u$TeXu denotes the remaining admissible source and operating choices (detector operating point, estimator settings, source-stability set point, and related controls). The quantity $\Ximeas(u)$TeX\Ximeas(u) denotes the controllable irreversibility increment associated with those within-campaign choices. It is a within-campaign control label and it does not redefine the isolation-depth index $s$TeXs. Absolute calibration is not required; the protocol uses only reproducible ordering within the same apparatus family.

definition: Minimal attainable irreversibility at isolation depth $s$TeXs. For an isolation depth $s$TeXs with admissible protocol set $\mathcal U(s)$TeX\mathcal U(s), define

\Xiirrmin(s)\equiv \inf_{u\in\mathcal U(s)}\Xiirr(u).

This is the operational quantity that the plateau/no-plateau scan probes.

Within any fixed Tier 1 campaign, $s$TeXs indexes the pre-registered isolation-control vector, while $u$TeXu indexes only the remaining admissible within-campaign operating settings. The quantity $\Xires$TeX\Xires is therefore a campaign-fixed residual-floor candidate label attached to the declared $(\Tint,L,B)$TeX(\Tint,L,B) tuple, $\Ximeas(u)$TeX\Ximeas(u) is the within-campaign controllable increment, and only $\Xiirrmin(s)$TeX\Xiirrmin(s) is probed operationally by the plateau/no-plateau scan.

definition: Normalized coherence proxy. Throughout the paper, $\V(u)$TeX\V(u) denotes a normalized coherence proxy within a declared apparatus family. For interferometric platforms it is the normalized fringe visibility; for echo platforms it is the normalized echo amplitude or fidelity proxy extracted from the same acquisition contract.

definition: Declared uncertainty budget. For any reported coherence proxy $X$TeXX within a fixed apparatus family and a fixed analysis convention, the declared tolerance is

\dX^2
= \sigma_{\mathrm{stat},X}^2
+ \sigma_{\mathrm{drift},X}^2
+ \sigma_{\mathrm{det},X}^2
+ \sigma_{\mathrm{fit},X}^2,

where the four terms denote, respectively, statistical uncertainty, revisit/interleave drift, detector-envelope uncertainty (dead time, afterpulsing, latching persistence, reset-state uncertainty, or declared response-envelope uncertainty), and any fixed fit or extraction uncertainty associated with the declared estimator. No plateau, no-plateau, or upper-limit statement is admissible without an explicit budget of the form reference.

definition: Family-fixed analysis convention. A declared apparatus family is paired with a fixed analysis convention

\Cfam=(\V,\Edet,\mathcal P_{\mathrm{PSD}},\Omega_{\mathrm{rev}},\Omega_{\mathrm{dir}},q,m,\dV,\Omega_{\mathrm{cont}}),

where $\V$TeX\V is the chosen normalized coherence proxy, $\Edet$TeX\Edet is the declared detector-envelope model and perturbation range, $\mathcal P_{\mathrm{PSD}}$TeX\mathcal P_{\mathrm{PSD}} fixes the PSD estimator and its full frequency-domain convention, $\Omega_{\mathrm{rev}}$TeX\Omega_{\mathrm{rev}} fixes the revisit cadence and pairing rule, $\Omega_{\mathrm{dir}}$TeX\Omega_{\mathrm{dir}} fixes the ascending/descending interleave rule, $q$TeXq fixes the robust estimator fraction in reference, $m$TeXm is the continuation-window length used in the Tier 1 decision, $\dV$TeX\dV is the declared coherence-budget tolerance, and $\Omega_{\mathrm{cont}}$TeX\Omega_{\mathrm{cont}} records the continuation rule for no-plateau and unresolved classifications. Changing any entry of reference defines a new family rather than a refinement within the same null test. This fixed convention determines the admitted observer family for the present paper, and all downstream comparison and transfer claims are read only relative to that declared family. The protocol uses only the declared observable $\V$TeX\V and detector-envelope model $\Edet$TeX\Edet supplied by this fixed convention; it does not rederive the underlying readout architecture from detector microdynamics.

proposition: Family-fixity and frozen convention. Tier 1 and Tier 2 inferences are admissible only within one declared family-fixed convention reference. Observable definition, detector-envelope model, PSD convention, revisit rule, scan-direction interleave rule, robust-estimator fraction, continuation-window rule, and uncertainty budget are fixed before data inspection. Changing any of these items defines a new family and invalidates within-family plateau, no-plateau, and upper-limit comparisons.

Why ratios rather than ``zero-irreversibility'' claims The protocol never requires a reachable $\Xiirr=0$TeX\Xiirr=0 point. All experimental content is phrased in terms of relative improvement, plateau detection, and upper limits. Operationally, this means that the experiment is treated as a decision problem between continued improvement and gate-stable saturation over the declared scan window, not as a search for a certified zero boundary. This is essential both operationally---because scans are finite and platform-limited---and conceptually, because the protocol is meant to remain agnostic about the microscopic origin of any floor.

Visibility and echo amplitudes as monotone probes The only structural assumption needed by the protocol is that the normalized coherence proxy decreases monotonically with the total operational irreversibility budget within a declared apparatus family:

\V(u)=\V_\star F\!\big(\Xiirr(u)\big),
\qquad
\Xiirr(u)=\Xires+\Ximeas(u),

with $0<F\le 1$TeX0<F\le 1, $F'(x)\le 0$TeXF'(x)\le 0, and $F(0)=1$TeXF(0)=1. The functional form of $F$TeXF is not decision-bearing in this paper. A simple exponential,

\V(u)=\V_\star\exp[-\alpha\,\Xiirr(u)],

may be used as an engineering representative within an apparatus family, but no plateau/no-plateau decision depends on estimating $\alpha$TeX\alpha.

Model-independence of the decision logic..

The protocol is defined entirely in terms of gate-passing estimates of $\V$TeX\V, robust aggregation into $\V_{\max}(s)$TeX\V_{\max}(s), interleaved revisits, and declared tolerances. Any choice of monotone $F$TeXF is interpretive rather than decisional.

Unless an independently declared family-fixed monotone transfer map from observable coherence deficit to irreversibility budget is supplied before the scan, all plateau and upper-limit outputs remain at the observable level of $\V$TeX\V, $\Delta$TeX\Delta, and $\Delta_{\mathrm{UL}}$TeX\Delta_{\mathrm{UL}}.

Ratio observable and platform-normalization cancellation Choose a reference setting $u_{\mathrm{ref}}$TeXu_{\mathrm{ref}} in the same apparatus family with the smallest attainable measurement-induced increment under the declared acquisition contract. Define

\Rrat(u\mid u_{\mathrm{ref}})\equiv \frac{\V(u)}{\V(u_{\mathrm{ref}})}.

Then

\Rrat(u\mid u_{\mathrm{ref}})=
\frac{F\!\big(\Xires+\Ximeas(u)\big)}{F\!\big(\Xires+\Ximeas(u_{\mathrm{ref}})\big)},

which removes the platform normalization $\V_\star$TeX\V_\star exactly while retaining sensitivity to protocol-induced increments. Equations reference--reference define the declared relative-information frame for the present paper; all downstream comparison and transfer claims are read only relative to this fixed frame on one declared campaign family. Under the exponential representative reference, the ratio simplifies to

\Rrat(u\mid u_{\mathrm{ref}})=\exp\!\big[-\alpha\big(\Ximeas(u)-\Ximeas(u_{\mathrm{ref}})\big)\big],

but again this simplification is optional and not part of the plateau decision.

Declared asymptotic saturation hypothesis under increased isolation A bounded-support asymptotic organizer consistent with a Tier 1 plateau is

\lim_{s\to\infty}\Xiirrmin(s)=\Xires>0,

which would imply

\lim_{s\to\infty}\V_{\max}(s)=\V_\star F(\Xires)<\V_\star.

The experimental question is therefore not whether one can reach ``perfect reversibility'', but whether the maximal attainable coherence admits a reproducible plateau on the scanned isolation window under increasing isolation.

Isolation depth as an action-indexed control Isolation depth $s$TeXs is not an arbitrary quality score. Each $s_k$TeXs_k must correspond to a declared control vector over the same platform class:

- vibration/acceleration isolation, - thermal stabilization, - vacuum or pressure control when relevant, - electromagnetic and electrical isolation, - detector operating-point stabilization.

Only the ordering of these pre-registered control vectors is used in the analysis. Any change in control definition or estimator convention starts a new campaign.

Declared bandwidth descriptor $B$TeXBB Tier 2 is bounded-support: it never overrides a Tier 1 plateau/no-plateau/unresolved declaration and it never rescues a failed Tier 1 gate. The protocol allows one declared bandwidth descriptor $B$TeXB per campaign:

- end-to-end electronics bandwidth of the readout chain, - passband width of the fixed digital filter used for fringe/echo extraction, - stabilization-loop bandwidth when the control loop itself limits the accessible coherence.

The choice must be recorded and frozen for the full scan. In the default protocol, $B$TeXB is retained as part of the campaign label and detector-envelope context; it is not a default sign-bearing Tier 2 axis unless a family-specific orientation for $B$TeXB is pre-registered before acquisition.

Minimal monotone scaling class Tier 2 compares campaigns with different fixed $(\Tint,L,B)$TeX(\Tint,L,B) values; within any one Tier 1 campaign, $\Xires$TeX\Xires is treated as the constant label $\Xires(\Tint,L,B)$TeX\Xires(\Tint,L,B). To keep the scaling layer operational, we do not commit to a unique microscopic formula for $\Xires$TeX\Xires. Instead, we admit a minimal monotone class

\Xires(\Tint,L;B,\ldots) \sim c_0 + c_T f_T(\Tint)+c_L f_L(L),

with $c_0>0$TeXc_0>0 and nonnegative monotone functions $f_T,f_L$TeXf_T,f_L over the declared scan window. The default Tier 2 sign-order test is therefore carried only by $\Tint$TeX\Tint and $L$TeXL at fixed declared $B$TeXB. The bandwidth descriptor $B$TeXB remains recorded with every campaign, but no default monotonicity claim in $B$TeXB is admitted without an independently pre-registered family-specific orientation. This is sufficient to generate falsifiable plateau-shift inequalities without committing to a microscopic fit class.

Concrete, falsifiable cases Two simple specializations are used for reporting purposes. They are declared reporting organizers within the admitted monotone class rather than unique microscopic laws.

Case A (interaction-window scaling)..

\Xires(\Tint)\approx c_0+c_T\,\Tint,

so that any asymptotic plateau deficit should be nondecreasing with increasing interaction window, up to the gate-defined uncertainty budget.

Case B (path-length scaling)..

\Xires(L)\approx c_0+c_L\,L,

so that longer propagation length should not improve the asymptotic plateau once confounders are fixed.

Minimal monotonicity predictions

definition: Admitted plateau estimator. For a fixed campaign label $(\Tint,L,B)$TeX(\Tint,L,B) that returns a Tier 1 plateau under the declared gate discipline, let $\Vplat(\Tint,L,B)$TeX\Vplat(\Tint,L,B) denote the robust plateau estimator extracted from the final admitted plateau window of that campaign. No extrapolation beyond the scanned isolation range is introduced in this notation.

Define the admitted plateau deficit

\Delta(\Tint,L;B)\equiv -\ln \Vplat(\Tint,L,B).

Under the same family-fixed convention, the minimal monotone class yields the following bounded-support ordering expectation:

\Delta(\Tint_2,L;B)\ge \Delta(\Tint_1,L;B)\quad (\Tint_2>\Tint_1),
\qquad
\Delta(\Tint,L_2;B)\ge \Delta(\Tint,L_1;B)\quad (L_2>L_1),

within the gate-defined uncertainty budget. These ordering relations are asserted only across campaigns reduced under the same normalized observable, the same detector-envelope convention, the same PSD convention, the same estimator, and the same continuation rule; they are bounded-support comparisons and not mechanism-identifying laws. No default monotonicity statement in $B$TeXB is admitted in the present protocol; $B$TeXB is retained as a declared campaign descriptor unless a separate family-level orientation for $B$TeXB is pre-registered before acquisition. A violation under interleaved, fail-closed control constrains the residual-floor scaling hypothesis for that platform class.

Platform classes and recommended implementation The protocol is intended for two platform classes.

- Primary demonstration class: fiber-based or integrated photonic Mach--Zehnder interferometers, where high statistics, explicit path-length control, phase locking, and stable visibility estimation are straightforward [citation]. - Secondary extension class: echo-based platforms (NMR Loschmidt echo, spin echo, Ramsey echo, neutron spin echo, or related reversal experiments) where the same logic can be applied to a normalized echo amplitude rather than a fringe contrast [citation].

The primary platform recommendation is pragmatic rather than exclusive. Minimal implementation sketches for both classes are given in Appendix reference.

Representative worked family: gated photonic Mach--Zehnder with SPAD readout To close one apparatus family end-to-end, fix a declared photonic family $\Fphot$TeX\Fphot consisting of an unbalanced fiber or integrated Mach--Zehnder interferometer, a phase-locking loop, a fixed source wavelength and pulse format, and gated InGaAs/InP SPAD readout operated under one declared quench/reset architecture [citation].

For $\Fphot$TeX\Fphot, the family-fixed analysis convention is

\Cfam_{\mathrm{MZI}}=(V_{\mathrm{fr}},\Edet^{(\mathrm{SPAD})},\mathcal P_{\mathrm{PSD}},\Omega_{\mathrm{rev}},\Omega_{\mathrm{dir}},q,m,\dV,\Omega_{\mathrm{cont}}),

where $V_{\mathrm{fr}}$TeXV_{\mathrm{fr}} is the normalized fringe visibility, $\Edet^{(\mathrm{SPAD})}$TeX\Edet^{(\mathrm{SPAD})} includes the declared gate width, dead-time setting, afterpulse-correction model, and reset/quench configuration, $\mathcal P_{\mathrm{PSD}}$TeX\mathcal P_{\mathrm{PSD}} fixes the full PSD convention, $\Omega_{\mathrm{rev}}$TeX\Omega_{\mathrm{rev}} fixes the revisit cadence and pairing rule, $\Omega_{\mathrm{dir}}$TeX\Omega_{\mathrm{dir}} fixes the ascending/descending interleave rule, $q$TeXq fixes the top-quantile estimator in reference, $m$TeXm fixes the continuation-window length, and $\dV$TeX\dV is built from the budget reference. For the worked SPAD family, $\Edet^{(\mathrm{SPAD})}$TeX\Edet^{(\mathrm{SPAD})} also records the admitted incident-rate or count-rate domain under which dead-time, afterpulse, and reset-state calibration were obtained; changing that domain defines a new detector-envelope context rather than a refinement within the same family. In this family, only the isolation vector $s$TeXs is scanned inside Tier 1; all entries of reference are fixed before acquisition. A detector-aware uncertainty budget is then reported as

\dV^2=\sigma_{\mathrm{stat}}^2+\sigma_{\mathrm{drift}}^2+\sigma_{\mathrm{det}}^2+\sigma_{\mathrm{fit}}^2,

where $\sigma_{\mathrm{det}}$TeX\sigma_{\mathrm{det}} is extracted from the declared detector-envelope perturbation scan and includes the sensitivity of the corrected visibility to dead time, afterpulse subtraction, and reset/quench settings [citation].

The worked family closes as follows. First, acquire gate-passing visibility blocks at ordered isolation depths $s_1<\cdots<s_n$TeXs_1<\cdots<s_n using the fixed convention reference. Second, perform a short detector-envelope perturbation scan at fixed $(L,\Tint,B,s_k)$TeX(L,\Tint,B,s_k) to determine $\sigma_{\mathrm{det}}$TeX\sigma_{\mathrm{det}} and reject device-ceiling plateaus. Third, apply the deterministic Tier 1 rule with the same $\mathcal P_{\mathrm{PSD}}$TeX\mathcal P_{\mathrm{PSD}}, the same $\Omega_{\mathrm{rev}}$TeX\Omega_{\mathrm{rev}}, the same $\Omega_{\mathrm{dir}}$TeX\Omega_{\mathrm{dir}}, the same $q$TeXq, the same $m$TeXm, and the same $\dV$TeX\dV across the entire campaign. If the final window is plateau-like, report the plateau estimator $\Vplat(\Tint,L,B)$TeX\Vplat(\Tint,L,B) together with revisit residuals and detector-envelope margins. If the final window is no-plateau or unresolved, report instead the detector-aware upper-limit package consisting of the largest gate-passing $\widehat{V}_{\max}^{(q)}$TeX\widehat{V}_{\max}^{(q)}, the observable-scale upper-limit statistic $\Delta_{\mathrm{UL}}$TeX\Delta_{\mathrm{UL}}, the final-window span, the revisit residuals, and the scanned detector-domain margins under the same fixed convention.

Protocol outputs by tier

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

Observable definitions

definition: Maximal attainable coherence at isolation depth $s$TeXs. For fixed apparatus settings $(\Tint,L,B)$TeX(\Tint,L,B), define

\V_{\max}(s)\equiv \max_{u\in\mathcal U(s)}\V(u;s),

where $\mathcal U(s)$TeX\mathcal U(s) is the admitted scan set at isolation depth $s$TeXs. Operationally, $\V_{\max}(s)$TeX\V_{\max}(s) is estimated only from gate-passing blocks.

Robust estimator..

To avoid a ``maximum-as-luck'' objection, the protocol reports both the raw maximum and a robust top-quantile estimator. Let $\{\V_i(s)\}_{i=1}^{N_s}$TeX\{\V_i(s)\}_{i=1}^{N_s} be the admissible coherence estimates at depth $s$TeXs. Fix a pre-registered fraction $q\in(0,1)$TeXq\in(0,1) and define

\widehat{\V}_{\max}^{(q)}(s)
\equiv
\frac{1}{\lceil qN_s\rceil}
\sum_{i\in \mathrm{Top}(q)} \V_i(s),

where $\mathrm{Top}(q)$TeX\mathrm{Top}(q) denotes the indices of the largest $\lceil qN_s\rceil$TeX\lceil qN_s\rceil admissible values. Unless stated otherwise, $\V_{\max}(s)$TeX\V_{\max}(s) refers to reference.

Confounder gate A plateau is admitted into inference only if the following channels pass a pre-registered gate:

- Path-balance / loss gate: no unresolved arm-loss or channel-imbalance drift large enough to mimic a visibility ceiling. - PSD / phase-noise gate: the same declared PSD convention, band limits, segment length, and drift cut are used at every $s$TeXs level [citation]. - Source-stability gate: source spectrum, pulse shape, and intensity statistics remain within declared tolerances. - Detector-linearity gate: no detector compression or saturation regime can account for the observed ceiling. - Detector-envelope gate: dead time, afterpulsing, latching persistence, quench/recovery conditions, reset architecture, and declared detector-response modeling do not generate the same apparent plateau structure that would otherwise be reported as a gate-stable residual-floor declaration on the declared scan window [citation]. - Revisit gate: interleaved revisits to earlier $s$TeXs levels reproduce earlier $\V_{\max}$TeX\V_{\max} values within the propagated uncertainty budget.

If any gate fails, the corresponding block is excluded from plateau inference rather than interpreted.

Detector-envelope perturbation control and scan-direction hysteresis The detector-envelope gate is not passive bookkeeping. At fixed $(L,\Tint,B,s)$TeX(L,\Tint,B,s) the protocol requires a short calibration sub-protocol in which detector operating point, declared dead time, quench/recovery setting, or reset conditions are perturbed within a safe operating interval. A plateau candidate is admissible only as a protocol-level residual-floor declaration if, after the same gate renormalization is applied, the corresponding $\V_{\max}$TeX\V_{\max} and final-window spread remain invariant within the propagated uncertainty budget under these perturbations. Strong drift with dead time, quenching, reset, afterpulse envelope, or latching persistence is evidence for a device ceiling rather than for a gate-stable residual-floor declaration on the declared scan window.

In addition, the isolation scan is not trusted in a single direction only. Ascending- and descending-isolation scans must be interleaved or alternated under the same gate convention. A plateau candidate that depends on scan direction beyond the declared tolerance is rejected as a hysteretic technical ceiling rather than interpreted as a protocol-admitted residual-floor declaration.

Detector-envelope validity condition..

The deterministic Tier 1 plateau/no-plateau rule is admitted only when dead time, afterpulsing, latching persistence, and reset memory are either negligible on the declared event window after the detector-envelope correction or already absorbed into the fixed detector model $\Edet$TeX\Edet used in the uncertainty budget. If detector-memory effects cannot be stabilized within the declared perturbation range, the corresponding family is reported only through upper limits and gate failures rather than through a plateau declaration.

remark: Detector-envelope scope. Detector-side readout structure enters this paper only through the declared envelope model $\Edet$TeX\Edet, its perturbation range, and the fixed family convention $\Cfam$TeX\Cfam. No statement in this protocol promotes dead time, afterpulsing, reset memory, latching persistence, or quench dynamics into a microscopic detector theory, a detector source law, or an objectivity criterion.

Tier 1: Saturation test protocol Choose fixed $(L,\Tint,B)$TeX(L,\Tint,B) and increasing isolation depths $s_1<\dots <s_n$TeXs_1<\dots <s_n. For each $s_k$TeXs_k, scan admissible protocol settings $u\in\mathcal U(s_k)$TeXu\in\mathcal U(s_k), estimate $\V_{\max}(s_k)$TeX\V_{\max}(s_k), and apply the same gates, revisit rule, and scan-direction hysteresis check.

Deterministic plateau test..

Choose integers $m\ge 3$TeXm\ge 3 and an absolute tolerance $\dV>0$TeX\dV>0 in advance. For the declared family, $\dV$TeX\dV is the detector-aware budget reference when the worked photonic family is used, or the corresponding budget reference for any other declared family. The same $m$TeXm, the same estimator fraction $q$TeXq, and the same detector-envelope convention are held fixed across the entire campaign. A plateau is declared over the last $m$TeXm steps only if

- all $m$TeXm points pass the same gate convention; - the revisit condition $|\V_{\max}(s_k)-\V_{\max}(s_{k-2})|\le \delta_{\V}$TeX|\V_{\max}(s_k)-\V_{\max}(s_{k-2})|\le \delta_{\V} holds on the corresponding revisit pairs under the same fixed convention; - the drift across the last $m$TeXm steps is bounded: equation \max_jn-m+1,,n\V_(s_j)-\min_jn-m+1,,n\V_(s_j) \delta_, equation - the same last-window decision survives the ascending/descending scan interleave within the same tolerance.

A no-plateau outcome is declared when the same final window exhibits persistent improvement,

\V_{\max}(s_j)-\V_{\max}(s_{j-1})\ge \delta_{\V}
\qquad
\text{for all }j\in\{n-m+1,\dots,n\},

with revisit consistency retained. If neither the plateau rule nor the persistent-improvement rule is satisfied, the Tier 1 outcome is unresolved. In that case the scan is extended or the campaign is reported only through its current upper-limit sensitivity and gate record.

Optional Bayesian decision support The primary plateau/no-plateau declaration is always the deterministic rule above. As a secondary decision aid, one may compare two low-complexity families on the same gate-passing dataset $\mathcal D_G$TeX\mathcal D_G: a saturating monotone family $M_{\mathrm{sat}}$TeXM_{\mathrm{sat}} and a persistently improving monotone family $M_{\mathrm{imp}}$TeXM_{\mathrm{imp}}. Define the optional evidence ratio

\mathcal B_{\mathrm{sat/imp}} \equiv \frac{p(\mathcal D_G\mid M_{\mathrm{sat}})}{p(\mathcal D_G\mid M_{\mathrm{imp}})}.

This ratio is not itself a plateau declaration and it is not a microscopic mechanism fit. Its role is narrower: to report whether the gate-passing dataset offers auxiliary support for a saturating or persistently improving monotone description under the same fail-closed protocol. The evidence ratio is auxiliary only and never overrides the deterministic Tier 1 state.

Tier 2: Relative plateau-shift tests Tier 2 does not optimize a microscopic fit. Its purpose is to test whether relative plateau shifts remain stable in sign and ordering under the same gate discipline when one varies interaction window or path length across campaigns that have already returned an admitted Tier 1 plateau at fixed declared $B$TeXB. The declared comparison class for Tier 2 therefore consists only of campaigns that have already returned a Tier 1-admitted plateau within one fixed family and at one fixed declared bandwidth descriptor $B$TeXB; all downstream sign and ordering claims are read only relative to that class. For a baseline setting $(\Tint_0,L_0)$TeX(\Tint_0,L_0) in the same campaign family and at the same declared $B$TeXB, define

\Delta_{\mathrm{rel}}(\Tint,L\mid \Tint_0,L_0;B)
\equiv
-\ln\!\left[\frac{\Vplat(\Tint,L,B)}{\Vplat(\Tint_0,L_0,B)}\right].

In Tier 2, $\Vplat(\Tint,L,B)$TeX\Vplat(\Tint,L,B) again denotes the admitted plateau estimator returned by Tier 1 at fixed $(\Tint,L,B)$TeX(\Tint,L,B), not a free extrapolation beyond the scanned isolation range. Equation reference is the declared frame-transfer observable for cross-campaign sign and ordering tests; all downstream transfer claims are read only relative to this observable, the declared comparison class, and the admitted observer family fixed above. The decisive content is whether $\Delta_{\mathrm{rel}}$TeX\Delta_{\mathrm{rel}} preserves the monotone constraints of reference under interleaving and gate control, as expected for a non-removable background contribution rather than for a platform-specific ceiling. The default protocol does not assign a sign-bearing monotonicity claim to $B$TeXB; bandwidth changes may define a new campaign family or a separately pre-registered comparison, but they do not enter the default Tier 2 inequality test. Tier 2 is bounded-support: it organizes relative plateau shifts only after Tier 1 has admitted a plateau, and it never overrides a Tier 1 no-plateau or unresolved classification.

proposition: Operational falsification criteria for the declared fail-closed plateau protocol. For a declared platform class, the residual-floor declaration is constrained or rejected if any of the following occurs:

- no admissible plateau declaration survives the Tier 1 plateau test and the final scan window satisfies the symmetric no-plateau rule reference; - an apparent plateau disappears under interleaved revisits or under a change of PSD convention that should have been irrelevant within the declared gate tolerance; - the same apparent ceiling is explained by detector-side readout-envelope limits (dead time, afterpulsing, latching persistence, reset architecture), or moves under detector-envelope perturbation control rather than with the isolation scan itself; - the relative plateau shifts in Tier 2 violate the monotonicity constraints of reference within the propagated uncertainty budget.

Sensitivity and upper limits A no-plateau or unresolved outcome is not empty. It yields an observable-scale upper-limit statement tied to the scanned range and the declared tolerances.

definition: Observable-scale plateau-deficit upper limit. For a no-plateau or unresolved campaign with largest gate-passing robust estimator $\widehat{\V}_{\max}^{(q)}$TeX\widehat{\V}_{\max}^{(q)} and coherence-budget tolerance $\dV$TeX\dV, define

\Delta_{\mathrm{UL}}(L,\Tint;B)\equiv
-\ln\!\Big(\max\{\widehat{\V}_{\max}^{(q)}-\dV,\varepsilon\}\Big),

where $\varepsilon>0$TeX\varepsilon>0 is a fixed numerical floor declared only for stable reporting.

Equation reference is an observable-scale upper limit on any unresolved plateau deficit within the scanned isolation domain. It is transported to an irreversibility-budget bound only when a family-fixed monotone transfer map from coherence deficit to irreversibility is declared independently of the scan.

The report must include, at minimum,

- the largest gate-passing robust estimator $\widehat{\V}_{\max}^{(q)}$TeX\widehat{\V}_{\max}^{(q)}, the chosen top-quantile fraction $q$TeXq, and the corresponding $\Delta_{\mathrm{UL}}$TeX\Delta_{\mathrm{UL}} value; - the final-window tolerance $\delta_{\V}$TeX\delta_{\V} and the final-window span used in the plateau/no-plateau test; - the revisit residuals for the last accepted revisits; - the scanned $(\Tint,L,s)$TeX(\Tint,L,s) domain, the declared $B$TeXB, and the corresponding detector operating-point domain; - any gate failures or exclusions and the retained detector-envelope margin; - when used, the optional evidence ratio $\mathcal B_{\mathrm{sat/imp}}$TeX\mathcal B_{\mathrm{sat/imp}}.

These items define the unresolved plateau-deficit bound for the declared campaign. The upper-limit statement is meaningful only relative to the same family-fixed convention reference; changing the detector envelope, the PSD convention, revisit pairing, direction interleave, estimator, or continuation rule produces a different campaign family rather than a stronger limit inside the same one. This reporting style follows the use of detector-aware Bayesian sensitivity and upper-limit statements in current interferometric and measurement-characterization practice [citation].

Interpretation of a positive result A positive result would mean only this: on the declared scanned domain and within the declared platform class and gate discipline, the data admit a gate-stable nonzero residual-floor declaration and stable monotone plateau-shift trends with controllable laboratory parameters under a fixed analysis convention. Without an independently declared transfer map, the reported plateau deficit remains an observable-scale quantity rather than a direct estimator of $\Xires$TeX\Xires. It would not by itself supply a detector source law, identify a unique microscopic mechanism, or certify objectivity or semantic closure. Environmental decoherence, intrinsic decoherence, realistic-clock effects, collapse models, and other universal channels would remain distinguishable only by further platform-specific analysis.

A fail-closed residual-floor protocol has been specified for interference and echo observables on declared scan windows. Its admissible outputs are plateau/no-plateau/unresolved decisions, detector-envelope discrimination, explicit uncertainty budgets, detector-aware observable-scale upper limits, and Tier 2 relative plateau-shift tests under one declared family-fixed convention. The present analysis is a fail-closed protocol on declared coherence observables. It does not assume a reachable zero-irreversibility boundary and it does not identify a unique microscopic mechanism from a positive plateau. It distinguishes continued improvement from a gate-stable plateau declaration over the declared scan window. Residual-floor language is admitted only when the candidate plateau survives PSD-convention control, revisit stability, scan-direction hysteresis tests, and detector-envelope perturbations; otherwise the output is a constrained upper-limit statement. The admissible output of the present analysis is a protocol-level plateau/no-plateau/unresolved classification on declared coherence observables, together with detector-aware upper limits when no plateau is admitted; without an independently declared transfer map, no direct irreversibility-budget estimator is produced. The analysis does not identify a microscopic irreversibility mechanism, it does not replace detector microdynamics or supply a detector source law, and it does not certify detector-side objectivity, semantic closure, or shared facts.

For every campaign, the following items are frozen before acquisition:

- the PSD estimator (Welch or equivalent), frequency band $(f_{\min},f_{\max})$TeX(f_{\min},f_{\max}), segment length $\tau_{\mathrm{seg}}$TeX\tau_{\mathrm{seg}}, taper/window choice, and drift-cut rule; - the arm-balance or channel-balance tolerance; - the source-stability tolerance; - the detector operating-point and linearity tolerance; - the detector-envelope tolerance for dead time, afterpulsing, quench/recovery, and reset persistence; - the revisit cadence, hysteresis pass criterion, and perturbation-control pass criterion.

A block is admissible only if all six gate channels pass under the same frozen convention.

The Tier 1 output for each campaign consists of:

- the family-fixed convention reference together with the chosen detector-envelope model; - the robust estimator $\widehat{\V}_{\max}^{(q)}(s)$TeX\widehat{\V}_{\max}^{(q)}(s) for every scanned depth; - the plateau witness defined by the last-window spread in reference; - the no-plateau witness defined by the minimum final-window improvement in reference; - the revisit residuals and their comparison to $\delta_{\V}$TeX\delta_{\V}; - the scan-direction hysteresis residuals and perturbation-control residuals; - the gate pass/fail table for every scanned depth; - the explicit uncertainty budget $\dV^2=\sigma_{\mathrm{stat}}^2+\sigma_{\mathrm{drift}}^2+\sigma_{\mathrm{det}}^2+\sigma_{\mathrm{fit}}^2$TeX\dV^2=\sigma_{\mathrm{stat}}^2+\sigma_{\mathrm{drift}}^2+\sigma_{\mathrm{det}}^2+\sigma_{\mathrm{fit}}^2 used in the decision; - the final classification state (plateau, no-plateau, or unresolved); - when Tier 1 does not admit a plateau, the observable-scale upper-limit statistic $\Delta_{\mathrm{UL}}$TeX\Delta_{\mathrm{UL}} of reference; - the upper-limit checklist items listed in Sec. reference.

When no plateau is seen or the final window remains unresolved, the campaign must report the best achieved coherence and the corresponding sensitivity floor rather than a post-hoc interpretation of ``almost saturated'' behavior.

For a declared family-fixed convention, the Tier 1 decision order is fixed and exhaustive:

- discard all gate-failing blocks; - compute the robust estimator $\widehat{\V}_{\max}^{(q)}(s_k)$TeX\widehat{\V}_{\max}^{(q)}(s_k) from the retained blocks only; - evaluate revisit residuals under the same fixed convention; - evaluate ascending/descending hysteresis under the same fixed convention; - if any gate, revisit, or hysteresis check fails, return gate-failure or upper-limit only; - otherwise test the plateau rule reference; - otherwise test the no-plateau rule reference; - if neither rule is satisfied, return unresolved.

No post-hoc re-ordering of these tests is admissible within the same declared family.

Minimal scan-log schema..

Each admitted or rejected acquisition block is recorded by one row containing

(\texttt{scan\_id},\Tint,L,B,s_k,u_i,\texttt{gate\_pass},\texttt{detector\_state},

\qquad \widehat{\V}_i,\delta_{\V,i},\texttt{revisit\_tag},\texttt{direction\_tag},\texttt{classification}),

where detector_state minimally records the dead-time setting, afterpulse-correction model, quench/reset architecture, and the admitted incident-rate or count-rate domain used by the detector-envelope model.

Minimal Tier 1 reporting template..

A report that claims plateau, no-plateau, or unresolved status must include: the frozen family convention; the gate table; the ordered list of $\widehat{\V}_{\max}^{(q)}(s_k)$TeX\widehat{\V}_{\max}^{(q)}(s_k) values; the revisit and hysteresis residuals; the explicit uncertainty budget; the final Tier 1 classification; and, when Tier 1 does not admit a plateau, the best achieved coherence together with $\Delta_{\mathrm{UL}}$TeX\Delta_{\mathrm{UL}} and the corresponding detector-aware upper-limit statement.

Tier 2 is intentionally narrower than a microscopic fit. Its admissible outputs are chosen to test cross-regime sign and ordering stability in $\Tint$TeX\Tint and $L$TeXL at fixed declared $B$TeXB rather than to identify a unique mechanism:

- the plateau-ratio statistic $\Delta_{\mathrm{rel}}$TeX\Delta_{\mathrm{rel}} of reference; - the sign of the monotonicity inequalities in reference; - whether the same trend survives interleaving and gate control; - when used, the evidence ratio $\mathcal B_{\mathrm{sat/imp}}$TeX\mathcal B_{\mathrm{sat/imp}} as auxiliary decision support.

Bandwidth metadata remain part of the campaign record, detector-envelope context, and companion source summary, but they are not a default sign-bearing Tier 2 axis in the present paper. Parametric fits may be shown as descriptive summaries, but no claim in this paper depends on selecting one microscopic model over another. Tier 2 remains bounded-support and never rescues a failed Tier 1 gate.

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-role/status table for the central equations of the residual-floor protocol. center

Photonic Mach--Zehnder primary class..

For a photonic Mach--Zehnder implementation [citation], $s$TeXs denotes the ordered isolation vector over vibration, thermal stabilization, shielding, and source stabilization; $L$TeXL is the controlled path-length imbalance or equivalent propagation length; $\Tint$TeX\Tint is the acquisition or integration window used to estimate visibility; and $B$TeXB is the declared detector/electronics or digital-filter bandwidth descriptor that is frozen per campaign and reported with the detector-envelope model. The detector-envelope record also includes the admitted incident-rate or count-rate domain under which dead-time, afterpulse, and reset-state calibration were obtained. A revisit is a blinded return to an earlier $s$TeXs level under the same arm-balance, PSD, detector settings, and admitted count-rate domain. Detector-envelope failure means that the apparent plateau shifts under a declared change in detector dead time, afterpulse regime, quench/recovery setting, reset condition, or admitted count-rate domain.

Echo / neutron-spin-echo extension class..

For echo implementations [citation], $s$TeXs denotes the ordered isolation vector over field stability, thermal drift control, shielding, pulse-program stability, and detector or readout stabilization; $L$TeXL is the declared propagation or encoding-length proxy used for the campaign; $\Tint$TeX\Tint is the measurement window used to estimate the normalized echo amplitude; and $B$TeXB is the declared detection or filtering bandwidth descriptor recorded with the campaign rather than used as a default sign-bearing Tier 2 axis. A revisit is a blinded return to an earlier isolation level under the same pulse sequence, estimator convention, and detector settings.

Funding and competing interests..

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