Specification acceptance

Acceptance of v1.0 means the proposed audit structure is well-defined: independently specify R from non-loop channels with explicit feature-disjointness rules in omics settings; predeclare the formalism (single hysteron, Preisach density, continuous Preisach analog, bistable basin-geometry / Kramers-type model) and the scalar response coordinate y = f(x) before loop area is computed; estimate the predicted slope c (or, in the continuous regime, the predicted A_static curve) from independent calibration, mechanistic parameter estimates, simulation truth, or a held-out training loop family disjoint from the test loop family; verify major-loop validity from pretest threshold calibration or held-out switching measurements rather than from the loop being tested; report sweep rate(s) and the rate-extrapolated quasi-static loop area where applicable; compare the audited-R loop-area curve A_qs(R) (continuous regime) or A(R) (discrete regime) against the predicted slope or curve. The structural specification can be accepted independently of whether the diagnostic outperforms standard hysteresis-summary statistics on any specific dataset.

Performance acceptance

Acceptance of the protocol's performance — a stronger claim than acceptance of the specification — requires the frozen synthetic demonstrations of §4 and a downstream real-data application that satisfies the dataset prerequisites listed in §5. v1.0 presents the synthetic demonstrations at release quality. Real-data execution against the rice transcriptome heat-stress/recovery dataset (Jones 2026h, GSE74793) is the work of a future T16 demonstration paper, with the limitation that the rate audit (Step 3) is not executable on that dataset.

Rejection criterion

The protocol should be revised or rejected if R, the formalism choice, the response coordinate y, or the predicted curve A_pred cannot be specified without circular use of the loop being explained; if controlled synthetic cases fail to distinguish record-driven hysteresis from the named confounds (rate-driven apparent loops, instrument hysteresis, minor-loop violation, circular R-fit, additive offset confounds, record-modulated rate lag); if the response coordinate used to compute loop area is selected from the same data used to test loop-area scaling, unless a held-out validation design is used; or if the continuous-regime formulation cannot detect record-modulated rate lag (a system with positive finite-rate loop area but A_qs(R) = 0) when sweep-rate extrapolation is available. Acceptance does not require that all confounds be disambiguated from a single sweep alone; some require multi-rate sweeps, external R proxies, or external response-coordinate calibration, and the protocol's sharpness is conditional on those external channels being available.

2.1 Discrete-state regime: record-shifted thresholds imply linear scaling

Setup. Let x_t ∈ {−1, +1} be a binary state evolving under a continuous constraint signal K_t ∈ ℝ. Let R ≥ 0 be a scalar summary of accumulated record state, held fixed during a single sweep; multi-sweep regimes in which R updates as records accumulate are out of scope for this protocol (Jones 2026e §3). Let θ₀ ≥ 0 and α ≥ 0 be parameters with θ₀ + αR > 0 in the nondegenerate case. The state evolves under the switching rule x_{t+} = +1 if K_t ≥ θ_+(R) := θ₀ + αR; x_{t+} = −1 if K_t ≤ θ_−(R) := −θ₀ − αR; x_{t+} = x_t otherwise. The interval [θ_−(R), θ_+(R)] is the memory zone. A closed major sweep is a quasi-static trajectory in which K increases monotonically from K_min ≤ θ_−(R) to K_max ≥ θ_+(R), then decreases monotonically back to K_min, with initial state x(0) = −1 and no additional reversals. The loop area is A(R) := |∮ x dK| under such a sweep.

Theorem 1 (Jones 2026e). Under the setup above with θ₀ ≥ 0, α ≥ 0, R ≥ 0, and any closed major sweep K(t) with initial state x(0) = −1, the absolute loop area satisfies A(R) = 4θ₀ + 4αR, with slope dA/dR = 4α, strictly positive whenever α > 0. In the limit θ₀ = 0 and αR = 0, the memory zone collapses and A = 0, recovering reversible response. The (K, x) trajectory traces a counterclockwise rectangle of horizontal extent 2θ₀ + 2αR and vertical extent 2; the absolute enclosed area is 4θ₀ + 4αR. Full geometric and signed-integral derivation is in Jones 2026e §2.

Two identifiable objects. Two parameters organize the discrete-regime diagnostic. The intrinsic loop area A₀ := 4θ₀ fixes the loop area at zero record state. The record-coupling slope c := 4α = dA/dR fixes the rate at which loop area grows with audited R. The discrete-state falsifier compares an empirical fit ĉ from A(R) ≈ Â₀ + ĉR over the audited R range against an independently determined c — from mechanistic calibration, simulation truth, or a held-out training loop family disjoint from the test loop family. Slope-independence rules are specified in §3.4. The intrinsic-area object Â₀ is reported alongside ĉ but is not itself the falsifier; a fitted Â₀ that disagrees with an independent estimate of 4θ₀ is informative as a consistency check but does not by itself reject record coupling, since Â₀ also absorbs additive instrument-hysteresis offsets and rate-driven contributions at the nominal sweep rate (§4.1 regimes B and C).

Heterogeneous and Preisach extensions. The single-hysteron lemma is the base case for the general Preisach model (Mayergoyz 2003; Brokate & Sprekels 1996), which represents a system as a weighted superposition of independent hysterons indexed by threshold pair (θ_+, θ_−) against a Preisach density ρ(θ_+, θ_−). Under nonnegative density and a record update that monotonically widens active threshold gaps within the swept range, aggregate loop area inherits monotone growth from the single-hysteron case, and the predicted slope is obtained by integrating against ρ. Arbitrary translations, scalings, or asymmetric reweightings of the density need not preserve monotonicity without further assumptions. In the Preisach case the diagnostic compares ĉ against the density-integrated predicted slope, not against 4α; the principle is identical, the predicted scalar differs.

Out-of-scope regimes. Critical-disorder and avalanche regimes — random-field Ising model near critical disorder (Dahmen & Sethna 1996), crackling-noise systems (Sethna, Dahmen & Myers 2001) — produce power-law avalanche distributions and scaling behavior in which loop area does not satisfy a linear lemma; the discrete-state diagnostic does not apply unmodified. Return-point memory (Sethna et al. 1993) is consistent with the discrete-state lemma but neither implies it nor is implied by it; the diagnostic does not require a return-point-memory check, and many continuous-state systems exhibit S3 without exhibiting return-point memory in any strict form.

2.2 Continuous-state regime: rate-extrapolated quasi-static loop scaling

Setup. Let x(t) ∈ ℝ^p be a continuous (possibly high-dimensional) state evolving under a continuous constraint signal K(t) ∈ ℝ. Let ν > 0 denote the sweep rate (operationally ν := |dK/dt| for a triangular sweep, or any operationally defined rate parameter for the chosen sweep family). Let R ≥ 0 be a scalar summary of accumulated record state, fixed during a single sweep.

Response coordinate. For p > 1, the line integral ∮ x dK is vector-valued and does not admit a scalar loop-area interpretation directly. The protocol therefore requires a predeclared scalar response coordinate y(t) := f(x(t)) ∈ ℝ, with f fixed before the loop-area test is executed. The scalar loop area is then A(R, ν) := |∮ y dK| under a closed sweep at rate ν. Construction rules for f are in §2.3 (admissibility) and §3.2 (declaration); two-dimensional embedding alternatives (e.g., area in a predeclared (PC₁, PC₂) plane) are noted as scope-limited extensions.

Decomposition. The protocol assumes the loop area admits a static-plus-dynamic decomposition

A(R, ν) = A_static(R) + D(R, ν), with lim_{ν→0} D(R, ν) = 0,

where A_static(R) is the rate-independent component of the loop area, identified with the record-driven contribution under the chosen formalism, and D(R, ν) collects all rate-driven contributions, including both record-independent rate lag (e.g., a system with constant relaxation timescale) and record-modulated rate lag (e.g., a system with R-coupled relaxation timescale τ(R)). The S3 diagnostic is specifically a claim about A_static(R), operational via the quasi-static extrapolation A_qs(R) := lim_{ν→0} A(R, ν). Where the decomposition holds, A_qs(R) → A_static(R) as the sweep is slowed.

Important caveat — record-modulated rate lag is a separate phenomenon. A system in which the relaxation timescale couples to record state, τ(R), can produce loop area at finite sweep rate that scales with R, but with A_static(R) = 0. This is record-modulated rate lag — a real and meaningful phenomenon, but structurally distinct from the persistent threshold structure that the S3 lemma describes. The S3 diagnostic treats record-modulated rate lag as a negative control (zero quasi-static residual) and is intentionally insensitive to it. A diagnostic for record-modulated rate lag would test the fitted τ(R) curve against an independent prediction; that is a different protocol and not the subject of this paper.

Limit caveat. The quasi-static limit may fail to exist in some systems (e.g., glassy relaxation with no rate-independent endpoint) or may contain a non-record rate-independent contribution (e.g., topological hysteresis from boundary geometry). In such cases the correct outcome is "protocol not applicable" or "record-coupling model rejected for this system," not automatic S3 failure.

Three model classes for A_static(R). Three model classes supply candidate predicted curves A_pred(R) = A_static(R) under genuine record coupling, each appropriate for a different domain. The diagnostic compares A_qs(R) against A_pred(R) under the class committed to before the test.

(i) Continuous-state Preisach analog. For systems with a continuous distribution of bistable elements indexed by threshold pair (θ_+, θ_−), the aggregate quasi-static loop area is the integral of single-hysteron contributions against a Preisach density ρ(θ_+, θ_−; R) that may itself depend on accumulated record state. Under R-coupled density shift (e.g., density translation or width modulation with R) within a fixed swept range, A_static(R) inherits the integral structure of the discrete case and admits a model-predicted slope obtained by integrating dρ/dR weighted by the per-hysteron loop contribution. The discrete-state lemma A_static ∝ 4α is recovered when ρ is concentrated on a single hysteron pair.

(ii) Bistable basin-geometry model (Kramers / first-passage-time class). For systems where state transitions are escape events from metastable basins (continuous bistable gene regulation, Pájaro et al. 2019; chemical reactions in the Kramers regime; cellular fate commitment), record coupling enters as basin-geometry modification with R. In the deterministic quasi-static limit, the system tracks the stable branch until the saddle-node bifurcation, where the metastable branch vanishes; A_static(R) follows from the R-dependence of the bifurcation thresholds θ_±(R). With branch separation normalized to 2 in symmetric cases, A_static(R) = 2[θ_+(R) − θ_−(R)]; for the symmetric form θ_±(R) = ±(θ₀ + αR), this reduces to 4(θ₀ + αR), recovering the discrete-hysteron base case. With finite noise or finite sweep rate, Kramers / mean-first-passage-time corrections (Hänggi, Talkner & Borkovec 1990) shift the effective switching points away from the deterministic bifurcation thresholds. These corrections vanish only in the deterministic, zero-noise quasi-static limit; at finite noise, slow sweeps can still produce pre-bifurcation escape, so deterministic-quasi-static and finite-noise-quasi-static are not the same regime and the protocol's A_qs estimate tracks whichever is realized in the data. This class is the natural framing for the rice transcriptome target in §5.

(iii) Soft bistable continuous-state model. For systems where x(t) is a continuous variable in a bistable potential V(x; K, R) with R-coupled threshold structure, a closed sweep traces a smooth version of the discrete-hysteron loop. In the limit of fast intra-branch relaxation, A_static(R) → 4(θ₀ + αR), recovering the discrete lemma. This class is used as the genuine synthetic case in §4.2.

Confound class — pure linear viscoelastic relaxation with possibly R-coupled τ. A linear viscoelastic system with relaxation dy/dt = −(y − K)/τ(R) has zero quasi-static loop area at every R, regardless of whether τ depends on R. Such a system produces finite loop area at finite sweep rate and can show R-dependence in that finite-rate area when τ depends on R, but A_qs(R) = 0. This is record-modulated rate lag, not S3, and is used as the negative-control synthetic case in §4.2.

Falsifier. Under the decomposition above and an independently specified A_pred(R) from one of the three model classes (or a domain-specific alternative), the falsifier is A_qs(R) ≉ A_pred(R) within declared tolerance. Class commitment must be made before the test; comparing a single dataset against the best-fitting class chosen post hoc is descriptive, not falsificatory. A separate negative-control falsifier is A_qs(R) ≈ 0 with positive finite-rate loop area — this rejects S3 in favor of record-modulated rate lag or pure rate lag, regardless of class choice.

2.3 The circular-R trap and the circular-projection trap

Two structurally similar circularity failures threaten the diagnostic. Both are instances of the general "double dipping" problem (Kriegeskorte et al. 2009): using the same data for variable selection and selective analysis distorts descriptive statistics and invalidates inference.

Circular-R failures (inadmissible R constructions):

(a) R := A_observed / 4α — defining R as a function of the loop area being tested produces tautological linear scaling with no falsification content.

(b) R := coercive field, remanence, or any other geometric feature derived from the same loop — same failure mode.

(c) R := PC₁ score fit on the same genes and samples used to compute the loop coordinate y — the omics version of the same trap. Even if the gene panel is ostensibly different, leakage through correlated samples reproduces the failure.

(d) R := latent variable inferred from the same loop family being tested — without a held-out family, this is exploratory rather than confirmatory.

Admissible R constructions:

(a) Prior-exposure logs, environmental history, intervention history — events that occurred before the loop being tested and were recorded outside the loop measurement system.

(b) Predeclared molecular marker panels — gene sets, protein sets, or metabolite sets named in advance from prior literature, with feature-disjointness from the response coordinate y.

(c) Independent phenotype measurements — physiological, morphological, or behavioral measures collected before, after, or during the sweep but not used to construct y.

(d) Tissue-history or cell-history markers measured on disjoint samples or disjoint feature sets.

(e) Simulation truth (in synthetic demonstrations) — the ground-truth R is known by construction and is independent of any observed quantity.

(f) Pretest threshold calibration on a held-out sweep family — relevant for discrete-state systems where θ_±(R) is needed for major-loop verification.

Circular-projection failures (inadmissible y constructions):

When state x is high-dimensional, the response coordinate y = f(x) is itself a modeling choice that can be circularly tuned. The same family of failures applies: (a) y selected to maximize the observed loop area on the test family; (b) y selected as a discriminant between high-R and low-R samples in the test family; (c) y selected by post-hoc feature inspection of the test loops. Admissible y constructions: predeclared scalar (e.g., "sum of heat-shock-protein expression"); predeclared linear projection from prior literature; PC₁ or other unsupervised summary fit on a training family disjoint from the test family; held-out validation design with explicit train/test split documented.

Feature disjointness in omics settings. When R is also derived from molecular features (e.g., a tissue-history marker panel) and y is derived from molecular features (e.g., a heat-shock response sum), the two feature sets must be disjoint. If they are not disjoint, an explicit leakage analysis must be reported showing that the shared features do not drive the observed scaling.

Why this section is load-bearing. The rest of the protocol assumes R, y, and the predicted slope are independent of the loop area being explained. Without operational rules for that independence, the protocol degrades into descriptive curve fitting and provides no falsification content. §3 implements these rules as checklist items; §5 demonstrates them in a real-data setting with explicit feature-disjointness reporting.

2.4 Diagnostic comparison

Table 1 summarizes the parallel structure across regimes. The choice of regime is determined by state-variable structure, not by framework or preference; discrete-regime statistics do not generalize directly to continuous-state data, and continuous-regime statistics do not generalize directly to discrete switching systems.

Table 1. Diagnostic comparison across regimes. The discrete-regime falsifier is the fitted slope ĉ versus 4α (single hysteron) or ρ-integrated slope (Preisach); the continuous-regime falsifier is A_qs(R) versus A_pred(R) under one of the three model classes of §2.2, with A_qs(R) ≈ 0 at positive finite-rate area as a separate negative-control rejection (record-modulated rate lag). Mixing regime statistics is a category error analogous to Jones 2026f's classification-versus-continuous warning.

3.1 Step 1: Independent record-state characterization

Specify R from non-loop channels. Acceptable inputs are listed in §2.3 (admissible R constructions). The audit must not use any feature of the loop being tested and must satisfy feature-disjointness from the response coordinate y in omics settings. Multi-sweep accumulation regimes (Jones 2026e §3) require a separate model and are out of scope for this protocol.

Step 1 reporting checklist. (a) R-source (admissible category from §2.3); (b) measurement timing relative to the test sweep; (c) scalar / vector / categorical structure of R; (d) independence relation to the loop statistic and to y; (e) leakage controls; (f) missingness rule; (g) confirmation that R is fixed during each test sweep.

3.2 Step 2: Response-coordinate declaration and loop-area curve construction

(2a) Declare the response coordinate y = f(x). For p = 1, f is the identity. For p > 1, f must be specified before loop area is computed and must satisfy the admissibility rules in §2.3 (predeclared, prior-literature-fit, training-family-fit, or held-out validation). The declaration must be documented and timestamped before §3.2(b) is executed.

(2b) Construct the loop-area curve. Drive closed major sweeps at multiple values of audited R. Compute A(R) := |∮ y dK| at each R. For continuous-state systems, sweep at multiple rates ν where the dataset permits and report A(R, ν) at each rate.

Step 2 reporting checklist. (a) loop statistic (signed line integral, absolute value, or RMS); (b) scalar response coordinate f and source (predeclared / prior-literature / training-fit / held-out); (c) sweep bounds K_min, K_max; (d) sweep direction and closure rule (see §5 for the open-trajectory closure convention); (e) interpolation method between samples; (f) numerical integration method (trapezoid, Simpson, Green's-theorem area); (g) replicate-aggregation rule; (h) per-R uncertainty estimate (bootstrap with stated unit, parametric, or replicate-based).

3.3 Step 3: Sweep-rate audit and major-loop verification

Sweep-rate audit (where executable). Vary ν at fixed R and report A(R, ν); extrapolate to the quasi-static limit ν → 0 to obtain A_qs(R) under the §2.2 decomposition. The convergence criterion is that the relative change between the two slowest rates falls below a declared tolerance (default ε = 0.05); if convergence is not achieved, the rate audit is reported as "partial" with the achieved tolerance stated.

Where the rate audit is not executable (single fixed protocol, time-course data with no rate variation, or a dataset where rate cannot be operationally defined), the step is reported as "not executable on this dataset" and the analysis explicitly names which confound classes therefore remain unaudited (record-modulated rate lag, record-independent rate lag, thixotropic recovery, frequency-dependent dissipation).

Major-loop verification independence. If thresholds θ_±(R) (discrete-state) or saturation extents (continuous-state) are needed for the major-loop check, they must be obtained from pretest threshold calibration, independent switching measurements, or a held-out sweep family. If they are inferred from the same loop-area curve being tested, the verification must be reported as internal consistency rather than an independent audit.

Step 3 reporting checklist. (a) sweep-rate values ν₁ < ν₂ < … (or "not executable"); (b) extrapolation model (linear in ν, polynomial, exponential, model-class-specific); (c) quasi-static convergence criterion and achieved tolerance; (d) executability label per step (executable / partially-executable / not-executable) with named unaudited confound classes if not executable; (e) major-loop verification source (pretest calibration, held-out family, or internal-consistency-only flag).

3.4 Step 4: Falsifier comparison

Slope-independence rule. The predicted slope c (or, in continuous regime, the predicted curve A_pred(R)) must be obtained from independent calibration, mechanistic parameter estimates, simulation truth, or a training loop family disjoint from the test loop family. If α (single-hysteron case), the Preisach density (superposition case), or the continuous-state model class (continuous case) is fit from the same A(R) curve being tested, the comparison is descriptive only and cannot serve as the sharp falsifier.

Discrete-state. Fit A(R) ≈ A₀ + cR over the audited R range; compare fitted slope ĉ against predicted 4α (single hysteron) or model-predicted slope (Preisach density).

Continuous-state. Compare A_qs(R) (where the rate audit was executable) or A(R) at the slowest available sweep rate (where it was not) against A_pred(R).

Fixed-rate caveat. When Step 3 is not executable, comparison of A(R) at the slowest or only available sweep rate is conditional or exploratory unless independent evidence rules out rate-driven confounds (record-modulated rate lag, record-independent rate lag, thixotropic-analog dynamics, frequency-dependent dissipation). Such a result may be reported as S3-consistent under stated caveats — naming the specific unaudited confound classes — but not as a full rate-audited S3-positive result. The protocol does not permit a confirmatory S3-positive claim from a fixed-rate dataset alone.

Tolerance band — synthetic demonstrations. When the predicted slope c_pred is simulation truth, the default decision rule is |ĉ − c_pred| < Z·SE(ĉ) for declared Z (Z = 1.96 for two-sided 95% CI), or analogous in the continuous regime. This is the rule used in §4.

Tolerance band — empirical applications. When the predicted slope or curve is obtained from independent calibration, mechanistic estimates, or a held-out training family, the SE-only rule is generally insufficient: a large standard error can let almost any ĉ pass, and a small standard error can flag substantively trivial discrepancies as failures. The empirical rule must combine statistical uncertainty with a pre-declared substantive equivalence tolerance δ:

|ĉ − c_pred| < max(δ, Z·SE(ĉ)),

or, more conservatively, |ĉ − c_pred| < δ with the confidence interval reported separately. Pre-registration of δ, Z, and SE construction (bootstrap with stated resampling unit, profile-likelihood, or analytic) is required for the decision to count as confirmatory. The choice of δ is domain-specific and must be justified in terms of what discrepancy from c_pred would be substantively meaningful for the claim being tested; it is not a free parameter to be tuned to the observed ĉ.

Step 4 reporting checklist. (a) predicted curve A_pred(R) (or scalar c) with derivation source; (b) fitted curve and confidence interval (bootstrap with stated resampling unit, profile-likelihood, or analytic); (c) tolerance band (synthetic-demo SE rule, or empirical max(δ, Z·SE) rule, with declared values of δ and Z); (d) decision rule (accept / reject / exploratory); (e) named confound signature where deviation is observed; (f) sensitivity analysis across alternative admissible model classes from §2.2 (continuous regime) or alternative Preisach-density specifications (discrete-Preisach regime).

4.1 Discrete-state regime: single hysteron with five confound structures

Five regimes are simulated with θ₀ = 0.4, α = 1.0, K_max = 2.0 (for non-minor-loop cases), R ∈ [0, 1] in nine increments, sweep rates ν ∈ {0.5, 1.0, 2.0, 4.0}, and 2,000 bootstrap resamples for slope CIs. (A) Genuine: θ_±(R) = ±(θ₀ + αR), rate-independent. (B) Rate-driven: θ_±(R) = ±(θ₀ + γν) with γ = 0.20 — the rate-driven increment vanishes as ν → 0, leaving the intrinsic baseline A = 4θ₀ at quasi-static, with no R-coupling at any rate. (C) Instrument hysteresis: θ_± = ±θ₀ fixed, no R-coupling, no rate dependence. (D) Minor-loop violation: same as (A) but K_max = 0.7 < θ_+(R) for R > 0.3. (E) Circular-R: same dynamics as (A), but R is operationally defined as A_observed / 4.

Table 2. §4.1 discrete-regime results. Predicted slope is 4α = 4.000; predicted intrinsic area is 4θ₀ = 1.600. Regime (A) recovers both within tight CI. Regime (B) shows zero R-slope with high rate-dep statistic; the rate-driven increment grows with ν and vanishes as ν → 0, leaving the intrinsic baseline 4θ₀ at quasi-static. Â₀ for regime (B) is 4(θ₀ + γν) at ν = 1, reported as a consistency check; the predicted Â₀ = 4θ₀ at ν → 0 is recovered by extrapolation in the continuous regime but not directly tested here. Regime (C) shows zero R-slope with no rate dependence and Â₀ matching 4θ₀, the diagnostic signature of instrument hysteresis with no record coupling. Regime (D) shows saturation at high R: the fit slope is dominated by the truncated high-R points where the sweep cannot reach θ_+(R), invalidating the major-loop assumption rather than the lemma. Regime (E) gives a perfect slope by construction but the test is tautological since R was derived from A; INADM. = inadmissible by §2.3.

Figure 1. §4.1 discrete-regime demonstration. Left: A(R) at nominal sweep rate ν = 1, with predicted line from Theorem 1. Regime A tracks the prediction; regimes B and C are flat in R (no R-coupling); regime D rises until K_max < θ_+(R), then collapses. Right: rate-dependence at fixed R range. Regime A is rate-independent (record-driven); regime B shows growing loop area with ν (rate lag, vanishing as ν → 0); regimes C and D are rate-independent but flat in R. Generated by methods_s3_v10_figures.py.

4.2 Continuous-state regime: bistable system with three confound structures

Three regimes are simulated with θ₀ = 0.4, α = 1.0, K_max = 2.0, R ∈ [0, 1] in seven increments, and sweep rates ν ∈ {0.05, 0.1, 0.2, 0.5, 1.0}. A_qs(R) is estimated by linear extrapolation in ν over the three slowest rates. (A) Genuine bistable + small fixed lag: y(t) relaxes with timescale τ_A = 0.02 toward a hysteron-target switching at θ_±(R) = ±(θ₀ + αR). The static loop area is 4(θ₀ + αR) by §2.2 model class (iii), recovered as ν → 0. (B) Pure rate lag (negative control): no hysteron; dy/dt = (K − y)/τ(R) with τ(R) = τ₀ + λR (τ₀ = 0.05, λ = 0.25). Quasi-static loop area is zero at every R. (C) Combined: hysteron-target as in (A), but with R-coupled relaxation timescale τ_C(R) = τ_A + δR (δ = 0.10). Both A_static(R) > 0 and an additional R-coupled rate-lag contribution are present.

Table 3. §4.2 continuous-regime results. Quasi-static loop area A_qs(R) is obtained by linear ν → 0 extrapolation from the three slowest rates. Regime (A) recovers A_static(R) = 4(θ₀ + αR) within tight CI on the slope and at both endpoints, with very small rate-dep statistic — a clean PASS for genuine record-driven static residual. Regime (B), the negative control, shows A_qs ≈ 0 at all R despite finite-rate loop area at higher ν (rate-dep ≈ 2.5, the dominant feature of this regime); the diagnostic correctly rejects S3 in favor of record-modulated rate lag. Regime (C) recovers the same A_static slope as (A) — record coupling is intact — but the rate-dep statistic is approximately three times larger, indicating a residual R-coupled rate-lag component on top of the static record structure. The Step-3 sweep-rate audit is what disambiguates (A) from (B) and (A) from (C).

Figure 2. §4.2 continuous-regime demonstration. Left: A(R, ν) curves at five sweep rates per regime (faded lines), with the ν → 0 extrapolated A_qs(R) shown as bold curves with markers. Genuine (A) and Combined (C) sit on the predicted line A_static = 4(θ₀ + αR); Pure rate lag (B) sits at zero across all R despite having visible finite-rate loop area at faster sweeps. Right: sweep-rate audit at R ≈ 0.5. The genuine and combined cases have nearly rate-independent loop area near 4(θ₀ + 0.5α) = 3.6 across all rates with star (★) extrapolation hitting the predicted value. The pure-rate-lag case grows from near zero at slow ν to ~1.3 at ν = 1, with the ν → 0 extrapolation correctly returning to zero. Generated by methods_s3_v10_figures.py.

6.1 Domains

Direct applications: stress–recovery in soft tissue (Fung 1967; Zhang et al. 2007); ecosystem regime shifts (Scheffer et al. 2001; Beisner et al. 2003); developmental fate commitment under cue reversal (Pájaro et al. 2019); institutional and policy lock-in (Arthur 1989; Pierson 2000); ferromagnetic and ferroelectric Preisach systems (Mayergoyz 2003); rate-dependent viscoelastic and thixotropic systems (Wang & Ewoldt 2023, useful as a comparison case where the rate audit is the primary diagnostic and the protocol's negative-control regime is the relevant outcome); memory and belief systems where prior commitments alter response to subsequent evidence (Lord, Ross & Lepper 1979). Choice of regime — discrete or continuous — is determined by state-variable structure, not framework preference.

6.2 The independence audit as the load-bearing step

Neither the single-hysteron lemma nor the static-plus-dynamic decomposition is novel in itself. What this protocol contributes is the pairing of these textbook predictions with an independently audited R proxy, an independently declared response coordinate, and a sweep-rate audit (where executable) that distinguishes static record-driven structure from rate-driven contributions including record-modulated rate lag. The audit converts a descriptive loop-area statistic into a falsifiable record-state claim. Without independent R, independent y, and independent slope, the protocol can verify the form of loop scaling (linear, sublinear) but not whether the scaling is record-driven.

6.3 Comparison to alternative diagnostics

Standard hysteresis-summary statistics — loop area, coercive field, remanence, energy dissipation per cycle (Mayergoyz 2003) — are single-condition descriptive summaries. They share the limitation, structurally similar to the one Bland and Altman (1986) flagged for method-comparison correlation in clinical measurement, that genuine record-coupling and rate-driven apparent hysteresis can produce indistinguishable single-condition values under plausible parameter substitutions. The agreement-curve approach of Auditing Independence (Jones 2026f) and the latency-curve approach of Auditing Constraint Asymmetry (Jones 2026g) addressed the same structural problem — comparing observed scaling against an independently specified prediction — for S1 and S2 respectively; the present protocol is the S3 analog.

Preisach density estimation (Mayergoyz 2003; Brokate & Sprekels 1996) is more diagnostic in principle but is typically calibrated against the same loop family being audited, reproducing the circular-fit failure mode (Varma & Simon 2006). The protocol of §3 differs in requiring the predicted slope or curve to come from a non-test source. Where a Preisach density is genuinely available from independent calibration (mechanistic micromagnetics, separate validation experiments), the protocol incorporates it as the predicted slope in §3.4.

Return-point memory tests (Sethna et al. 1993) are complementary but address a different signature; the present protocol does not require a return-point-memory check, since many continuous-state systems exhibit S3 without exhibiting return-point memory in any strict form. Avalanche-scaling diagnostics (Dahmen & Sethna 1996; Sethna, Dahmen & Myers 2001) apply to a different regime — critical disorder — where the linear lemma does not hold and a separate scaling-form diagnostic is appropriate.

Rheology-aware diagnostics for thixotropic and viscoelastic systems (Wang & Ewoldt 2023) overlap most directly with the rate-audit component of this protocol and are complementary in continuous-state physical systems. The present protocol's contribution is the pairing of a rate audit with an independently audited R-scaling check, allowing record-coupling claims to be separated from record-modulated rate lag in systems that may exhibit both.

Where the protocol is most useful: settings with multiple R levels, an independently constructible record-state proxy, and either a multi-rate sweep design or external grounds for ruling out rate-driven contributions. Where it is least useful: single-condition material characterization, systems where record state is intrinsically inferable only from loop geometry, and critical-disorder regimes where the linear lemma does not apply.