Finite-Threshold First-Passage Ratios Retain Excess Entropy Information in Hidden-State Dynamics

Authors: Sterling Geisel, QBist Lab; Dr. Hideo Tanaka, QBist Lab

Abstract

Izaak Neri gives a thermodynamic estimator built from first-passage data of one measured current. The estimator uses a mean stopping time, a splitting probability, and two thresholds. It can lower-bound entropy production even when the measured current is not itself stochastic entropy production. This Working Paper applies Temporal Softening to that finite-threshold setting. The claim is narrow. Pudding Theory does not rederive Neri’s bounds. It predicts one extra scaling feature: in hidden-state nonequilibrium systems, finite thresholds should show an excess captured entropy fraction over matched Markovian surrogates, and that excess should scale with stopping-time breadth. The test is operational in a six-state driven jump process with known entropy production, fixed thresholds, and surrogate dynamics matched to the one-current statistics available before stopping. Failure of that excess across the grid falsifies the Postulate in this domain.

Source Synopsis

Neri studies how to estimate the mean entropy production rate of a nonequilibrium process when only one fluctuating current is measured. This is a common experimental limitation. Heat and microscopic entropy production are not usually observed directly. A molecular motor, colloidal particle, or coarse-grained biochemical network may give only a displacement, a step count, or a partial flux.

The method uses first-passage observables. A current \(J(t)\) is monitored until it leaves an interval \((-\ell_-,\ell_+)\). Each run gives a stopping time \(T_J\) and a sign indicating whether the positive or negative threshold was crossed first. Across many runs, the observer estimates the mean first-passage time \(\langle T_J\rangle\) and the negative splitting probability \(p_-\). For positive mean current, Neri writes the finite-sample first-passage estimator in the form

\[ \widehat{\dot{s}}_{\rm FPR}(\ell_+,\ell_-) = \frac{\ell_+}{\ell_-} \frac{|\ln \widehat{p}_-|}{\widehat{T}_J}. \]

Earlier first-passage results showed asymptotic bounds at large thresholds. Neri extends the analysis to the finite regimes used in experiments: finite thresholds, finite sample numbers, and currents not proportional to stochastic entropy production.

The paper compares first-passage ratios with thermodynamic uncertainty ratios and Kullback-Leibler based estimates. Variance-based uncertainty ratios can become weak far from equilibrium because typical current fluctuations may miss rare dissipative structure. Kullback-Leibler estimates can be powerful but are difficult under partial observation when the measured process is nonMarkovian. First-passage ratios use stopping statistics directly, so they retain information from rare crossings and memory effects in the coarse-grained current.

The result is an inference method. It does not claim direct microscopic access. It gives a practical lower-bound estimator that can capture a finite fraction of total dissipation when more familiar current-fluctuation estimators capture little.

Postulate Lens

The applicable Postulate is Temporal Softening. The fit is exact enough to be useful, but only if it is made quantitative. The source paper is organized around thresholds and stopping times. A thermodynamic estimate is recovered not from an instantaneous current value, but from the time required for a fluctuating current to find a boundary.

In Pudding Theory, Temporal Softening states that susceptibility is time-dependent. Barriers that are effectively rigid in a short window become statistically permeable as fluctuation exposure accumulates. Neri’s paper supplies a clean mathematical arena for this claim. It treats boundary crossing as information-bearing. The elapsed time is not discarded as a nuisance parameter. It is part of the thermodynamic signal.

The Postulate adds only one prediction beyond standard first-passage inference. In a system with hidden slow variables, the finite-threshold first-passage ratio should retain an excess entropy signal relative to a Markovian surrogate that has been matched on the observable one-current record available before stopping. Removing the Postulate removes that scaling claim. It does not remove Neri’s estimator.

Pudding Theory Prediction

Pudding Theory predicts a finite-threshold excess in hidden-state nonequilibrium dynamics. Consider a continuous-time Markov jump process with six states arranged as two coupled three-state cycles. One cycle is observed through a net winding current \(J(t)\). The second cycle is hidden and modulates transition rates through slow switching. The full generator is known, so the total entropy production rate \(\dot{s}_{\rm tot}\) can be computed from steady-state probabilities and transition-rate affinities.

For each threshold pair \((\ell_+,\ell_-)\), run \(N=10^5\) trajectories from steady state until \(J(t)\notin(-\ell_-,\ell_+)\). Use the fixed grid

\[ (\ell_+,\ell_-) \in \{(2,2),(4,4),(8,8),(16,16),(32,32)\}. \]

For each grid point compute

\[ C_{\rm FPR}(\ell)= \widehat{\dot{s}}_{\rm FPR}(\ell,\ell)/\dot{s}_{\rm tot}. \]

Construct a Markovian surrogate on the observed current increments by matching the empirical pre-stopping one-current marginal statistics: stationary increment distribution, mean current, variance, autocorrelation through lag 10, and the empirical distribution of \(J(t)\) values before absorption. The surrogate is then run with the same thresholds and sample count. Define

\[ E(\ell)=C_{\rm FPR}^{\rm hidden}(\ell)-C_{\rm FPR}^{\rm surrogate}(\ell). \]

Temporal Softening predicts \(E(\ell)>0\) in an intermediate threshold window, with the strongest effect at \(\ell=4\) or \(\ell=8\) for parameter sets where the hidden switching time is comparable to the observed first-passage time. The excess should scale monotonically with a preregistered breadth statistic of the stopping-time distribution, for example \(\mathrm{CV}[T_J]\) or the interquartile width of \(\ln T_J\). It should weaken when hidden switching is made fast enough that the observed current becomes effectively Markovian, or slow enough that most trajectories stop before sampling the hidden barrier.

This is not a claim that entropy is violated. The prediction is about estimator content. Finite stopping times should preserve a trace of hidden dissipative organization that is erased by surrogate matching of ordinary one-current marginals.

Falsifiable Observable

The observable is the preregistered excess captured entropy fraction \(E(\ell)\) on the threshold grid \(\ell\in\{2,4,8,16,32\}\), with uncertainty defined by 1,000 bootstrap resamples of complete first-passage trajectories. The rejection criterion is \(E(\ell)\le 0\) for all five thresholds, with the upper edge of the 95 percent bootstrap interval below \(0.01\), in at least three hidden-state parameter sets satisfying \(\mathrm{CV}[T_J]\ge 1.5\). If the preregistered excess captured entropy fraction were measured to be zero by that criterion, this Postulate would be falsified.

Editorial Dialogue

Tanaka: The paper is better, but the danger remains. Neri already has the mechanism. Hidden variables alter stopping statistics. That is ordinary stochastic thermodynamics.

Sterling: Yes. The source result is ordinary stochastic thermodynamics. The Postulate is not being used to explain why first-passage ratios work. It is being used to predict a residual scaling after surrogate matching.

Tanaka: Then the surrogate is the key. If it is too weak, the excess is trivial.

Sterling: The surrogate must be strong. It matches the one-current marginal record before stopping, including finite-lag autocorrelation and the empirical pre-absorption distribution. It does not know the hidden cycle affinities.

Tanaka: That still may leave nonMarkovian structure. A statistician would call the excess memory, not temporal softening.

Sterling: That is acceptable. The Postulate earns no immunity from standard language. It is falsified if the memory-sensitive excess fails under the fixed grid and matched surrogate construction.

Tanaka: And if it appears?

Sterling: Then the Postulate has not proven a new field. It has supplied a correct scaling rule for where finite thresholds retain hidden dissipation. That is enough for this paper.

Discussion

This application is deliberately narrow. Neri’s result stands without Pudding Theory. The present claim lives only in the finite-threshold comparison between hidden dynamics and matched observable surrogates. That comparison is useful because it removes the weak version of the claim, namely that stopping times matter. Neri already proves that.

The main limitation is model dependence. A six-state jump process is clean, but it is not a molecular motor or a driven colloid. The next test should use an experimental system where independent calibration gives \(\dot{s}_{\rm tot}\), or at least a high-confidence full-state estimate. The surrogate construction also matters. If stronger surrogate matching removes the excess, the Postulate loses content in this domain.

The conclusion would change under three conditions: no positive intermediate-threshold excess, no scaling with stopping-time breadth, or equal performance by surrogates that lack hidden dissipative structure. Any one would weaken the interpretation. All three would end it.

References

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