From Composite Thermodynamic Bounds to an Exact Two-State Markov Erasure Test: A Controlled TSTOEAO Examination of Error, Current Fluctuations, Entropy Production, and the Proposed Precision-Speed Crossover

From Composite Thermodynamic Bounds to an Exact Two-State Markov Erasure Test

A Controlled TSTOEAO Examination of Error, Current Fluctuations, Entropy Production, and the Proposed Precision-Speed Crossover

John Swygert

Ivory Tower Publishing

September 12, 2026

Research Note — September 12, 2026

This paper is the direct device-level follow-up to the preceding TSTOEAO relational-synthesis paper. Independent review identified a serious compatibility issue in the earlier general treatment: a steady-state thermodynamic uncertainty relation cannot simply be combined with a transient finite-time erasure protocol. The present paper therefore freezes one explicit two-state continuous-time Markov model and derives its dynamics, net-current statistics, entropy production, and thermodynamic speed-limit quantities from the same model. It also tests, rather than assumes, the proposed precision-speed crossover. The purpose is not to protect the previous crossover idea. The purpose is to determine whether it survives exact model-level analysis.

Abstract

The preceding TSTOEAO paper combined Landauer-type erasure cost, thermodynamic uncertainty relations, and finite-time speed limits into a composite constraint framework. That analysis correctly emphasized non-overcounting and domain compatibility, but it left the key operational quantities abstract. Independent critique then exposed the decisive next requirement: all candidate bounds must be evaluated inside one explicit stochastic memory model whose assumptions are mutually compatible.

The present paper performs that test using a two-state continuous-time Markov memory. The memory begins unbiased, p1(0)=1/2, and relaxes under fixed bidirectional transition rates k01 and k10 toward a biased stationary state q=k01/(k01+k10). The reset interval ends at p1(τ)=ε, with q<ε<1/2. The master equation is solved exactly, giving the full relation among the rate scale, stationary bias, target error, and operation time.

p₁(t) = q + (1/2 - q)exp(-λt),     λ = k₀₁ + k₁₀

exp(-λτ) = (ε - q)/(1/2 - q)

For the exact net reset current J=N₁→₀-N₀→₁, the two-state topology gives J=X(0)-X(τ). Consequently:

⟨J⟩ = 1/2 - ε

Var(J) = (1/2)(1-r) - (1/2-ε)²,     r=exp(-λτ)

For detailed-balance relaxation, the total dimensionless entropy production is exactly:

σ = Σ/k_B = D(p(0)||q) - D(p(τ)||q)

The Shiraishi-Funo-Saito classical speed limit can also be evaluated from the same model. The crucial test concerns the proposed precision term. The familiar steady-state expression 2⟨J⟩²/Var(J) is not a valid universal finite-time TUR for this transient relaxation. A concrete parameter choice q=0.05 and ε=0.10 gives an exact entropy production σ≈0.810 while the naive steady-state-style precision term is ≈1.125. Treating the latter as a mandatory lower bound would therefore falsely exclude an exactly solved trajectory.

The simple crossover obtained by equating that naive precision term to the speed-limit term does not survive the controlled model test. This is not a failure of finite-time TUR theory: generalized TURs for arbitrary initial states and time-dependent driving exist. It is a failure of the simplified identification used in the provisional crossover argument.

The paper therefore reaches a sharper result. The two-state model validates the missing-edge warning of the previous work: target logical error ε does not determine precision cost by itself. The current variance and entropy production depend additionally on the stationary bias q, rate structure, observable choice, and protocol. A legitimate precision-speed seam must be derived from a finite-time TUR that is valid for the exact protocol and observable under study.

The proposed simple crossover is rejected; the relational-synthesis program survives by using the rejection to define the correct next calculation.

01 Research Continuation

The research sequence has progressively moved from broad relational hypotheses to increasingly controlled tests. The preceding paper reached the first explicit composite thermodynamic constraint but deliberately stopped short of a device-level calculation. The present paper begins where that paper ended.

The question is no longer whether speed and precision can both matter. They can. The question is whether a single explicit erasure process supports the specific functional bounds required to produce a genuine crossover.

02 Why the Earlier General Result Was Not Yet Enough

The earlier composite form

W ≥ k_B T[ln 2 - H_b(ε) + max(B_precision, B_speed)]

is valid only when both B_precision and B_speed are legitimate lower bounds on the same entropy-production quantity for the same physical process.

A standard steady-state TUR and a transient erasure speed limit do not automatically satisfy that requirement. The compatibility condition is therefore not a footnote. It decides whether the synthesis is physics or merely algebra.

03 Model Selection

We choose the smallest continuous-time stochastic memory that still contains error, reversibility, transitions, entropy production, and finite-time relaxation: a two-state Markov process with states 0 and 1.

The logical reset target is state 0. State 1 is the error state.

04 State Probabilities

Let p(t)=p₁(t) be the probability of logical state 1 at time t. Then p₀(t)=1-p(t).

p(0)=1/2

p(τ)=ε,     0≤ε<1/2

05 Transition Rates

Let k₀₁ denote the excitation rate 0→1 and k₁₀ the reset-favoring rate 1→0. Both are positive.

λ = k₀₁ + k₁₀

q = k₀₁/λ

The stationary error probability is q. A genuine finite-time relaxation toward state 0 requires q<ε<1/2.

06 Local Detailed Balance Interpretation

If the two states differ by an energy gap ΔE while coupled to a bath at inverse temperature β=1/(k_B T), local detailed balance gives

k₀₁/k₁₀ = exp(-βΔE)

and therefore

q = 1/[1+exp(βΔE)]

This interpretation is useful but not required for solving the Markov dynamics.

07 Exact Master Equation

The probability p(t) obeys

dp/dt = k₀₁(1-p) - k₁₀p

or

dp/dt = λ(q-p)

08 Exact Solution

With p(0)=1/2, the solution is

p(t) = q + (1/2-q)exp(-λt)

This is exact for the chosen protocol.

09 Exact Time-Error Relation

Imposing p(τ)=ε gives

ε = q + (1/2-q)exp(-λτ)

and therefore

r ≡ exp(-λτ) = (ε-q)/(1/2-q)

τ = (1/λ) ln[(1/2-q)/(ε-q)]

This already displays an important missing variable: the same final error ε can be reached with different stationary biases q and rate scales λ.

10 Feasibility Boundary

The logarithm requires q<ε<1/2 for a finite positive relaxation time in this constant-rate model.

q < ε < 1/2

Perfect ε=0 erasure cannot be achieved at finite time with strictly positive reverse rate k₀₁. It is approached only by taking a limiting bias q→0 and/or an infinite-time limit.

11 The Exact Net Reset Current

Let N₁→₀ count transitions from 1 to 0 and N₀→₁ count transitions from 0 to 1 during [0,τ]. Define the signed net reset current

J = N₁→₀ - N₀→₁

For a two-state chain every forward jump changes the state by -1 and every reverse jump changes it by +1. Therefore the accumulated net current telescopes exactly:

J = X(0) - X(τ)

where X(t)=1 when the memory is in state 1 and X(t)=0 when it is in state 0.

12 Exact Mean Current

Taking the expectation gives

⟨J⟩ = p(0)-p(τ) = 1/2-ε

Define the erased probability mass

d ≡ 1/2-ε > 0

so that ⟨J⟩=d.

13 Transition Probabilities

For the constant-rate two-state process, the endpoint transition probabilities over time τ are

P(1→0) = (1-q)(1-r)

P(0→1) = q(1-r)

r=exp(-λτ)

14 Exact Current Distribution

Because the initial state is unbiased,

P(J=+1) = (1/2)(1-q)(1-r)

P(J=-1) = (1/2)q(1-r)

P(J=0) = 1 - (1/2)(1-r)

The current is therefore not a Poisson variable. It is a bounded endpoint current taking only -1, 0, or +1.

15 Why the Poisson Approximation Is Dangerous Here

A provisional derivation suggested Var(J)≈⟨J⟩ under rare reverse transitions. That is not the exact net-current statistic of the two-state memory. Multiple microscopic jumps can occur, but they cancel in the signed net current because the two-state topology makes J equal to the endpoint difference.

This is the first place where the exact model changes the proposed crossover calculation.

16 Exact Current Variance

Since J²=1 exactly when the endpoint states differ,

⟨J²⟩ = (1/2)(1-r)

and therefore

Var(J) = (1/2)(1-r) - d²

Using r=(ε-q)/(1/2-q), this becomes

Var(J) = d/[2(1/2-q)] - d²

17 Exact Relative Uncertainty

The relative current uncertainty is

R_J = Var(J)/⟨J⟩²

R_J = 1/[2(1/2-q)d] - 1

This expression depends on both ε and q. Therefore ε alone does not determine current precision.

18 Exact Proof of the Missing Edge

The previous paper identified the missing edge ε↔R_J as a research target. The exact model now proves why the edge is not universal.

For fixed ε, changing q changes R_J while leaving the logical target error unchanged.

ε fixed, q varied  ⇒  R_J varied

Thus no universal function R_J=F(ε) exists even in this minimal family unless additional protocol information is supplied.

19 Entropy Production for Detailed-Balance Relaxation

For a time-homogeneous detailed-balance Markov process relaxing toward stationary distribution q, the total dimensionless entropy production equals the decrease in relative entropy to equilibrium:

σ ≡ Σ/k_B = D(p(0)||q) - D(p(τ)||q)

20 Binary Relative Entropy

For a Bernoulli probability p relative to stationary error q,

D(p||q) = p ln(p/q) + (1-p)ln[(1-p)/(1-q)]

Hence

σ = D(1/2||q) - D(ε||q)

This is exact for the selected relaxation.

21 Positivity

Because detailed-balance relaxation monotonically decreases the relative entropy to the stationary state,

σ ≥ 0

with equality only when there is no thermodynamic relaxation over the interval.

22 Exact Dynamical Activity

The instantaneous dynamical activity is the total expected jump rate

A(t) = k₁₀p(t) + k₀₁[1-p(t)]

Using λ and q:

A(t) = λ[2q(1-q) + 2(1/2-q)²exp(-λt)]

23 Time-Averaged Activity

Define

Ā = (1/τ)∫₀^τ A(t)dt

Then

Ā = 2λq(1-q) + [2(1/2-q)²(1-r)]/τ

24 Endpoint Distance

The L1 distance between the initial and final probability distributions is

L₁ = Σ_i |p_i(τ)-p_i(0)| = 2d = 1-2ε

The usual total-variation distance is d=L₁/2.

25 Classical Thermodynamic Speed Limit

For classical Markov processes satisfying the relevant local-detailed-balance assumptions, the Shiraishi-Funo-Saito speed limit can be written in the form

τ ≥ L₁²/(2σĀ)

or equivalently

σ ≥ B_speed = L₁²/(2τĀ) = 2d²/(τĀ)

This bound is now evaluated from quantities belonging to the same exact two-state process.

26 The Earlier Proposed Precision Bound

The provisional crossover calculation used the familiar steady-state-style expression

B_precision,naive = 2/R_J = 2⟨J⟩²/Var(J)

and treated it as a lower bound on σ.

That step must now be tested rather than assumed.

27 Why This Test Is Decisive

If the inequality

σ ≥ 2⟨J⟩²/Var(J)

fails for an exact trajectory of this transient process, then the simple precision term cannot be used to define the proposed crossover for this model. This would not disprove thermodynamic uncertainty relations. It would show only that the wrong TUR form was imported.

28 Concrete Exact Test

Choose the dimensionless rate scale λ=1, stationary error q=0.05, and target error ε=0.10.

r = (0.10-0.05)/(0.50-0.05) = 1/9

τ = ln 9 ≈ 2.19722

d = 0.40

29 Exact Numerical Current Statistics

For this model:

Var(J) = 0.284444...

R_J = 0.284444... / 0.16 = 1.777777...

2/R_J = 1.125

30 Exact Numerical Entropy Production

The exact entropy production is

σ = D(0.5||0.05) - D(0.10||0.05) ≈ 0.809711

Therefore

σ ≈ 0.809711 < 1.125 = 2/R_J

31 Result: The Naive Precision Bound Fails

The inequality σ≥2/R_J is violated by the exactly solved transient trajectory.

σ < B_precision,naive

This proves that the steady-state-style precision expression cannot serve as the required finite-time precision bound for this erasure protocol.

32 The Speed Limit Still Survives

For the same parameters, the exact time-averaged activity gives approximately

Ā ≈ 0.258843

and the classical speed-limit lower bound is

B_speed = 2d²/(τĀ) ≈ 0.562651

which satisfies

σ ≈ 0.809711 ≥ 0.562651

as required.

33 Fate of the Proposed Crossover

The provisional crossover was obtained by solving

B_speed = B_precision,naive

But one side of that equality is not a valid lower bound for this transient model. Therefore the resulting crossover time is not established physics.

τ* from the naive 2/R_J construction is rejected for this model

34 This Is a Useful Failure

The failure identifies the exact source of the problem. The speed relation was evaluated within a transient Markov process. The precision term was imported from an incompatible simplified TUR form.

The model therefore confirms the earlier rule:

common variable is not enough; common validity domain is mandatory

35 Finite-Time TURs Do Exist

The rejection of the naive precision expression must not be confused with rejection of TUR theory. Generalized TURs have been derived for arbitrary initial states, arbitrary time-dependent driving, and broader classes of observables.

Those generalized results include additional response, protocol, kinetic, or activity information absent from the simple steady-state formula. That extra information is exactly what the present model shows is needed.

36 What the Exact Model Says About Precision

In this model, precision is controlled not only by how much probability is erased, d=1/2-ε, but also by the stationary bias q and the rate protocol.

R_J = R_J(ε,q)

while

σ = σ(ε,q)

and

τ = τ(ε,q,λ)

The hoped-for universal one-dimensional precision edge collapses into a higher-dimensional relational surface.

37 A More Accurate Relational Picture

The operational state of this model is minimally described by

(ε, q, λ, τ, R_J, σ, Ā)

with exact constraints linking them.

The device is not governed by a universal curve in ε alone. It occupies a constrained surface in this larger space.

38 TSTOEAO Interpretation

From the TSTOEAO perspective, the failure is informative because it identifies a boundary of transport between theorem forms.

The shared quantity σ did not justify transporting a steady-state precision law into a transient protocol. The mapping failed because the temporal and statistical structure was not preserved.

39 Refined Compatibility Principle

Relational Compatibility Principle: two constraints may be synthesized only when the physical process, observable definition, initial-state class, driving class, and entropy-production convention lie inside a common validity domain.

This is stronger than merely requiring the same variable symbols.

40 Refined Missing-Edge Principle

A missing relation must not be replaced by a convenient approximation unless that approximation is validated inside the target model.

The provisional Poisson-current step violated this rule for the exact net current.

41 What Would Constitute a Legitimate Crossover

A legitimate precision-speed crossover in this model requires a finite-time precision inequality of the form

σ ≥ B_precision^(FT)(ε,q,λ,τ,observable,protocol)

that is proven valid for the same process.

Only then may one solve

B_precision^(FT) = B_speed

42 The Correct Next Mathematical Task

The next calculation is therefore not to adjust the rejected crossover formula. It is to choose one specific finite-time TUR applicable to continuous-time Markov dynamics from nonstationary initial conditions, evaluate every term for this exact two-state process, and compare it against B_speed.

This preserves the prediction-lock discipline while removing the compatibility defect.

43 Why Koyuk-Seifert-Type Relations Matter

Finite-time TURs for arbitrary time-dependent driving explicitly address the weakness exposed here: the precision bound depends on more than the steady-state current mean and variance. They provide a legitimate route for building B_precision^(FT) without pretending that 2/R_J remains universally valid.

Likewise, arbitrary-initial-state TURs show that transient boundary information changes the form of the uncertainty relation.

44 Why a Unified TUR-Speed-Limit Framework Matters

Later stochastic-thermodynamics work has shown that speed limits and generalized uncertainty relations can arise from closely related mathematical structures. This is important for TSTOEAO because it warns against treating B_precision and B_speed as automatically independent constraints.

If they descend from the same deeper inequality, the correct synthesis may not be max(B_precision,B_speed) at all; it may be one tighter unified bound.

45 Consequence for the Previous Paper

The previous composite-bound paper remains useful as a structural map, but its B_precision and B_speed terms must be understood as placeholders until a common model and theorem family are specified.

The current paper supplies that correction explicitly.

46 Status of the Landauer Term

The present constant-rate relaxation is not claimed to be a globally optimal finite-time erasure protocol. The Landauer information term remains the reversible informational floor under its standard assumptions, but the exact model-level test here is directed at the irreversible precision-speed structure.

This separation avoids confusing a controlled stochastic relaxation with an optimally driven memory-reset protocol.

47 Exact Model as a Baseline

The simplicity of the model is an advantage. Any more complicated erasure architecture should reduce to this kind of bookkeeping locally: identify the state probabilities, rates, currents, entropy production, activity, and theorem assumptions before combining bounds.

A method that cannot survive the two-state case should not be trusted in a many-state device.

48 What Has Been Demonstrated

The paper has produced five exact results from one model:

• the time-error relation;

• the exact net-current mean and variance;

• the exact entropy production;

• the exact dynamical activity and speed-limit bound;

• the failure of the naive steady-state precision bound for the transient reset.

49 What Has Not Been Demonstrated

This paper has not established a new universal thermodynamic law. It has not established a historically novel precision-speed crossover. It has not shown that TSTOEAO replaces stochastic thermodynamics.

What it has shown is that relational synthesis can be used adversarially: a proposed relation was carried into an exact model and rejected because one of its imported edges was invalid.

50 Why That Still Counts as Progress

A scientific framework gains credibility when it can tell its own attractive idea 'no.'

The previous work proposed a candidate seam. The present work did not protect it. It tested the seam and found that the simplified precision side was incompatible with the chosen transient dynamics.

proposal → exact model → contradiction → boundary identified

51 New Research Standard

Future TSTOEAO synthesis papers should not stop at common variables. They should record a compatibility vector for every imported relation:

C_i = (dynamics, initial states, driving, observable, entropy convention, time regime, approximation class)

Two relations may be combined only after the relevant entries are shown compatible.

52 A Device-Level Relational Graph

For the exact model, the reliable graph contains the edges

(q,λ,τ) → ε

(ε,q) → R_J

(ε,q) → σ

(ε,q,λ) → Ā

(ε,q,λ) → B_speed

The precision-TUR edge remains theorem-dependent and must be supplied by a finite-time relation valid for this graph.

53 The New Missing Edge

The missing edge is now more precise than before.

(ε,q,λ,τ,R_J,protocol) → B_precision^(FT)

This is no longer a vague request for an error-current relation. It is a concrete theorem-evaluation problem.

54 The Next Locked Experiment

The next experiment should freeze this same two-state model and select one published finite-time TUR valid for arbitrary initial states or arbitrary driving.

Before solving it numerically, the TSTOEAO analysis should predict whether the finite-time precision bound and the speed-limit bound are independent, nested, or different projections of one stronger common inequality.

55 Three Possible Outcomes

• Independent bounds: a genuine max-type crossover may survive, but with the correct finite-time precision term.

• Nested bounds: one bound dominates or implies the other, so no independent crossover exists.

• Unified origin: the apparent speed and precision constraints are projections of a single tighter inequality, making the original max construction conceptually redundant.

56 Falsification Conditions

The next-stage hypothesis fails if the chosen finite-time TUR does not apply to this model, if its required observable differs from the defined net current, if the resulting bound is always weaker or algebraically identical to the speed limit, or if the predicted seam disappears under exact evaluation.

No post-hoc substitution of a different TUR should be used to rescue a failed locked prediction.

57 Scientific Novelty Standard

Even if a corrected crossover is found, it must then be compared with existing stochastic-thermodynamics literature. A relation can be mathematically correct and independently derived while still being historically known.

correct ≠ new

58 Current Classification

The current result is best classified as:

exact model-level falsification of a provisional relational crossover

and

successful identification of the compatibility condition needed for the next test

59 TSTOEAO Status After This Test

TSTOEAO remains a relational mapping, constraint-synthesis, and boundary-detection methodology under scientific test.

The strongest contribution in this paper is methodological discipline: the framework did not treat a visually appealing equality of bounds as evidence until both bounds survived one common physical model.

60 Central Result

The naive precision-speed crossover does not survive the exact two-state transient Markov test.

The reason is specific and identifiable:

σ ≥ 2/R_J is not a valid universal precision bound for this transient erasure protocol.

61 Stronger Surviving Result

The stronger surviving result is not a crossover formula but a structural requirement:

Any valid precision-speed seam must be derived from theorem forms that share the same dynamics, initial-state class, driving protocol, observable, and entropy-production definition.

62 Final Research Question

When a valid finite-time TUR is evaluated on this exact two-state reset, does it define an independent precision constraint, or does it collapse into the same thermodynamic-kinetic structure that already generates the speed limit?

CONCLUSION

The previous TSTOEAO paper identified a candidate precision-speed crossover in finite-time information erasure. That proposal was useful because it created a concrete target. But the target depended on a hidden assumption: that a familiar steady-state-style thermodynamic uncertainty expression could be carried directly into a transient erasure process.

The present paper tested that assumption inside one exact two-state continuous-time Markov memory.

The model begins with an unbiased bit, relaxes toward a biased stationary state q, and reaches final logical error ε after a finite time τ. Every major quantity required by the synthesis can be obtained exactly from the same process: endpoint probabilities, operation time, net reset current, current variance, entropy production, dynamical activity, and the classical speed-limit bound.

p₁(t) = q + (1/2-q)exp(-λt)

τ = (1/λ) ln[(1/2-q)/(ε-q)]

⟨J⟩ = 1/2-ε

Var(J) = (1/2)(1-exp(-λτ)) - (1/2-ε)²

σ = D(1/2||q) - D(ε||q)

B_speed = 2(1/2-ε)²/(τĀ)

The exact current calculation also revealed why the provisional Poisson approximation was inappropriate for the signed net current: in a two-state chain the accumulated net current telescopes to the endpoint difference X(0)-X(τ). Its statistics are therefore bounded endpoint statistics, not simple Poisson counting statistics.

Most importantly, the exact model supplies a direct falsification of the naive precision term. For q=0.05 and ε=0.10, the exact entropy production is approximately 0.809711, while 2/R_J is 1.125. The supposed lower bound is larger than the actual entropy production. It is therefore not a valid bound for this transient process.

0.809711 < 1.125

The simple crossover obtained by equating that quantity with the speed-limit penalty must therefore be rejected.

This result does not undermine finite-time thermodynamic uncertainty relations. It shows why the finite-time versions matter. Generalized TURs for arbitrary initial states and time-dependent protocols contain additional structure that the simple steady-state formula omits.

The exact two-state model therefore converts a broad 'missing edge' into a precise research problem. The next precision bound must depend on the actual transient process and observable. It cannot be inferred from ε alone.

The research program has consequently advanced by eliminating another attractive but unjustified shortcut. The next experiment is now better defined than the previous one: keep this exact two-state memory fixed, evaluate a legitimate finite-time TUR on it, and determine whether precision and speed remain independent constraints or are two projections of a deeper thermodynamic-kinetic inequality.

Do not force the crossover.

Derive the valid precision relation first.

Then let the exact model decide whether a seam exists.

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