John Swygert

Ivory Tower Publishing

September 12, 2026

Research Note — September 12, 2026

This paper is the direct continuation of the exact two-state Markov erasure test. That prior paper rejected a provisional precision-speed crossover because the imported steady-state-style uncertainty bound was invalid for the transient protocol. Independent review then sharpened the result further: in the two-state topology, the signed net current telescopes to the endpoint state change, so current precision and endpoint displacement are structurally entangled. The present paper therefore changes only one ingredient: topology. It introduces the smallest cyclic Markov network capable of supporting a divergence-free circulation current that is not fixed by endpoint probabilities. The purpose is to test whether this extra cycle degree of freedom is sufficient to separate precision from speed, or whether a deeper unified bound still prevents an independent crossover.

Abstract

The preceding TSTOEAO paper solved an exact two-state continuous-time Markov memory and showed that a naive steady-state thermodynamic uncertainty relation cannot be used as a finite-time precision bound for transient erasure. More importantly, the two-state topology made the net current J equal to the endpoint displacement X(0)-X(τ), collapsing current and state-change information onto the same observable.

The present paper performs the next controlled experiment by moving to the smallest network with a nontrivial cycle: a three-state continuous-time Markov ring. The state probabilities p=(p0,p1,p2) obey a master equation generated by bidirectional rates on the edges 0↔1, 1↔2, and 2↔0. Integrated edge currents J01, J12, and J20 satisfy a discrete continuity equation. Their divergence determines the change in state probabilities, while a one-dimensional cycle component lies in the kernel of the incidence matrix and can vary without changing the endpoints.

This produces the structural decomposition that was impossible in the two-state model:

J = J_grad + J_cyc

B J_cyc = 0

The endpoint transformation constrains only the gradient/divergence part. A circulating component can therefore contribute to current statistics, entropy production, and thermodynamic precision while leaving the final logical error unchanged. This establishes, at the level of exact network topology, that the two-state speed-precision degeneracy is not generic to all Markov memories.

The paper does not assume that this topological separation automatically creates a new thermodynamic crossover. It instead defines a locked three-state test. One finite-time speed limit is built from the endpoint probability displacement and dynamical activity. A compatible finite-time current/TKUR bound is applied to a chosen cycle-sensitive current. The decisive question is whether the two bounds constrain genuinely independent directions in the admissible trajectory space, are nested, or collapse again into a common stronger inequality.

The central prospective result is therefore conditional but precise: a necessary structural condition for independent speed and precision constraints is the existence of a nonzero cycle space whose current is not determined by endpoint probabilities. The three-state ring is the minimal Markov network in which that condition holds. Whether it is sufficient is left to exact finite-time analysis and comparison with established unified thermodynamic-kinetic bounds.

01 Research Continuation

The prior paper established an exact falsification of the provisional two-state precision-speed crossover. It also identified the next structural question: was the collapse caused by the particular theorem choice, or by the topology of the two-state memory itself?

This paper isolates the topological issue by replacing the two-state chain with the smallest graph containing a genuine cycle.

02 The Two-State Limitation

In a two-state chain, every forward jump changes the binary state by one unit and every reverse jump changes it back. The signed accumulated current therefore telescopes:

J = X(0) – X(τ)

Once the endpoints are fixed, the net current is fixed. Current displacement and logical state displacement are the same topological object.

03 Why a Third State Changes the Problem

A three-state ring contains one independent cycle. A trajectory can circulate around the loop and return to the same endpoint. Therefore endpoint probabilities do not determine all integrated currents.

0 ↔ 1 ↔ 2 ↔ 0

This is the smallest Markov network with a nontrivial cycle space.

04 The Three-State Memory

Let the logical reset target be state 0. States 1 and 2 represent non-target states. The probability vector is

p(t) = (p₀(t), p₁(t), p₂(t)),     p₀+p₁+p₂=1

An unbiased initial condition may be chosen as

p(0) = (1/3,1/3,1/3)

while a finite-time reset target may be represented by

p(τ) = (1-ε₁-ε₂, ε₁, ε₂)

05 Bidirectional Rates

Each edge carries forward and reverse rates. Denote them by k01,k10,k12,k21,k20,k02. They may be constant or explicitly time dependent in the later driven version.

The exact network topology, not a particular numerical rate choice, is the first object under test.

06 Master Equation

The continuous-time Markov dynamics can be written

dp/dt = W(t)p(t)

with probability-conserving generator W. In component form:

dp₀/dt = k₁₀p₁ + k₂₀p₂ – (k₀₁+k₀₂)p₀

dp₁/dt = k₀₁p₀ + k₂₁p₂ – (k₁₀+k₁₂)p₁

dp₂/dt = k₀₂p₀ + k₁₂p₁ – (k₂₀+k₂₁)p₂

07 Instantaneous Edge Currents

Define oriented probability currents

j₀₁ = k₀₁p₀ – k₁₀p₁

j₁₂ = k₁₂p₁ – k₂₁p₂

j₂₀ = k₂₀p₂ – k₀₂p₀

Positive orientation is taken around the ring 0→1→2→0.

08 Integrated Edge Currents

Over the operation interval:

J_e = ∫₀^τ j_e(t) dt

collecting the edge currents into

J = (J₀₁, J₁₂, J₂₀)^T

09 Incidence Matrix

Choose the oriented incidence matrix

B = [[-1,0,+1],[+1,-1,0],[0,+1,-1]]

The integrated continuity equation is

p(τ)-p(0) = B J

10 Endpoint Information Fixes Only the Divergence

The endpoint change Δp constrains BJ, not J itself. Any additional current c satisfying Bc=0 can be added without changing the endpoints.

B(J+c)=BJ     whenever     Bc=0

11 The Cycle Space

For the three-state ring the null space of B is one-dimensional:

ker(B) = span{(1,1,1)^T}

Therefore

J_cyc = γ(1,1,1)^T

for arbitrary integrated circulation γ compatible with the stochastic dynamics.

12 Current Decomposition

Every integrated edge-current vector can be decomposed into a part fixed by endpoint transport plus a divergence-free cycle part:

J = J_grad + J_cyc

B J_grad = Δp,     B J_cyc = 0

The decomposition can be made unique after choosing a gauge or minimum-norm representative for J_grad.

13 Why This Was Impossible with Two States

A connected two-state graph has no nontrivial cycle space. Once the single edge current is fixed by probability conservation, no divergence-free circulation remains.

The extra third state therefore creates a genuinely new relational degree of freedom.

14 Topological Separation of Endpoint Error and Circulation

Final logical error depends only on p(τ). A cycle current can change while p(τ) remains unchanged.

Δp fixed, γ varied  ⇒  endpoint error fixed, cycle current varied

This is the first exact structural reason to expect precision and speed to have room to decouple.

15 A Locked Structural Prediction

Before choosing a specific finite-time TUR, TSTOEAO locks the following structural prediction:

If precision is measured with a cycle-sensitive current while speed is measured by endpoint probability displacement, then the two observables are not topologically identical on a three-state ring.

This is weaker than predicting an independent thermodynamic bound. It is a necessary structural condition only.

16 Four Allowed Outcomes

• False: even with a cycle current, the valid finite-time precision bound collapses exactly to the endpoint speed limit.
• Correct but trivial: the separation follows immediately from standard graph decomposition and adds no useful thermodynamic consequence.
• Correct but known: established stochastic-thermodynamics literature already contains the equivalent precision-speed separation.
• Correct, nontrivial, and apparently unreported: the cycle degree of freedom produces a new operating boundary that survives exact dynamics and novelty review.

17 Local Detailed Balance

For thermodynamic interpretation, rates may satisfy local detailed balance along each edge:

ln[k_ij(t)/k_ji(t)] = β q_ij(t)

where q_ij represents the appropriate bath entropy or energy exchange convention. Exact sign conventions must be fixed in the device-level test.

18 Entropy Production Rate

For a Markov jump process the standard entropy-production rate can be written

σ̇(t) = (1/2) Σ_{i≠j} j_ij(t) ln[(k_ij p_i)/(k_ji p_j)] ≥ 0

with total dimensionless entropy production

σ = ∫₀^τ σ̇(t) dt

19 Cycle Affinity

The ring possesses a cycle affinity

A_cyc = ln[(k₀₁ k₁₂ k₂₀)/(k₁₀ k₂₁ k₀₂)]

for time-homogeneous rates. If A_cyc≠0, stationary circulation can persist even without endpoint probability change.

20 Equilibrium Versus Nonequilibrium Ring

If detailed balance holds globally, A_cyc=0 and no stationary cycle current exists. If A_cyc≠0, the network is driven out of equilibrium and can sustain circulation.

The first three-state experiment should therefore distinguish two cases rather than silently mix them.

21 Case A: Detailed-Balance Reset

In the equilibrium-like case, transient cycle components can still appear depending on initialization and protocol, but persistent stationary circulation is absent.

This case tests whether topology alone, without sustained nonequilibrium drive, is sufficient to produce useful precision-speed separation.

22 Case B: Driven Cyclic Reset

In the nonequilibrium case, a nonzero cycle affinity supports circulation that can contribute entropy production and current fluctuations while the endpoint reset quality is held fixed.

This is the stronger candidate for genuine separation.

23 Endpoint Speed Observable

Define an endpoint statistical distance D[p(0),p(τ)]. For a classical speed limit, the state-space transformation contributes a bound of generic form

σ ≥ B_speed(D, τ, activity, protocol)

The exact theorem form must be chosen so that all assumptions match the three-state process.

24 Cycle-Sensitive Precision Observable

Choose an integrated current that responds to circulation. A natural candidate is the oriented cycle current

J_c = (J₀₁ + J₁₂ + J₂₀)/3

which changes under J→J+γ(1,1,1) but is not fixed by BJ=Δp.

25 Precision Statistic

Define relative uncertainty

R_c = Var(J_c)/⟨J_c⟩²

when ⟨J_c⟩≠0. A legitimate finite-time TUR or TKUR may then yield a bound involving R_c, entropy production, and kinetic activity.

26 The Central Independence Test

The paper now asks whether fixing the endpoint displacement and operation time leaves enough freedom to vary the cycle-current precision independently.

Δp, τ fixed; vary cycle affinity or cycle conductance; test whether R_c changes independently of B_speed

27 Necessary Condition for Independent Bounds

An independent precision constraint requires more than a nonzero cycle space. The chosen current must have a component in that cycle space, and the admissible dynamics must allow that component to vary without forcing a proportional change in the endpoint speed functional.

projection_cycle(J_obs) ≠ 0

28 Why the Cycle Space Is Only Necessary, Not Sufficient

Unified thermodynamic-kinetic inequalities may still couple cycle precision and endpoint displacement through the same entropy-production and activity resources.

Therefore topology creates room for separation, but does not guarantee a new crossover.

29 The Role of Unified TUR-Speed-Limit Results

Previous work in stochastic thermodynamics shows that uncertainty relations, minimum dissipation, optimal transport, and speed limits can arise from common geometric inequalities. The three-state test must therefore compare not only separate bounds but also the strongest available unified theorem.

If one unified inequality simultaneously dominates both projections, the apparent independence may remain only kinematic rather than thermodynamic.

30 The Correct Locked Question

Does the three-state cycle create two genuinely independent thermodynamic directions, or only two observables constrained by one deeper resource inequality?

31 Input Ledger for the Experiment

• Three-state master equation and normalization.
• Local detailed balance or explicitly stated nonequilibrium driving convention.
• One endpoint-based finite-time speed-limit theorem.
• One finite-time TUR/TKUR valid for the chosen cycle-sensitive current.
• Exact definitions of entropy production, dynamical activity, current, and statistical distance.
• No use of a known final crossover formula during the synthesis stage.

32 Locked Prediction Ledger

The prospective prediction should be written before the full parameter sweep. It must specify one of the following:

• a nonempty region where the precision bound dominates the speed bound;
• a nonempty region where the speed bound dominates the precision bound;
• a crossover surface separating the two;
• or a proof that one unified bound dominates both everywhere.

33 Exclusion Ledger

The synthesis stage should explicitly withhold any literature containing the final three-state trade-off, if such literature exists, until after the candidate is locked. The withheld set should be documented by title and reason.

34 Minimal Constant-Rate Test Family

A tractable first family uses constant rates with one tunable cycle affinity and one overall rate scale. One possible parametrization is

k₀₁ = a exp(+A/6),    k₁₀ = a exp(-A/6)

k₁₂ = b exp(+A/6),    k₂₁ = b exp(-A/6)

k₂₀ = c exp(+A/6),    k₀₂ = c exp(-A/6)

so that the total cycle affinity is A while a,b,c control conductance asymmetry. This is a test parametrization, not a unique physical model.

35 Reset Bias Must Be Added Separately

A pure symmetric cycle affinity does not by itself make state 0 the reset target. A state-energy bias or edge-specific detailed-balance factors must also favor state 0.

The full device model should therefore separate two controls:

reset bias  +  cycle drive

36 Two-Control Model

Introduce a reset-control parameter h that lowers state 0 relative to states 1 and 2, and a cycle-driving parameter A that biases circulation around the loop.

This permits the key experiment:

hold endpoint error approximately fixed with h; vary A; observe precision and dissipation changes

37 Prospective Signature of Genuine Separation

A strong indication of independent directions would be the existence of two protocols with the same p(0), p(τ), and τ but different cycle-current precision and different entropy-production cost.

same endpoints + same duration + different R_c + different σ

38 Stronger Signature

An even stronger result would be a parameter region in which tightening cycle-current precision raises the finite-time TUR cost while leaving the endpoint speed-limit term essentially unchanged to first order.

That would create a true local separation of constraint gradients in parameter space.

39 Constraint-Gradient Test

Let θ denote model parameters. Define

g_speed = ∇_θ B_speed,     g_precision = ∇_θ B_precision

If these gradients are not collinear in a common validity region, the bounds constrain different local directions.

g_speed ∦ g_precision

40 Why This Is Better Than Comparing Numbers Alone

Two bounds can cross numerically even if they arise from the same underlying direction. Gradient independence asks a stronger structural question: do the constraints actually remove different degrees of freedom from the admissible set?

41 Jacobian-Rank Criterion

For two candidate constraints F1(θ) and F2(θ), local independence requires the Jacobian

J_F = [∇F₁; ∇F₂]

to have rank 2 at the point of interest. Rank 1 indicates local redundancy or nesting.

42 Proposed TSTOEAO Relational Test

TSTOEAO therefore refines the old crossover question. Instead of merely asking where B_precision=B_speed, it asks whether the two constraint surfaces are locally independent before looking for their intersection.

independence first; crossover second

43 Why This Is a Methodological Upgrade

The two-state work showed that a visually attractive crossover can be meaningless when one bound is invalid or redundant. The three-state protocol therefore tests validity, topological distinction, and differential independence before interpreting any numerical seam.

44 Falsification Condition One

If every valid finite-time cycle-current TUR can be algebraically reduced to the same unified inequality that produces the endpoint speed limit, the independence hypothesis fails.

45 Falsification Condition Two

If varying the cycle degree of freedom while holding the endpoint transformation fixed necessarily changes the speed functional in exact proportion, the apparent topological separation does not produce thermodynamic separation.

46 Falsification Condition Three

If a claimed crossover depends on theorem forms with incompatible observables, initial-state classes, or entropy conventions, it is rejected exactly as in the two-state paper.

47 Falsification Condition Four

If the cycle-sensitive current is chosen after inspecting the answer, the result is post-hoc. The observable must be preregistered.

48 Falsification Condition Five

If the final relation is already a direct corollary of established Schnakenberg network theory, finite-time TURs, or optimal-transport results, it is classified as known or methodological rediscovery rather than new physics.

49 What Would Count as Success

A meaningful success would require all of the following:

• one explicit three-state protocol with a common theorem-validity domain;
• a preregistered cycle-sensitive current;
• a locked prediction about constraint independence or a crossover surface;
• exact master-equation or trajectory-level verification;
• a documented literature search after the lock;
• and, ideally, a measurable experimental signature.

50 What Would Count as Stronger Scientific Content

The strongest outcome would be a quantitative boundary not explicitly contained in the supplied source relations, obtained by relational synthesis, independently verified by exact stochastic dynamics, and not located in a subsequent literature search.

That remains the standard. The paper does not claim that standard has yet been met.

51 TSTOEAO Interpretation

The two-state system taught that shared variables can hide redundancy. The three-state ring adds a new relational object: a cycle-space direction invisible to endpoint constraints.

In TSTOEAO language, the relational map has acquired an additional coordinate that is structurally real, mathematically defined, and experimentally interpretable.

52 Revised Relational Object

A useful synthesis object for the network is

I_R^(network) = (B, ker B, Δp, J_cycle, σ, activity, validity metadata)

where B is the incidence matrix and ker B records the degrees of freedom not visible in endpoint conservation alone.

53 Cycle-Space Principle

Cycle-Space Principle: endpoint constraints determine current divergence, not divergence-free circulation.

This is established network mathematics. Its value here is to identify the minimal topology where precision observables can escape endpoint determination.

54 Minimal-Topology Principle

A connected Markov network requires at least one independent cycle before a cycle-sensitive current can vary without being fixed by endpoint probability change.

For a simple graph, the cycle-space dimension is

dim ker(B) = E – V + 1

for one connected component.

55 Three-State Ring as the Minimal Test

For the ring, V=3 and E=3, so

E – V + 1 = 1

giving exactly one independent cycle degree of freedom. This makes the model the smallest nontrivial test beyond the two-state degeneracy.

56 Why a Four-State Model Is Not Yet Needed

A larger network would introduce multiple cycles and more freedom but would also complicate interpretation. The three-state ring isolates the single structural change we care about: the appearance of one circulation coordinate.

57 Experimental Analogue

Possible physical realizations include three-state molecular switches, quantum-dot charge states in the classical sequential-tunneling regime, enzymatic cycles, or colloidal-state networks. The present paper does not select one experimental platform yet.

58 Literature Boundary

The graph-theoretic decomposition and cycle-affinity formalism are established. Finite-time TURs, thermodynamic-kinetic uncertainty relations, and speed limits are also established. The unresolved question is not whether these ingredients exist, but whether their combination in the locked three-state test produces a distinct operational boundary not already explicit in the literature.

59 Status of the Prediction

The paper locks only the structural prediction that the three-state topology removes the exact current-endpoint identity of the two-state model.

two-state: current fixed by endpoint change

three-state ring: endpoint change + independent cycle current

Any stronger thermodynamic prediction remains to be calculated.

60 The Immediate Next Calculation

The next calculation should choose one theorem-compatible three-state protocol and compute, over a controlled parameter sweep:

• exact p(t);
• exact endpoint distance;
• exact cycle-current mean and variance;
• exact entropy production;
• exact dynamical activity;
• one valid finite-time TUR/TKUR bound;
• one valid speed-limit bound;
• and the Jacobian rank of the two constraint surfaces.

61 The Decision Rule

rank 2 + nonempty common validity region → candidate independent constraints

rank 1 or exact theorem reduction → nested/unified constraints

invalid assumptions → reject synthesis

62 Final Research Question

Does the smallest nontrivial cycle in a three-state Markov memory create a genuinely independent precision constraint beyond endpoint speed, or does unified stochastic thermodynamics collapse both back onto one deeper resource bound?

CONCLUSION

The exact two-state paper reached an important negative result. The proposed precision-speed crossover failed because the imported precision relation was invalid for the transient protocol, and because the two-state net current itself collapses to the endpoint state difference.

The present paper asks whether that collapse is a special consequence of two-state topology rather than a universal property of stochastic memories.

A three-state ring is the minimal network that contains a nontrivial cycle space. Its integrated edge currents obey the continuity equation Δp=BJ, but the incidence matrix has a one-dimensional kernel. As a result, an arbitrary circulation component proportional to (1,1,1) can exist without changing the endpoint probability vector.

p(τ)-p(0) = B J

ker(B) = span{(1,1,1)^T}

J = J_grad + J_cyc

This changes the scientific question. In the two-state model, endpoint speed and net-current precision were forced onto the same topological degree of freedom. In the three-state ring, a cycle-sensitive current can vary in a direction invisible to endpoint conservation.

That is a necessary structural condition for genuine precision-speed separation. It is not sufficient. Modern finite-time TURs, thermodynamic-kinetic uncertainty relations, and optimal-transport speed limits may still place both observables under one stronger common resource inequality.

The next test must therefore proceed adversarially. It should preregister one endpoint speed observable, one cycle-sensitive precision observable, one compatible speed-limit theorem, and one compatible finite-time TUR/TKUR. Then it should solve the exact three-state dynamics and determine whether the two constraint surfaces are locally independent, nested, or unified.

The correct criterion is no longer simply whether two numerical lower bounds cross. The stronger question is whether they constrain different directions in the admissible parameter space.

independence first; crossover second

If the Jacobian of the two constraint surfaces has rank two in a common validity region, the three-state topology has created a serious candidate for a genuine precision-speed seam. If the rank is one, or if both bounds reduce to the same unified inequality, the independence hypothesis fails. If theorem assumptions do not overlap, the synthesis is invalid and must be rejected.

The paper therefore does not claim a new thermodynamic law. It identifies the smallest topology in which the central question can be asked without the exact degeneracy that doomed the two-state crossover.

The research progression is now sharply defined: the two-state model established the failure mode; the three-state ring supplies the first topology capable of escaping it.

Do not assume the cycle creates independence.

Measure the cycle direction.

Then let the exact stochastic thermodynamics decide.

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