From Cycle-Enabled Separation to Rank-Two Operational Independence
An Exact Three-State Markov Test of Endpoint Speed, Cycle Precision, and the Limits of Additive Thermodynamic Decomposition
John Swygert
Ivory Tower Publishing
September 12, 2026
Research Note — September 12, 2026
This paper directly continues the three-state-cycle manuscript. Independent review agreed that the three-state ring is the smallest Markov topology capable of supporting a circulation current that is invisible to endpoint conservation, but disagreed on what that topological separation guarantees thermodynamically. One interpretation predicts genuinely independent speed and precision directions, potentially permitting additive excess and housekeeping costs. The more cautious interpretation holds that topology is only a necessary condition and that modern unified uncertainty/speed-limit theory may still collapse both constraints onto one deeper resource bound. The present paper therefore performs the next locked test: it fixes an explicit three-state rate model with two physically distinct controls, solves the master equation and cycle-current statistics, and evaluates the Jacobian rank of endpoint-speed and cycle-precision observables. It does not assume that rank-two operational separation proves an additive entropy-production law.
Abstract
The preceding paper proved that a three-state ring possesses a one-dimensional cycle space absent from the two-state chain. That result established a topological possibility: an integrated circulation current can vary without being determined by the endpoint probability change. The present paper asks whether that possibility becomes an explicit local two-dimensional structure in a fully specified stochastic model.
We introduce a continuous-time three-state Markov ring with two independent controls. A reset-bias parameter h lowers the energy of state 0 and primarily controls endpoint transport toward the logical target. A cycle-affinity parameter A_cyc drives nonequilibrium circulation around 0→1→2→0. The transition rates are chosen to satisfy a local-detailed-balance form with the cycle affinity distributed around the ring.
kᵢ⟶ⱼ = ν exp{[Eᵢ-Eⱼ+Fᵢⱼ]/2}, E₀=-h, E₁=E₂=0
F₀₁=F₁₂=F₂₀=A_cyc/3, Fⱼᵢ=-Fᵢⱼ
Starting from p(0)=(1/3,1/3,1/3), the master equation is solved exactly by matrix exponentiation for a fixed duration τ. Endpoint-speed structure is quantified by the Shiraishi-Funo-Saito-type lower-bound functional
B_speed = L₁²/(2τĀ)
where L₁ is the endpoint L1 distance and Ā is the time-averaged dynamical activity. Cycle precision is characterized directly from the full counting statistics of the integrated oriented cycle current C, using
P_cyc = ⟨C⟩²/Var(C)
as an exact precision indicator. P_cyc is deliberately not labeled a thermodynamic lower bound unless a compatible finite-time TUR is supplied.
At the benchmark point h=1, A_cyc=1, ν=1, τ=1, the exact calculation gives B_speed≈0.05293 and P_cyc≈0.05103. Finite-difference differentiation with respect to (h,A_cyc) yields
∂(B_speed,P_cyc)/∂(h,A_cyc) ≈ [[0.11133,-0.00234],[-0.00869,0.10140]]
det J ≈ 0.01127 ≠ 0
The Jacobian therefore has rank two. In this explicit model, reset bias and cycle affinity generate locally independent operational directions: the endpoint-speed functional responds primarily to h, while the cycle-precision indicator responds primarily to A_cyc.
This confirms the topological prediction at the level of exact dynamics and current statistics. It does not establish that entropy-production lower bounds add. The paper therefore separates two claims that had previously been conflated: operational independence is demonstrated; additive thermodynamic decomposition remains conditional upon a theorem that assigns the two bounds to independent dissipative channels under a common metric and protocol.
The main result is thus a successful rank-two kinematic/operational separation, together with a precise statement of what remains unproved.
01 Research Continuation
The research sequence has moved from a failed two-state crossover to an exact diagnosis of why it failed, and then to the minimal cyclic topology capable of avoiding the two-state telescoping collapse.
The present paper performs the calculation that the previous three-state manuscript deliberately deferred.
02 The Decisive Question
Does the smallest nontrivial cycle create genuinely independent operational directions for endpoint speed and cycle precision?
The answer must be obtained from one explicit model, not inferred from topology alone.
03 What Is Being Tested — and What Is Not
The present test has two levels.
Level 1: operational independence — do the endpoint-speed and cycle-precision observables respond independently to two physical controls?
Level 2: thermodynamic additivity — can valid lower bounds on entropy production be assigned independently to those directions and summed?
Level 1 can be tested directly from exact dynamics and full counting statistics. Level 2 requires a compatible finite-time TUR/TKUR and a justified dissipation decomposition. The paper does not treat Level 1 as automatic proof of Level 2.
04 Three-State Ring
0 ⇄ 1 ⇄ 2 ⇄ 0
The ring has V=3 vertices and E=3 edges. For a connected graph its cycle-space dimension is
dim ker(B) = E - V + 1 = 1
Thus one independent circulation current exists.
05 Incidence Matrix and Current Decomposition
Choose the oriented edges 0→1, 1→2, and 2→0. The incidence matrix B maps integrated edge currents to endpoint probability changes.
Δp = B J
A cycle current satisfies
B J_cyc = 0
so an integrated current may be written schematically as
J = J_grad + J_cyc
with the endpoint change insensitive to the cycle component.
06 Why This Was Impossible in the Two-State Model
For a connected two-state graph with one edge,
E - V + 1 = 1 - 2 + 1 = 0
There is no cycle space. Every signed net current is determined by the endpoints. The three-state ring is therefore the minimal topology in which the current can contain information not fixed by endpoint transport.
07 Explicit Control Coordinates
We now construct a two-parameter physical family. The first control is the reset bias h. The second is the cycle affinity A_cyc.
θ = (h, A_cyc)
The purpose is to determine whether these coordinates generate independent responses in the speed and precision observables.
08 Energy Assignment
Set dimensionless bath units k_B T=1 and define
E₀=-h, E₁=E₂=0
Increasing h favors the target state 0.
09 Cycle Affinity
Orient the ring 0→1→2→0 and distribute the nonequilibrium affinity uniformly:
F₀₁=F₁₂=F₂₀=A_cyc/3
F₁₀=F₂₁=F₀₂=-A_cyc/3
The total cycle affinity is therefore A_cyc.
10 Transition Rates
Choose the symmetric local-detailed-balance parametrization
kᵢ⟶ⱼ = ν exp{[Eᵢ-Eⱼ+Fᵢⱼ]/2}
so that
ln(kᵢ⟶ⱼ/kⱼ⟶ᵢ) = Eᵢ-Eⱼ+Fᵢⱼ
with ν setting the overall time scale.
11 Rate Matrix
The continuous-time generator L is constructed from the six directed transition rates. In column-vector convention,
dp/dt = Lp
with off-diagonal element Lⱼᵢ=kᵢ⟶ⱼ and diagonal escape rate
Lᵢᵢ = -Σⱼ≠ᵢ kᵢ⟶ⱼ
12 Initial State and Duration
For the benchmark test we choose
p(0)=(1/3,1/3,1/3), ν=1, τ=1
The exact endpoint is
p(τ)=exp(Lτ)p(0)
13 Logical Error
State 0 is the reset target. Define the total final logical error as
ε = 1-p₀(τ)
This quantity depends on both controls in general, although the intended design makes h the dominant endpoint-control direction.
14 Endpoint Distance
Define the endpoint L1 distance
L₁ = Σᵢ |pᵢ(τ)-pᵢ(0)|
This measures the amount of net redistribution of probability mass between the initial and final state distributions.
15 Dynamical Activity
The instantaneous activity is
A(t)=ΣᵢΣⱼ≠ᵢ kᵢ⟶ⱼ pᵢ(t)
and the time-averaged activity is
Ā=(1/τ)∫₀^τ A(t)dt
16 Endpoint-Speed Functional
For the exact process we evaluate the classical Markov speed-limit functional
B_speed = L₁²/(2τĀ)
This is a lower-bound functional associated with endpoint transport, entropy production, and activity under the assumptions of the relevant classical speed-limit theorem.
17 Oriented Cycle Current
Let each jump along 0→1, 1→2, or 2→0 contribute +1/3 to the integrated cycle current C, and each reverse jump contribute -1/3.
A complete forward revolution therefore contributes +1.
18 Full Counting Statistics
Introduce a tilted generator L_χ by multiplying each oriented off-diagonal transition rate by exp(χgᵢⱼ), where gᵢⱼ=±1/3 according to edge orientation.
The moment-generating function is
M(χ)=1ᵀ exp(L_χ τ)p(0)
and the cumulant-generating function is
K(χ)=ln M(χ)
19 Exact Current Mean and Variance
The first two cumulants are
⟨C⟩ = ∂χK|χ=0
Var(C)=∂χ²K|χ=0
These statistics contain the circulation information that was absent from the two-state endpoint current.
20 Cycle-Precision Indicator
Define the exact dimensionless precision indicator
P_cyc = ⟨C⟩²/Var(C)
Larger P_cyc means a more directed, less relatively noisy cycle current.
This quantity is not automatically a thermodynamic lower bound. It becomes one only through an independently valid finite-time TUR/TKUR.
21 Why This Distinction Matters
The previous two-state failure came from treating a convenient precision expression as a universal entropy-production bound when its theorem assumptions were not satisfied.
The present paper therefore uses exact current precision to test rank and independence, while withholding any unsupported thermodynamic inequality.
22 Exact Entropy Production
For completeness, the total entropy production of the jump process is evaluated as
σ = ∫₀^τ dt Σ_{i<j} [kᵢ⟶ⱼpᵢ-kⱼ⟶ᵢpⱼ] ln[(kᵢ⟶ⱼpᵢ)/(kⱼ⟶ᵢpⱼ)]
This is nonnegative under the standard stochastic-thermodynamic convention.
23 Locked Benchmark Point
The first benchmark is fixed before interpretation:
h=1, A_cyc=1, ν=1, τ=1
No parameter is tuned afterward to obtain rank two.
24 Exact Benchmark Output
Solving the model gives approximately:
Quantity | Value |
p₀(τ) | 0.561182 |
p₁(τ) | 0.230302 |
p₂(τ) | 0.208515 |
ε=1-p₀(τ) | 0.438818 |
L₁ | 0.455698 |
Ā | 1.961675 |
σ | 0.223639 |
B_speed | 0.052929 |
⟨C⟩ | 0.105136 |
Var(C) | 0.216619 |
P_cyc | 0.051028 |
25 Control-Axis Sanity Check
At A_cyc=0 the model satisfies detailed balance and the mean cycle current vanishes to numerical precision.
A_cyc=0 ⇒ ⟨C⟩≈0
At fixed h=1, increasing A_cyc from 0 to 2 produces a substantial cycle current while changing the endpoint-speed functional only modestly.
26 Exact Parameter-Sweep Examples
The following values come from the same exact model with ν=τ=1.
h | A_cyc | B_speed | ⟨C⟩ | P_cyc | σ |
1 | 0 | 0.054117 | ≈0 | ≈0 | 0.119106 |
1 | 1 | 0.052929 | 0.105136 | 0.051028 | 0.223639 |
1 | 2 | 0.049566 | 0.213001 | 0.200240 | 0.542786 |
2 | 1 | 0.215791 | 0.085979 | 0.038606 | 0.558023 |
2 | 2 | 0.202518 | 0.174319 | 0.149976 | 0.816090 |
27 Qualitative Separation Visible in the Sweep
Changing h strongly changes B_speed because it changes the endpoint redistribution toward state 0.
Changing A_cyc strongly changes cycle current precision while affecting B_speed much less at the benchmark scale.
This is the operational pattern predicted by the cycle-space argument.
28 The Jacobian-Rank Criterion
Define the map
F(h,A_cyc) = (B_speed, P_cyc)
Local operational independence requires the two outputs to have linearly independent gradients.
rank[∇B_speed; ∇P_cyc] = 2
29 Numerical Jacobian at the Locked Benchmark
Using symmetric finite differences around h=1, A_cyc=1 gives
J ≈ [[0.111329, -0.002341],[-0.008690, 0.101399]]
det J ≈ 0.011268
30 Rank Result
Because the determinant is nonzero,
rank J = 2
The endpoint-speed functional and the exact cycle-precision indicator are locally independent functions of the control parameters.
31 Interpretation of the Matrix
The first row shows that B_speed responds strongly to h and weakly to A_cyc near the benchmark.
The second row shows that P_cyc responds strongly to A_cyc and weakly to h.
The matrix is therefore close to—but not exactly—diagonal. The two controls are physically coupled, yet they generate independent local directions.
32 Robustness Check
The rank-two result is not confined to one isolated point. The determinant remains nonzero at several additional benchmarks.
(h,A_cyc) | det J | Rank |
(1,1) | 0.01127 | 2 |
(1,2) | 0.02014 | 2 |
(1.5,1) | 0.01492 | 2 |
(2,1) | 0.01536 | 2 |
(2,2) | 0.02676 | 2 |
33 What Has Now Been Proven
Within this explicit rate family, the three-state cycle does more than merely permit a hidden circulation mode abstractly.
It produces an exact nonzero cycle current, an independently varying precision statistic, and a rank-two response map with respect to reset bias and cycle affinity.
topological possibility → exact operational independence
34 What Has Not Been Proven
The rank-two result does not prove
that B_speed and a valid finite-time precision TUR are additive;
that total entropy production decomposes into two orthogonal lower-bounded pieces for this protocol;
that the correct synthesis is a sum rather than a maximum or unified inequality;
that any resulting bound is historically new.
35 Why the Additive Interpretation Is Plausible but Not Yet Established Here
Stochastic thermodynamics possesses established decompositions of total entropy production into housekeeping/adiabatic and excess/nonadiabatic contributions under specified definitions and assumptions. Network Hodge decompositions also separate gradient-like and cycle-like currents.
However, algebraic orthogonality of graph subspaces does not by itself prove that the specific speed-limit lower bound and a specific finite-time TUR lower bound saturate independent additive entropy-production channels.
operational orthogonality ≠ automatically thermodynamic additivity
36 Why the Cautious Interpretation Also Survives
Unified TUR/TKUR and optimal-transport formalisms show that endpoint displacement, current precision, activity, and dissipation can descend from common geometric inequalities.
Therefore it remains possible that a valid finite-time precision bound and the speed bound are different projections of a common resource functional even though their observable responses have rank two.
37 The Exact Resolution of the Independent Reviews So Far
The stronger interpretation is supported at the operational level: the three-state topology genuinely creates two independent control-response directions.
The caution remains valid at the thermodynamic-bound level: rank-two observable independence does not yet establish independent additive lower bounds on entropy production.
38 Revised Outcome Classification
The three-state experiment now separates two questions that should no longer be merged:
Question A — Are speed and cycle precision operationally independent? Answer: yes, locally, in the tested rate family.
Question B — Are their thermodynamic lower bounds independent additive costs? Answer: not yet established.
39 Relation to the Additive-Channel Rule
Paper 7 stated that lower bounds may be added only when the constrained physical quantity is demonstrably decomposable into corresponding independent additive contributions.
The current rank-two result satisfies a necessary operational condition for that rule but not the full thermodynamic condition.
rank 2 is evidence for separate channels; it is not proof of additive dissipation
40 Housekeeping and Excess Dissipation
For nonequilibrium Markov dynamics, established stochastic-thermodynamic frameworks distinguish dissipation associated with maintaining nonequilibrium steady circulation from dissipation associated with relaxation of the state distribution.
This suggests a natural next mapping:
cycle circulation ↔ housekeeping-like dissipation
endpoint redistribution ↔ excess/nonadiabatic-like dissipation
But the correspondence must be derived for the exact protocol rather than declared by analogy.
41 The Next Locked Thermodynamic Test
Keep the exact rate family fixed. Select one established entropy-production decomposition valid for this nonstationary three-state process and one finite-time cycle-current TUR/TKUR valid for the same observable and initial condition.
Then test whether
σ_total ≥ B_speed + B_precision
is actually justified, or whether the correct theorem gives a unified/nonadditive relation.
42 Stronger Alternative Hypothesis
A particularly important possibility is that the full theory yields
σ_total ≥ B_unified(B_speed, P_cyc, Ā, …)
with neither a simple sum nor a simple maximum.
If so, the three-state ring would still have rank-two operational structure while the thermodynamic resource remains unified.
43 Locked Prediction for the Next Paper
The next paper should not predict additivity. It should lock a weaker and falsifiable structural prediction:
A valid common-theorem evaluation will preserve rank-two control sensitivity even if the final entropy-production inequality is unified.
This follows from the exact observable response found here and can be tested against the selected theorem.
44 A Potential Route to Genuine Excess Content
If an established unified theorem is specialized to this model and the rank-two structure forces a previously unreported boundary in the (h,A_cyc) control plane, that boundary could become a candidate latent consequence.
Historical novelty would still require a dedicated literature search.
45 Parameter-Space Geometry
The operational map
(h,A_cyc) → (B_speed,P_cyc)
is locally invertible wherever det J≠0.
That means the two observables can, in principle, serve as local coordinates on the control surface in those regions.
46 Why Local Invertibility Matters
Local invertibility means that speed-related endpoint behavior and cycle precision encode genuinely different information about the device controls.
This is stronger than merely observing two nonzero quantities. It says neither observable is locally a function of the other alone.
47 Boundary of the Rank-Two Claim
The rank can drop on special submanifolds.
At A_cyc=0, the mean cycle current vanishes and the precision indicator loses sensitivity at leading order.
At parameter values where one control becomes dynamically irrelevant, the determinant may vanish.
Symmetries or singular rate limits can reduce the effective dimension.
48 A New Relational Boundary
The determinant
Δ_rank(h,A_cyc)=det ∂(B_speed,P_cyc)/∂(h,A_cyc)
defines a natural structural boundary.
Δ_rank=0 ⇒ local collapse
Δ_rank≠0 ⇒ local operational separation
49 Why This Boundary Is More Defensible Than the Old Crossover
The earlier crossover depended on equating two candidate lower bounds whose compatibility was uncertain.
The rank boundary uses directly calculated observables from one exact model. It therefore tests structure before importing thermodynamic theorem claims.
50 TSTOEAO Interpretation
In TSTOEAO terms, the three-state cycle changes the relational coordinate structure of the admissible process.
The two-state system had one effective transport direction. The three-state ring adds a cycle coordinate that survives endpoint projection.
endpoint projection removes J_cyc but does not remove its physical trajectory cost
51 The Role of G_T in This Paper
G_T is used only as an organizational relational representation. It does not supply new stochastic dynamics.
It helps distinguish which observables survive which projections and where additional degrees of freedom are hidden from endpoint measurements.
52 The Emerging General Rule
The sequence suggests a broader methodological statement:
Topology determines which relational degrees of freedom can remain invisible under a chosen projection.
In a tree, integrated currents are strongly constrained by boundary change. In a graph with cycles, divergence-free currents can survive while boundary change remains fixed.
53 Scientific Test Standard
A useful theory should not ask established mathematics to bend around it. It should expose a place where alternatives make different calculable predictions and then accept the result.
This paper therefore treats the rank test as the relevant discriminator, not the attractiveness of an additive interpretation.
54 Falsification Conditions
The current claim would be weakened or falsified if
the rank-two result disappeared under exact rather than numerical differentiation;
the result depended entirely on an arbitrary definition of the cycle observable;
the chosen control parametrization secretly changed the same physical parameter twice;
the result vanished throughout every physically admissible rate region.
55 Reproducibility Standard
All formulas required to reproduce the benchmark are specified: the rate law, energies, affinity allocation, initial state, duration, activity definition, tilted generator, and finite-difference control coordinates.
A computational supplement should eventually provide the numerical script used for the parameter sweep.
56 Scientific Classification
The result is classified as
exact model-level demonstration of rank-two operational separation
not as
a new thermodynamic law
57 What Is Established Mathematics
The following ingredients are established:
cycle-space dimension E-V+1 for connected graphs;
incidence-matrix current decomposition;
continuous-time Markov master equations;
full counting statistics through tilted generators;
classical thermodynamic speed limits;
stochastic entropy-production theory;
TUR/TKUR and optimal-transport unification frameworks.
58 What Is Specific to This TSTOEAO Test
The TSTOEAO-specific contribution is the staged use of those ingredients as an adversarial relational experiment: two-state collapse → minimal cycle insertion → locked rank test → exact parameter sweep → separation of operational independence from thermodynamic additivity.
59 Current Result
The smallest nontrivial cycle breaks the two-state operational degeneracy.
It does not yet prove an additive speed-plus-precision entropy bound.
60 Next Paper
The next paper should perform the common-theorem thermodynamic test on this exact three-state family.
It should calculate an established finite-time cycle-current TUR/TKUR and an established excess/housekeeping or optimal-transport decomposition under matching assumptions, then determine whether the final inequality is additive, nested, or unified.
61 Decisive Next Question
Does rank-two operational independence in the three-state ring survive as rank-two thermodynamic constraint independence, or does a unified entropy-production geometry recombine the two directions?
CONCLUSION
The previous three-state paper established a necessary topological condition for escaping the two-state precision-speed collapse: a nonzero cycle space. The present paper performs the exact operational test that topology alone could not answer.
A fully specified three-state continuous-time Markov ring was constructed with two independent physical controls. The reset bias h alters the energetic preference for the logical target. The cycle affinity A_cyc drives nonequilibrium circulation around the ring. Exact master-equation dynamics, dynamical activity, entropy production, and full counting statistics of the oriented cycle current were then calculated from the same model.
F(h,A_cyc)=(B_speed,P_cyc)
At the locked benchmark h=1 and A_cyc=1, the response Jacobian is approximately
J ≈ [[0.111329,-0.002341],[-0.008690,0.101399]]
det J ≈ 0.011268 ≠ 0
The rank is therefore two. The result persists across several additional benchmark points.
This resolves one part of the independent-review disagreement. The three-state cycle does create genuinely independent operational directions. Endpoint speed and cycle precision are no longer the same observable in disguise.
But the calculation also preserves the more cautious conclusion. Rank-two operational independence is not identical to thermodynamic additivity. Modern stochastic thermodynamics can decompose dissipation in several useful ways, and unified TUR/speed-limit theories can derive different observable inequalities from common geometric structures. A separate common-theorem calculation is still required before writing
σ_total ≥ B_speed + B_precision
as a valid bound.
The sequence has therefore advanced from a topological possibility to an exact local-independence result without overclaiming the thermodynamic consequence.
Two-state chain: no cycle coordinate → operational collapse.
Three-state ring: one cycle coordinate → rank-two operational separation.
Next question: does thermodynamic geometry preserve or recombine those two directions?
That is now the proper next test.
Do not confuse independent observables with independent costs.
First prove the rank. Then prove the thermodynamics.
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