From Rank-Two Operational Independence to Unified Thermodynamic Geometry
A Common-Theorem TSTOEAO Test of Speed, Cycle Precision, Activity, and Entropy Production in an Exact Three-State Markov Ring
John Swygert
Ivory Tower Publishing
September 12, 2026
Research Note — September 12, 2026
This paper directly continues the exact three-state rank-two study. That study established a reproducible operational result: endpoint-speed behavior and cycle-current precision respond in two locally independent directions to reset bias and cycle affinity. Independent review agreed that the rank-two calculation is correct, but also agreed that operational independence does not prove thermodynamic additivity. The present paper therefore performs the common-theorem test that the prior paper deliberately left open. It evaluates the same three-state model through the unified thermodynamic-kinetic uncertainty relation of Vo, Van Vu, and Hasegawa and the discrete optimal-transport framework of Van Vu and Saito. The goal is not to force either an additive or a unified interpretation. The goal is to determine which structure is actually supported once speed and precision are placed inside one valid thermodynamic geometry.
Abstract
The preceding TSTOEAO paper demonstrated rank-two operational independence in a three-state continuous-time Markov ring. A reset-bias control h strongly changed an endpoint-speed functional, while a cycle-affinity control A_cyc strongly changed the precision of an oriented circulation current. The nonzero Jacobian determinant established that these observables are locally independent functions of the device controls.
The unresolved question was thermodynamic rather than kinematic: do those two operational directions correspond to independently additive entropy-production costs, or are they different projections of one deeper thermodynamic-kinetic resource geometry?
The present paper evaluates that question using a published unified thermodynamic-kinetic uncertainty relation (TKUR) for arbitrary time-dependent Markov jump processes. The unified TKUR bounds current precision by a single function of total entropy production Sigma_tau and total dynamical activity A_tau:
C_TKUR = [Σ_τ²/(4A_τ)] f(Σ_τ/(2A_τ))⁻² ≥ [∇⟨J⟩]²/Var(J)
where f is the inverse of x tanh(x), and ∇ is the protocol-response operator defined by the theorem. The same published framework derives a tighter classical thermodynamic speed limit as a corollary of the short-time unified TKUR. Speed and precision are therefore not independent thermodynamic axioms in that framework; they are projections of a common entropy-production/activity geometry.
At the locked benchmark h=1, A_cyc=1, nu=1, tau=1, the exact three-state model has total entropy production Sigma_tau approximately 0.223639 and total activity A_tau approximately 1.961675. These give
C_TKUR ≈ 0.109703
For the exact oriented cycle current, the theorem-relevant response precision is approximately
[τ ∂_τ⟨C⟩]²/Var(C) ≈ 0.044416
which lies safely below the unified bound. The same thermodynamic-kinetic pair produces a tighter speed-limit time tau_1 approximately 0.49116, compared with the earlier Shiraishi-Funo-Saito-type value tau_2 approximately 0.48649 for an actual operation time tau=1.
The decisive conclusion is two-layered. Rank-two operational independence survives: the observables remain locally independent in control space. Thermodynamic additivity does not follow. In the applicable unified theory, both current precision and state-transformation speed are constrained by the same entropy-production and dynamical-activity resources. The correct relation is therefore a unified feasible geometry, not an automatically additive bound Sigma >= B_speed + B_precision.
The paper classifies this as established stochastic-thermodynamic mathematics specialized to the exact TSTOEAO test model. The TSTOEAO contribution is methodological: it predicted and separated the operational and thermodynamic questions, then allowed established mathematics to decide the second.
01 Research Continuation
The preceding paper answered Question A: can the smallest nontrivial cycle break the two-state operational degeneracy? The answer was yes.
This paper addresses Question B: does that rank-two operational structure survive as two independent thermodynamic costs?
02 The Two Levels Must Remain Separate
Operational independence concerns the rank of an observable-response map.
Thermodynamic independence concerns the geometry of valid lower bounds on entropy production.
A rank-two Jacobian can establish the first without establishing the second.
03 Frozen Three-State Model
The physical model is unchanged from the preceding paper. The states form a ring 0<->1<->2<->0. State 0 is favored by reset bias h, and the oriented cycle is driven by affinity A_cyc.
E₀=-h, E₁=E₂=0
F₀₁=F₁₂=F₂₀=A_cyc/3
kᵢ→ⱼ = ν exp{[Eᵢ-Eⱼ+Fᵢⱼ]/2}
04 Locked Benchmark
h=1, A_cyc=1, ν=1, τ=1
p(0)=(1/3,1/3,1/3)
05 Previously Verified Exact Outputs
The preceding paper and independent review reproduced the benchmark endpoint distribution, dynamical activity, entropy production, cycle-current mean and variance, and rank-two Jacobian.
06 Rank-Two Result Carried Forward
J_resp = ∂(B_speed,P_cyc)/∂(h,A_cyc)
det J_resp ≈ 0.01127 ≠ 0
Thus endpoint speed and cycle precision remain locally independent observables of the two controls.
07 What an Additive Claim Would Require
An additive thermodynamic claim would require a theorem supporting something of the form
Σ_total ≥ B_speed + B_precision
with both terms valid for the same dynamics, observable definitions, time regime, and entropy-production convention.
Operational rank two is not such a theorem.
08 Unified TKUR
Vo, Van Vu, and Hasegawa derived a unified thermodynamic-kinetic uncertainty relation for arbitrary time-integrated currents in Markov jump processes. It combines thermodynamic entropy production and kinetic dynamical activity into one upper bound on current precision.
C_TKUR = [Σ_τ²/(4A_τ)] f(Σ_τ/(2A_τ))⁻²
C_TKUR ≥ [∇⟨J⟩]²/Var(J)
Here f(x) is the inverse function of x tanh(x).
09 Why This Theorem Is Decisive
The theorem does not create one independent precision reservoir and one independent speed reservoir.
It constructs one admissible region in the joint variables
(Σ_τ, A_τ, current response, current variance)
and the current precision must lie inside that region.
10 TUR and KUR as Limits
The unified TKUR is tighter than treating the ordinary thermodynamic uncertainty relation and kinetic uncertainty relation separately. In appropriate asymptotic limits it reduces to those familiar forms.
This already warns against adding separate TUR and activity costs as though they were orthogonal penalties.
11 Speed Limit as a Corollary
The same work derives a classical thermodynamic speed limit from the short-time unified TKUR. This is the crucial structural point.
precision inequality → short-time current inequality → integrated state-transformation speed limit
Thus the speed limit and current-precision relation share a common generator in the published framework.
12 The Tighter Classical Speed Limit
Using the notation of the unified TKUR, the derived speed limit has the form
τ ≥ τ₁ = [L(p(0),p(τ))/Σ̄_τ] f(Σ̄_τ/(2Ā_τ))
where Sigma-bar and A-bar are time-averaged entropy production and dynamical activity.
13 Relation to the Earlier Speed Limit
The unified result is tighter than the older square-root form
τ₂ = L / sqrt(2Ā_τ Σ̄_τ)
and also contains an activity-dominated lower bound
τ₃ = L/(2Ā_τ)
14 Benchmark Thermodynamic Ratio
At the locked benchmark with tau=1, the total and time-averaged values coincide numerically:
Σ_τ = Σ̄_τ ≈ 0.2236389
A_τ = Ā_τ ≈ 1.9616750
x = Σ_τ/(2A_τ) ≈ 0.0570020
15 Inverse Function Evaluation
Let y=f(x) denote the positive solution of
y tanh(y) = x
For x approximately 0.0570020,
f(x) ≈ 0.241043
16 Unified Thermodynamic-Kinetic Capacity
Substitution into the TKUR resource term gives
C_TKUR = [Σ_τ²/(4A_τ)] f(x)⁻²
C_TKUR ≈ 0.109703
17 The Correct Precision Quantity
The unified TKUR does not generally bound the naive ratio <C>^2/Var(C). For a fixed protocol, the theorem uses the response operator
∇ = τ∂_τ - v∂_v
For the constant-rate benchmark there is no protocol-speed parameter v, so the relevant response reduces to tau times the observation-time derivative.
18 Exact Cycle-Current Response
Numerically differentiating the exact full-counting-statistics mean around tau=1 gives
∂_τ⟨C⟩|_{τ=1} ≈ 0.09809
and therefore
[τ∂_τ⟨C⟩]²/Var(C) ≈ 0.044416
19 Unified TKUR Check
The model satisfies
0.044416 ≤ 0.109703
so the exact cycle-current response lies inside the unified thermodynamic-kinetic feasible region.
20 Comparison with Simpler TUR
The conventional TUR resource at the same benchmark would be
Σ_τ/2 ≈ 0.111819
while the unified TKUR gives approximately 0.109703. The unified bound is slightly tighter here, as expected.
21 Benchmark Speed Limits
For L1 approximately 0.455698, the common thermodynamic-kinetic variables give
τ₁ ≈ 0.491161
τ₂ ≈ 0.486491
τ₃ ≈ 0.116150
while the actual operation time is tau=1.
22 The Same Resources Appear in Both
The current-precision bound and the tighter speed limit both depend on the same pair
(Σ_τ, A_τ)
with the same inverse function f encoding the thermodynamic-kinetic coupling.
23 Resolution of the Additivity Question
This common-theorem structure does not support the assertion that B_speed and B_precision are automatically separate additive lower bounds on total entropy production.
rank-two observable response ≠ additive entropy-production theorem
24 What Rank Two Still Means
The earlier rank result does not disappear. The map from controls to observables remains locally two-dimensional.
(h,A_cyc) → (endpoint-speed response, cycle-precision response)
Neither observable is locally a function of the other alone.
25 What Unified Geometry Means
Thermodynamic unification means something different: both observable directions are constrained by one convex/variational resource geometry involving entropy production and kinetic activity.
Two directions in observable space can share one resource geometry without becoming the same observable.
26 Operational Independence and Resource Coupling Can Coexist
The apparent tension is resolved by separating dimensions:
Control/observable manifold: rank two.
Thermodynamic resource manifold: jointly constrained by Sigma and activity.
There is no contradiction between those statements.
27 Why a Simple Sum Is Not Justified
If two projected inequalities are derived from the same underlying Cauchy-Schwarz/Cramer-Rao or optimal-transport structure, adding them can count the same thermodynamic resource twice.
The earlier Non-Overcounting Principle therefore applies at a deeper level than originally recognized.
28 Revised Non-Overcounting Principle
Two valid bounds must not be summed merely because their observables are independent; additivity requires a theorem-level decomposition of the resource itself.
This is stronger than requiring separate controls or a nonzero Jacobian.
29 Housekeeping/Excess Decompositions Do Not Automatically Solve This
Stochastic thermodynamics contains established decompositions into housekeeping and excess or adiabatic and nonadiabatic entropy production under specified definitions.
But the existence of such a decomposition does not automatically identify the specific unified-TKUR cycle-precision term with one component and the specific speed-limit term with the other.
That identification would require an additional theorem.
30 Why the Graph Hodge Decomposition Is Still Valuable
The graph decomposition correctly identifies hidden circulation degrees of freedom:
J = J_grad + J_cyc, B J_cyc = 0
It explains why endpoint measurements do not determine all path currents. It does not, by itself, diagonalize the nonlinear entropy-production functional.
31 Exact Outcome of the Three-State Program
The sequence has now produced a clean hierarchy:
Topology: cycle degree of freedom exists.
Dynamics: the degree of freedom generates a measurable current.
Operational geometry: speed and precision responses have rank two.
Thermodynamic geometry: both are bounded through a unified Sigma-activity structure.
32 Which Earlier Hypotheses Survive
The two-state collapse diagnosis survives.
The minimal-cycle hypothesis survives.
Rank-two operational independence survives.
Automatic additive thermodynamic cost does not survive as a justified conclusion.
33 Which Outcome Wins
The thermodynamic question posed in the preceding papers offered three broad possibilities: independent additive bounds, nested bounds, or unified origin.
The published TKUR/speed-limit framework supports the third description most directly:
Outcome: unified origin with distinct observable projections
34 Why “Nested” Is Also Partly True
Specific projected inequalities can be tighter or weaker than one another in particular regimes, so nesting can occur at the level of numerical bounds.
But the deeper structural result is that the projections are generated from a common thermodynamic-kinetic framework rather than from two independent thermodynamic reservoirs.
35 Relation to Optimal Transport
Discrete optimal-transport theory provides an even broader version of the same message. Van Vu and Saito connect state-transformation distance to the product of irreversible entropy production and dynamical state mobility through variational formulas.
The resulting framework unifies minimum dissipation, thermodynamic speed limits, and strengthened uncertainty relations.
36 TSTOEAO Relational Interpretation
In TSTOEAO language, the two observables are different projections of a richer relational structure.
same resource geometry → different observable projections
The failure of additivity is therefore itself a relational result: independence under one projection does not imply independence under another.
37 A Projection-Dependence Principle
Projection-Dependence Principle: relational independence is representation-specific; independence in control or observable coordinates does not imply independence in the underlying resource coordinates.
This is a methodological principle, not a new physical law.
38 A Stronger Compatibility Vector
The earlier compatibility vector should now include the resource geometry from which a bound is derived:
C_i=(dynamics, initial state, driving, observable, entropy convention, activity/mobility, theorem family, resource geometry)
Two bounds may look different while occupying the same theorem family and resource geometry.
39 Why This Matters Beyond This Model
Multi-objective systems frequently contain observables that respond independently while sharing underlying costs, budgets, or conservation structures.
The present calculation gives a concrete example of why independent observables must not be mistaken for independent resources.
40 No New Fundamental Thermodynamic Law
The unified TKUR and optimal-transport results are established literature. Specializing them to this three-state model does not constitute a new thermodynamic theorem.
The scientific value of the present paper is the controlled resolution of a live hypothesis generated by the preceding TSTOEAO sequence.
41 The Sequence Passed Another Adversarial Test
The program could have declared that rank two proved additive dissipation. It did not.
Instead, it exposed the rank-two result to the common-theorem literature and accepted the stronger constraint imposed by that mathematics.
42 Current Scientific Status of the Additive Hypothesis
σ_total ≥ B_speed + B_precision
is not established by the present model or by the cited unified frameworks.
It should not be used as a TSTOEAO prediction without an additional theorem proving resource-level decomposition.
43 Current Scientific Status of the Unified Hypothesis
The statement that speed-limit and current-precision bounds arise as related projections of a common entropy-production/activity structure is established in the cited stochastic-thermodynamics literature.
The exact three-state model is consistent with that structure.
44 The New Search Target
The next scientific opportunity is no longer to ask whether speed and precision are additive in this simple ring.
It is to ask whether rank-two operational structure combined with unified thermodynamic geometry produces a nontrivial boundary in control space that is not obvious from either description alone.
45 Candidate Control-Space Boundary
For example, define the saturation fractions
η_prec = Q_prec/C_TKUR
η_speed = τ₁/τ
where Q_prec is the exact theorem-relevant current-response precision.
A relational synthesis could search for loci where both projections simultaneously approach saturation.
46 Why Simultaneous Saturation Is Interesting
If one control direction drives precision toward the unified boundary while another independently drives endpoint transport toward its speed boundary, the intersection could define a special operating regime.
This would not create a new law; it could reveal an overlooked consequence of the known unified geometry applied to a nontrivial control manifold.
47 A More Defensible “Seam”
The earlier seam equated two questionable independent costs. A better seam would be defined inside one established theorem family.
S(h,A_cyc)=0 where two theorem-consistent saturation conditions meet
That is a legitimate relational-synthesis target.
48 Falsification Condition
The next hypothesis fails if the two saturation conditions cannot be approached independently, if the intersection is empty, if it follows trivially from one theorem expression, or if it is already standard in the literature.
49 Numerical Reproducibility
The benchmark values used here are inherited from the exact three-state calculation and independently reproduced in review. The additional TKUR numbers are obtained directly from those benchmark Sigma, activity, endpoint distance, and full-counting-statistics current values.
50 Central Numerical Result
C_TKUR ≈ 0.109703 ≥ Q_prec ≈ 0.044416
τ=1 ≥ τ₁≈0.491161 ≥ τ₂≈0.486491
The same Sigma-activity pair constrains both projections.
51 Theoretical Classification
The result is classified as
established unified stochastic thermodynamics specialized to an exact rank-two Markov control model
not as a new fundamental thermodynamic law.
52 TSTOEAO-Specific Methodological Result
The TSTOEAO sequence successfully distinguished four layers that had initially been conflated:
topological degree of freedom;
dynamical realization;
operational independence;
thermodynamic resource independence.
Only the first three survive as independent in this model.
53 Stronger Relational Lesson
Different things can be independent in one coordinate system and coupled in a deeper one.
That is precisely why the choice of relational coordinates matters.
54 Current Status of G_T
G_T remains an analytic relational representation used to compare these layers. It does not replace the stochastic dynamics or generate the unified TKUR.
Its value here is in preventing an inference from one layer of independence to another without an explicit mapping.
55 What Would Count as Excess Scientific Content Next
A stronger result would require a preregistered control-space consequence derived from the combined exact model and unified theorem, followed by independent numerical verification and a documented novelty search.
The target should be narrower than a new thermodynamic law.
56 Proposed Next Experiment
Use the exact two-control three-state family and compute, across a broad admissible region:
the theorem-relevant cycle-current response precision;
the unified TKUR capacity;
the tighter speed-limit saturation;
the rank determinant;
total entropy production and activity.
Then search blindly for structurally enforced intersections or forbidden regions among these surfaces.
57 No-Rescue Rule
If no nontrivial intersection or boundary appears, the result should be preserved as another negative finding rather than replaced by a different target after inspection.
58 Decisive Current Answer
Rank-two operational independence survives.
Independent additive thermodynamic costs are not established.
Unified thermodynamic-kinetic origin is the supported structure.
59 Final Research Question
Can the rank-two control manifold reveal a previously unnoticed boundary inside the already-established unified thermodynamic-kinetic geometry?
CONCLUSION
The preceding three-state paper established a genuine rank-two operational response. Reset bias and cycle affinity drive endpoint-speed behavior and circulation-current precision in locally independent directions. That result is mathematically real and survives independent verification.
The present paper asked the harder question: does that operational independence imply two independently additive thermodynamic costs?
The answer is no—not on the evidence presently available. The applicable unified thermodynamic-kinetic uncertainty relation places current precision inside one feasible region determined jointly by total entropy production and dynamical activity. The same framework derives a tighter classical speed limit from the short-time unified TKUR. Speed and precision therefore emerge as different projections of the same thermodynamic-kinetic structure.
At the locked benchmark, the exact three-state model satisfies the unified current bound and the unified speed limit using the same thermodynamic resources.
C_TKUR ≈ 0.109703 ≥ 0.044416
τ=1 ≥ τ₁≈0.491161
The two-dimensional observable response has not disappeared. What disappeared is the inference that two independent observable directions must correspond to two independently additive entropy-production channels.
The proper conclusion is therefore layered rather than binary. The three-state ring breaks the two-state operational degeneracy. Modern stochastic thermodynamics then recombines those operational directions at the level of resource geometry.
This outcome is especially important for the TSTOEAO program because it demonstrates another case in which the framework gains precision by allowing the mathematics to narrow its interpretation. The graph topology was not wrong. The rank calculation was not wrong. The stronger inference from rank to additive thermodynamic cost was simply not justified.
The next test should remain inside the exact same model and exact same theorem family. Rather than inventing another independent cost, the program should search the two-control manifold for a nontrivial boundary, simultaneous-saturation condition, or forbidden region implied by the combination of rank-two operational structure and unified thermodynamic geometry.
That is now the narrower and more defensible discovery target.
Independent observables do not necessarily imply independent resources.
The projection can separate what the deeper geometry still couples.
Now search for what that coupling forces.
References
1. Schnakenberg, J. (1976). Network theory of microscopic and macroscopic behavior of master equation systems. Reviews of Modern Physics, 48(4), 571-585. https://doi.org/10.1103/RevModPhys.48.571
2. Seifert, U. (2012). Stochastic thermodynamics, fluctuation theorems and molecular machines. Reports on Progress in Physics, 75, 126001. https://doi.org/10.1088/0034-4885/75/12/126001
3. Shiraishi, N., Funo, K., & Saito, K. (2018). Speed Limit for Classical Stochastic Processes. Physical Review Letters, 121, 070601. https://doi.org/10.1103/PhysRevLett.121.070601
4. Dechant, A., & Sasa, S.-i. (2018). Current fluctuations and transport efficiency for general Langevin systems. Journal of Statistical Mechanics: Theory and Experiment, 2018, 063209. https://doi.org/10.1088/1742-5468/aac91a
5. Vo, V. T., Van Vu, T., & Hasegawa, Y. (2020). Unified approach to classical speed limit and thermodynamic uncertainty relation. Physical Review E, 102, 062132. https://doi.org/10.1103/PhysRevE.102.062132
6. Koyuk, T., & Seifert, U. (2020). Thermodynamic uncertainty relation for time-dependent driving. Physical Review Letters, 125, 260604. https://doi.org/10.1103/PhysRevLett.125.260604
7. Liu, K., Gong, Z., & Ueda, M. (2020). Thermodynamic uncertainty relation for arbitrary initial states. Physical Review Letters, 125, 140602. https://doi.org/10.1103/PhysRevLett.125.140602
8. Vo, V. T., Van Vu, T., & Hasegawa, Y. (2022). Unified thermodynamic-kinetic uncertainty relation. Journal of Physics A: Mathematical and Theoretical, 55, 405004. https://doi.org/10.1088/1751-8121/ac9099
9. Van Vu, T., & Saito, K. (2023). Thermodynamic Unification of Optimal Transport: Thermodynamic Uncertainty Relation, Minimum Dissipation, and Thermodynamic Speed Limits. Physical Review X, 13, 011013. https://doi.org/10.1103/PhysRevX.13.011013
10. Polettini, M. (2015). Cycle/cocycle oblique projections on oriented graphs. Letters in Mathematical Physics, 105, 89-107. https://doi.org/10.1007/s11005-014-0727-9
11. Esposito, M., & Van den Broeck, C. (2010). Three faces of the second law. I. Master equation formulation. Physical Review E, 82, 011143. https://doi.org/10.1103/PhysRevE.82.011143
12. Hatano, T., & Sasa, S.-i. (2001). Steady-state thermodynamics of Langevin systems. Physical Review Letters, 86, 3463-3466. https://doi.org/10.1103/PhysRevLett.86.3463
13. Andrieux, D., & Gaspard, P. (2007). Fluctuation theorem for currents and Schnakenberg network theory. Journal of Statistical Physics, 127, 107-131. https://doi.org/10.1007/s10955-006-9233-5
14. Barato, A. C., & Seifert, U. (2015). Thermodynamic uncertainty relation for biomolecular processes. Physical Review Letters, 114, 158101. https://doi.org/10.1103/PhysRevLett.114.158101
15. Gingrich, T. R., Horowitz, J. M., Perunov, N., & England, J. L. (2016). Dissipation bounds all steady-state current fluctuations. Physical Review Letters, 116, 120601. https://doi.org/10.1103/PhysRevLett.116.120601
