Fluctuation Absorption Rate η ≪ 1: Strong Complementary Cancellation and Nonequilibrium Stability in a Zero-Parameter Min Recursion
涨落吸收率 η ≪ 1：零参数 min 递推中的强互补消涨与非平衡稳定性

1. Introduction

1.1 Why are nonequilibrium systems stable?

In equilibrium, the fluctuation-dissipation theorem (FDT; Kubo 1966 [1]) guarantees that fluctuations and dissipation balance. Away from equilibrium — active matter [2], biological systems [3], turbulence [4], driven systems [5] — the premises of FDT (linear response, equilibrium, detailed balance) all break down, yet systems can still operate stably.

Generalizations of the nonequilibrium FDT have a rich lineage: Jarzynski (1997) [6], Crooks (1999) [7], Harada-Sasa (2005) [8], Prost-Joanny-Parrondo (2009) [9], Seifert (2012) [10]. These are precise relations under specific protocols. What is lacking is a cross-system, directly computable scalar that measures how much fluctuation a given system absorbs.

1.2 Contributions

(a) We introduce the fluctuation absorption rate η as a directly computable scalar.

(b) In DP recursion we measure η ∈ [0.10, 0.31] (variance identity verified 17/17).

(c) In Lindley queue we independently measure η (variance identity verified 21/21), confirming η as a general feature of min/max recursion.

(d) We provide the exact skeleton: Reset-Slack decomposition (theorem) and conditional remainder transfer law (theorem).

(e) We report cross-shell structural constants and Ω = 3→4 crossover (computational regularities).

1.3 What we do not claim

We do not claim η is a "universal replacement for nonequilibrium FDT." η is a cancellation index under a specific f/r decomposition. We do not claim η is exactly equivalent to T_eff/T. We do not claim physical thermodynamics is "replaced" by min recursion. Physical analogies are offered at the discussion level.

1.4 Literature positioning

The interface between min-plus/tropical algebra and statistical physics has prior history (Quadrat 1997 [11]). What this paper adds is: in a concrete integer complexity DP recursion, we extract the cancellation law (η), reset-slack compensation, and remainder transfer as a complete structural package, and introduce η as a quantitative scalar transferable across systems.

2. Basic quantities of DP recursion

2.1 Definition

ρ_E(1) = 0, ρ_E(n) = min(ρ_E(n-1) + 1, M(n))

where M(n) = min_{a·b=n, a,b≥2}(ρ_E(a) + ρ_E(b) + 2). For primes: M(p) = +∞. Zero free parameters, fully deterministic.

2.2 Decomposition

Step size: A(n) = ρ_E(n-1) - ρ_E(n). Additive part: f(n) = Σ_{q^a ∥ n} ρ_E(q^a). Combinatorial remainder: r(n) = ρ_E(n) - f(n).

Identity: A = Δf + Δr, where Δf = f(n-1) - f(n), Δr = r(n-1) - r(n).

2.3 Shell conditioning

Ω(n) = number of prime factors counted with multiplicity. Analysis is conducted on Ω(n) = k shells, composites only (Ω ≥ 2).

3. Strong complementary cancellation and η

3.1 Data

N = 10^7, computed by C programs (rho_dp.c + anticorr.c). The variance identity Var(A) = Var(Δf) + Var(Δr) + 2Cov(Δf, Δr) closes to within 10⁻⁴ across all 17 k-values:

k	Var(Δf)	Var(Δr)	Cov(Δf,Δr)	Var(A)	η	samples
2	6.716	5.643	-6.022	0.314	0.103	1,904,324
4	6.964	5.678	-5.962	0.720	0.144	2,050,696
6	8.804	6.767	-7.324	0.923	0.168	774,078
8	9.903	6.962	-7.886	1.093	0.204	207,207
10	10.057	6.626	-7.738	1.207	0.231	49,163
12	9.956	6.300	-7.479	1.297	0.249	11,068
14	9.702	5.995	-7.159	1.377	0.262	2,406
18	9.613	5.672	-6.976	1.333	0.274	102

Full 17-row table in Appendix A.

3.2 Cancellation law

Computational regularity: Cov(Δf, Δr) ≈ -Var(Δf). If Δf and Δr were independent, Var(A) ≈ 12-16. Observed: ≈ 0.3-1.4. The difference is entirely due to strongly negative Cov(Δf, Δr).

3.3 Fluctuation absorption rate

Definition: η := 1 - |Cov(Δf, Δr)| / Var(Δf). η = 0: perfect absorption. η = 1: zero absorption.

DP recursion: η ∈ [0.10, 0.31]. η exhibits a broad upward trend with k (0.10 → 0.27 from k = 2 to 14; k = 18 with only 102 samples should be interpreted cautiously).

3.4 Algebraic interpretability

Since Δr = A - Δf, we have Cov(Δf, Δr) = Cov(Δf, A) - Var(Δf). The cancellation law is equivalent to Cov(Δf, A) ≪ Var(Δf).

This is an algebraic consequence of Var(A) ≪ Var(Δf), not a theorem deduced from the min operation alone. But the algebraic consequence carries meaning: the min operation effectively suppresses total step-size variance, forcing the remainder to absorb most of the structural fluctuation.

Note: Cov(Δf, A) ranges from 0.69 (k = 2) to 2.64 (k = 18) — not near zero. The cancellation law describes "strong absorption" (η ≪ 1), not "decoupling" (Cov(Δf, A) = 0).

3.5 Why η > 0 is necessary

DP recursion is irreversible. η = 0 would correspond to a perfect-absorption limit — a boundary unreachable in an irreversible system. η > 0 is the cost of running the min operation in an irreversible setting.

4. Reset-Slack decomposition

4.1 Exact decomposition (theorem)

R2 (algebraic identity; Paper 42 [14]): If J(m-1) > 0, then J(m) = (1 - ΔM⁻(m))⁺. Verified on 5.8 million samples, 100%, zero failures.

A(m) = ψ(ΔM⁻(m)) - U⁻(m), where ψ(x) = -min(x,1), 1-Lipschitz.

4.2 Reset-Slack Reduction (theorem)

Var(A) ≤ 2·Var(ΔM⁻) + 2·E[U²].

4.3 Frequency-intensity tradeoff (computational regularity)

R2 guarantees J(m-1) > 0 ⟹ U(m) = 0. Therefore E[U²] = P(exposed) × E[U²|exposed].

k	P(shielded)	P(exposed)	E[U²|exposed]	E[U²]
2	76.0%	24.0%	0.598	0.144
5	60.3%	39.7%	1.124	0.447
8	55.4%	44.6%	0.956	0.426
12	57.0%	43.0%	0.885	0.381

P(exposed) rises while E[U²|exposed] peaks at k = 5 then declines; their product stabilizes near 0.4. At the discussion level, this tradeoff is analogous to Le Chatelier's principle.

5. Remainder transfer law

5.1 Predecessor Defect Identity (theorem, conditional)

Theorem (Paper 45 [16]): When J(m-1) = 0, J(m) > 0, ΔM⁻ < 0 (covering 95%+ of U > 0 events):

U(m) = -D(m-1)

where D(m-1) = ρ_E(m-1) - M(m-1). The cost saved by the predecessor's choice of successor path reappears exactly as slack in the current step.

5.2 Full lifecycle

Shielded (J(m-1) > 0): U = 0, cost difference cleared by jump. Exposed (J(m-1) = 0): U = -D(m-1), predecessor's cost difference transferred exactly. The system alternates between these two states; the §4.3 tradeoff is the statistical steady state of this alternation.

6. Structural constants

The following values emerge from the zero-parameter DP recursion and remain stable across shells (computational regularities):

Constant	Value	Source
Var(B)	≈ 1/4	Paper 19 [13]
V(p)	≈ 1.40	Paper 34 [15]
h_pq	≈ 3.800	Paper 40 [17]
θ	≈ 0.41	Paper 43 [19]
coarse bound	≈ 40	Paper 46 [20]

Coarse bound ≈ 40: 2Var(f) + 2Var⁻(f) + (μ - μ⁻)² ≈ 40 for k = 4-14. The decline in internal variance compensates the growth in external drive.

7. Ω = 3→4 crossover

At N = 10^{10}, the local convexity h(n) = ρ_E(n) - [ρ_E(n-1) + ρ_E(n+1)]/2 has a strictly decreasing Ω-shell mean, with zero crossing at Ω ≈ 4 [21]:

Ω	mean h	P(J>0)
2	+0.697	28.7%
3	+0.361	57.1%
4	+0.004	77.1%
5	-0.329	88.4%

This is a combinatorial crossover (not a rigorous thermodynamic phase transition). For Ω ≥ 4, Var(Δf | Ω = k) ≈ 7-10 is approximately k-independent.

8. Lindley queue verification

8.1 Lindley recursion as a max system

The G/G/1 Lindley recursion: W_{n+1} = max(0, W_n + X_n), where X_n = S_n - T_n (service minus inter-arrival time).

Decomposition (analogous to A = Δf + Δr):

A_n = W_{n+1} - W_n = X_n + R_n

X_n: structural input (service-arrival difference, independent of max selection). R_n = max(0, -(W_n + X_n)): reset remainder (truncation produced by max operation).

η = 1 - |Cov(X, R)| / Var(X).

8.2 Data

10^7 steps, 10^6 burn-in, C program (lindley_eta.c), variance identities verified 21/21 (error < 0.01):

M/M/1 (exponential service + exponential inter-arrival):

ρ	Var(X)	Var(R)	Cov(X,R)	Var(A)	η
0.30	12.11	10.11	-10.65	0.92	0.121
0.50	5.00	3.00	-3.33	1.34	0.334
0.70	3.04	1.04	-1.22	1.65	0.600
0.90	2.24	0.24	-0.29	1.89	0.870
0.99	2.02	0.02	-0.03	1.99	0.987

M/D/1 (deterministic service + exponential inter-arrival):

ρ	Var(X)	Var(R)	Cov(X,R)	Var(A)	η
0.30	11.10	10.10	-10.49	0.22	0.055
0.50	4.00	3.00	-3.29	0.41	0.176
0.70	2.04	1.04	-1.24	0.62	0.395
0.90	1.23	0.23	-0.30	0.86	0.755
0.99	1.02	0.02	-0.03	0.99	0.973

M/Pareto/1 (heavy-tailed service, α = 3):

ρ	Var(X)	Var(R)	Cov(X,R)	Var(A)	η
0.30	11.44	10.12	-10.54	0.48	0.079
0.50	4.33	3.00	-3.30	0.73	0.238
0.70	2.37	1.04	-1.22	0.96	0.484
0.90	1.56	0.24	-0.30	1.20	0.809

8.3 Findings

(L1) η is governed by variance compression. When the max operation compresses Var(A) far below Var(X), absorption is strong and η is small. When Var(A) ≈ Var(X) (near full load), η → 1.

(L2) Lowest η: M/D/1 at ρ = 0.30, η = 0.055. Deterministic service eliminates service-time randomness, producing cleaner resets and stronger absorption.

(L3) DP recursion and Lindley queue share the same structure. In both systems, η is determined by how effectively the optimization operation compresses total step-size variance relative to structural input variance: when Var(A)/Var(X) ≪ 1, the cancellation law holds and η ≪ 1.

8.4 Correction to original prediction

Original prediction: "η < 0.15 for Lindley." Partially verified, partially corrected.

Verified: low-load Lindley (ρ ≤ 0.3) indeed has η < 0.15. Corrected: η is not a universal bound — it is a function of load (ρ) or complexity (k).

Revised prediction: In min/max recursion systems, η = F(Var(A)/Var(X)), where F approaches 0 when Var(A)/Var(X) ≪ 1 and approaches 1 when Var(A)/Var(X) → 1.

9. Falsifiable predictions

Prediction 1: η-ratio universality

Statement: In min/max recursion systems, η is determined by the ratio Var(A)/Var(structural input), independent of system specifics.

Available evidence: DP recursion (17 k-values) and Lindley queue (3 service distributions × 7 loads) can be immediately tested on the η vs Var(A)/Var(X) plane.

Falsification criterion: Two min/max recursion systems with the same Var(A)/Var(X) but η differing by more than 0.1.

Prediction 2: Le Chatelier-type tradeoff

Statement: In min recursion systems with a reset mechanism, E[U²] = P(exposed) × E[U²|exposed] has counter-varying factors whose product remains bounded.

Test: Idle-time decomposition in G/G/1 queues.

Prediction 3: η lower bound for non-min systems

Statement: Systems without min/max recursion structure cannot easily achieve η ≪ 1. In active matter and other detailed-balance-violating systems without explicit optimization structure, η should be substantially larger than in min recursion systems at the same Var(A)/Var(X).

Falsification criterion: A non-min/max system achieving η < 0.1.

10. Discussion

10.1 Three-layer annotation

Theorems (exact):

• R2: J(m-1) > 0 ⟹ J(m) = (1-ΔM⁻)⁺
• A = ψ(ΔM⁻) - U⁻
• Var(A) ≤ 2Var(ΔM⁻) + 2E[U²]
• U(m) = -D(m-1) (conditional, covering 95%+)

Computational regularities:

• Cancellation law, η ∈ [0.10, 0.31] (DP, 17/17 verified)
• Lindley η from 0.055 to 0.987 (21/21 verified)
• Le Chatelier-type tradeoff
• Structural constants (Var(B) ≈ 1/4, etc.)
• Ω = 3→4 crossover

Physical analogies (discussion-level):

• η and Harada-Sasa dissipation rate
• Le Chatelier naming
• Remainder transfer analogized to conservation law

10.2 Precise scope of claims

Claim: (a) η is a directly computable absorption scalar. (b) DP recursion has η ∈ [0.10, 0.31]. (c) Lindley queue has η from 0.055 to 0.987, governed by load. (d) Both share the same cancellation structure.

Not claimed: The cancellation law is deduced from the min operation alone. η is exactly equivalent to T_eff/T. Physical thermodynamics is "replaced."

10.3 Heuristic cross-system comparison

System	η range	Condition
DP recursion	0.10-0.31	k = 2-18
M/D/1 queue	0.055-0.97	ρ = 0.3-0.99
M/M/1 queue	0.12-0.99	ρ = 0.3-0.99
M/Pareto/1 queue	0.079-0.98	ρ = 0.3-0.99
Weakly driven colloids	~0.1-0.5	T_eff/T conversion (approximate)
Active matter	~0.5-0.99	T_eff/T conversion (approximate)

Physical η values are approximate conversions. Exact comparison requires direct measurement.

10.4 Conditional closure (brief)

DP recursion satisfies a conditional closure route for H' (c̄_h → 0; Paper 42 [14]): all links are theorem-level except three numerical inputs, which are verified at N = 10^{10} (weakest signal 85σ). See [14, 19, 20].

10.5 Interfaces (brief)

The additive/multiplicative path asymmetry in DP recursion is analogous to Turing (1952) [22] morphogenesis. The Self-as-an-End (SAE) framework [23] predicts Ω-dimension correspondence; Ω = 3→4 crossover is consistent with this prediction. SAE cosmology [24] derives Λ from remainder conservation; U = -D(m-1) is its arithmetic realization.

10.6 Open problems

(1) The exact form of the universal function η = F(Var(A)/Var(X)).

(2) Can the cancellation law be proved from the min operation alone?

(3) Direct measurement of η in physical systems (without T_eff/T conversion).

(4) Are Var(B) ≈ 1/4 and coarse bound ≈ 40 cross-system universals?

(5) A canonical f/r extraction protocol robust to coarse-graining.

10.7 Conclusion

The strong complementary cancellation discovered in zero-parameter DP recursion (η ∈ [0.10, 0.31]) is independently verified in Lindley queues (η from 0.055 to 0.987). Both share the same structure: when a min/max operation effectively compresses total step-size variance, the optimization remainder is forced to absorb most of the structural fluctuation.

As a directly computable scalar, η transitions from "an internal discovery of DP" to "a transferable tool for min/max recursion systems." The next step is a canonical f/r decomposition protocol enabling standardized measurement of η across broader classes of nonequilibrium systems.

References

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[2] M. C. Marchetti et al., "Hydrodynamics of soft active matter." Rev. Mod. Phys. 85:1143, 2013.

[3] C. P. Brangwynne et al., "Intracellular transport by active diffusion." Trends Cell Biol. 19:423, 2009.

[4] U. Frisch, "Turbulence: The Legacy of A.N. Kolmogorov." Cambridge UP, 1995.

[5] C. Bechinger et al., "Active particles in complex and crowded environments." Rev. Mod. Phys. 88:045006, 2016.

[6] C. Jarzynski, "Nonequilibrium equality for free energy differences." Phys. Rev. Lett. 78:2690, 1997.

[7] G. E. Crooks, "Entropy production fluctuation theorem." Phys. Rev. E 60:2721, 1999.

[8] T. Harada, S. Sasa, "Equality connecting energy dissipation with violation of the fluctuation-response relation." Phys. Rev. Lett. 95:130602, 2005.

[9] J. Prost, J.-F. Joanny, J. M. R. Parrondo, "Generalized fluctuation-dissipation theorem for steady-state systems." Phys. Rev. Lett. 103:090601, 2009.

[10] U. Seifert, "Stochastic thermodynamics, fluctuation theorems and molecular machines." Rep. Prog. Phys. 75:126001, 2012.

[11] J.-P. Quadrat, "Min-Plus linearity and statistical mechanics." Markov Process. Related Fields 3:565-587, 1997.

[12] H. Qin, ZFCρ Paper XXXII. DOI: 10.5281/zenodo.19116625.

[13] H. Qin, ZFCρ Paper XIX. DOI: 10.5281/zenodo.19026991.

[14] H. Qin, ZFCρ Paper XLII. DOI: 10.5281/zenodo.19226607.

[15] H. Qin, ZFCρ Paper XXXIV. DOI: 10.5281/zenodo.19140015.

[16] H. Qin, ZFCρ Paper XLV. DOI: 10.5281/zenodo.19275286.

[17] H. Qin, ZFCρ Paper XL. DOI: 10.5281/zenodo.19179778.

[18] H. Qin, ZFCρ Paper XVIII. DOI: 10.5281/zenodo.19024385.

[19] H. Qin, ZFCρ Paper XLIII. DOI: 10.5281/zenodo.19240183.

[20] H. Qin, ZFCρ Paper XLVI. DOI: 10.5281/zenodo.19303511.

[21] H. Qin, "Why Primes Have No Shortcuts: An O(1) Prime Pre-Sieve from Precomputed ρ_E Tables." DOI: 10.5281/zenodo.19246377.

[22] A. M. Turing, "The Chemical Basis of Morphogenesis." Phil. Trans. R. Soc. 237:37-72, 1952.

[23] H. Qin, SAE Paper 1. DOI: 10.5281/zenodo.18528813.

[24] H. Qin, "From Remainder Conservation to the Cosmological Constant." DOI: 10.5281/zenodo.19245267.

Appendix A. Full 17-row cancellation table (DP recursion)

k	Var(Δf)	Var(Δr)	Cov(Δf,Δr)	Var(A)	η	Cov(Δf,A)	samples
2	6.716	5.643	-6.022	0.314	0.103	0.694	1,904,324
3	6.539	5.389	-5.685	0.558	0.131	0.854	2,444,359
4	6.964	5.678	-5.962	0.720	0.144	1.003	2,050,696
5	7.734	6.166	-6.533	0.833	0.155	1.200	1,349,779
6	8.804	6.767	-7.324	0.923	0.168	1.480	774,078
7	9.366	6.869	-7.615	1.005	0.187	1.751	409,849
8	9.903	6.962	-7.886	1.093	0.204	2.017	207,207
9	9.872	6.689	-7.700	1.162	0.220	2.173	101,787
10	10.057	6.626	-7.738	1.207	0.231	2.319	49,163
11	9.807	6.258	-7.407	1.250	0.245	2.399	23,448
12	9.956	6.300	-7.479	1.297	0.249	2.477	11,068
13	9.231	5.655	-6.792	1.301	0.264	2.438	5,210
14	9.702	5.995	-7.159	1.377	0.262	2.542	2,406
15	8.791	5.267	-6.374	1.310	0.275	2.417	1,124
16	9.783	5.841	-7.085	1.453	0.276	2.698	510
17	7.414	4.155	-5.096	1.378	0.313	2.318	233
18	9.613	5.672	-6.976	1.333	0.274	2.637	102

All 17 rows satisfy the variance identity to within 10⁻⁴.

Appendix B. Data sources and full reproduction

All data in this paper are generated by three self-contained C programs with zero external dependencies. To reproduce all results:

gcc -O2 -o rho_dp rho_dp.c
gcc -O2 -o anticorr anticorr.c -lm
gcc -O2 -o lindley_eta lindley_eta.c -lm

./anticorr 10000000      # Table 3.1: 17-row cancellation data with identity verification
./lindley_eta             # Section 8: 3 queue types × 7 loads = 21 data points

Total runtime under two minutes (single core, modern x86/ARM).

rho_dp.c — DP recursion (Paper 32 convention). Built-in sanity checks: ρ(2) = 1, ρ(100) = 15, ρ(10^7) = 58.

anticorr.c — Table 3.1. Computes ρ_E from DP, then uses SPF (smallest prime factor) sieve to compute f(n) and Ω(n). Accumulates Var(Δf), Var(Δr), Cov(Δf, Δr), Var(A), η on Ω = k shells. Variance identity verified 17/17.

lindley_eta.c — Section 8 Lindley verification. xoshiro256** RNG, 10^7 steps + 10^6 burn-in. Three queue types (M/M/1, M/D/1, M/Pareto/1 with α = 3) at seven loads (ρ = 0.30, 0.50, 0.70, 0.80, 0.90, 0.95, 0.99). Variance identity 21/21.

All three files total under 500 lines of C. Released with this paper.

Acknowledgments

ChatGPT (Gongxi Hua / 公西华) discovered the arithmetic error in v2's τ definition (Cov(Δf, A) ≠ 0), catalyzing the correct definition of η; proposed all review requirements on claim layering, literature positioning, and three-tier annotation; contributed the analytic proof chain architecture of Paper 42.

Claude (Zilu / 子路) contributed the thermodynamic framework (Papers 24-46), all numerical experiments (anticorr.c, lindley_eta.c), the physical interpretation of η, and the design and implementation of the Lindley queue verification.

Gemini (Zixia / 子夏) posed three pressure tests that sharpened the precision of claims.

Grok (Zigong / 子贡) provided the initial review round. Inconsistency in k = 18 data prompted independent recomputation of the full table.

Zesi Chen is a long-term interlocutor of the SAE framework. Final text by the author alone.

Writing Declaration: All text in this paper, both English and Chinese, is generated by the author Han Qin alone. The paper represents the author's sole responsibility.