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The exact lower endpoint of unicyclic skewness minus kurtosis

Abstract

For every finite cycle size at least two and every strictly positive real rate vector, S − K is at least C = 3(107 − 51√17)/2048; C is the greatest lower bound of all actual values.

Definition 1.1 (The detailed source’s lower endpoint).

Formalization. D5/S3/StatisticalMechanics/UnicyclicSkewnessMinusKurtosisLowerBound.lowerBound (✓ std3).

Citation. Krzysztof Ptaszyński (2022). Bounds on skewness and kurtosis of steady-state currents. DOI: 10.1103/PhysRevE.106.024119. URL: https://arxiv.org/abs/2204.08421v4.

Commentary.

The primary arXiv v4 main.tex display at line 431 and Figure 6 concern S − K. The summary at line 133 instead labels the interval K − S. This sign conflict is retained: the present endpoint follows the detailed display and figure. Line 424 evaluates the asymptotic exceptional-rate family a = A/N with A = (5 + √17)/4. Line 417 and Appendix line 792 describe the universal extremal topology as numerical/conjectural; no such topology is a hypothesis.

Definition 1.2 (Every positive vector and the entire actual value set).

Formalization. D5/S3/StatisticalMechanics/UnicyclicSkewnessMinusKurtosisLowerBound.claim (✓ std3).

Source. Repository-derived.

Acknowledgement. Krzysztof Ptaszyński (2022). Bounds on skewness and kurtosis of steady-state currents. DOI: 10.1103/PhysRevE.106.024119. URL: https://arxiv.org/abs/2204.08421v4.

Commentary.

The first clause includes every natural n ≥ 2 and every k : Fin n → ℝ with all coordinates strictly positive. The original skewness and kurtosis use the same vector, with repeated coordinates counted with multiplicity. The IsGLB clause uses exactly the values realized by these finite vectors, with unbounded n. It imposes neither a topology classification nor an abstract-moment realizability assumption.

Theorem 1.3 (The universal bound and its sharp actual approximation).

Proof. Machine-checked in Lean as D5/S3/StatisticalMechanics/UnicyclicSkewnessMinusKurtosisLowerBound.result (✓ std3). ∎

Resolves. Problems/ptaszynski-skewness-minus-kurtosis-lower-bound (proved) by D5/S3/StatisticalMechanics/UnicyclicSkewnessMinusKurtosisLowerBound.result.

Source. Repository-derived.

Acknowledgement. Krzysztof Ptaszyński (2022). Bounds on skewness and kurtosis of steady-state currents. DOI: 10.1103/PhysRevE.106.024119. URL: https://arxiv.org/abs/2204.08421v4.

Commentary.

Write Rⱼ = ∑ᵢ kᵢ⁻ʲ. The original definitions are S = 3R₂²/R₁⁴ − 2R₃/R₁³ and K = 15R₂²/R₁⁴ − 20R₃/R₁³ + 6R₄/(R₁²R₂). They are scaled current cumulant ratios C₃/C₁ and C₄/C₂. They are not standardized central moments. The original normalized_simplex and normalized_difference apply to xᵢ = kᵢ⁻¹/R₁. With p = ∑xᵢ², q = ∑xᵢ³, r = ∑xᵢ⁴ and u = √p, they give ∑xᵢ = 1, 0 < p, 0 ≤ xᵢ ≤ u ≤ 1 and D = S − K = −12p² + 18q − 6r/p.

For u ≤ 1/3 the nonnegative sum ∑xᵢ²(u − xᵢ)(u + xᵢ − 3p) equals p[D − f(u)]/6, where f(u) = −12u⁴ + 18u³ − 6u². The factorization f(u) − C = (u − (9 − √17)/16)² [−12u² + (9 + 3√17)u/2 + (15 + 9√17)/32] is nonnegative on this interval. For u ≥ 1/3 put d = (3p − u)/2. The nonnegative sum ∑xᵢ(u − xᵢ)(xᵢ − d)² equals p[D + (3/2)u(1 − u)³]/6. Finally 8/81 − u(1 − u)³ = (u − 1/3)[4/27 + (u − 1/3)(1 − u)(4/3 − u)] ≥ 0, so D ≥ −4/27 ≥ C. These are estimates on the original common finite vector.

For sharpness let α = (9 − √17)/16, which lies strictly between zero and one. For every natural m take n = m + 2 and rates k₀ = 1/α and kᵢ = (m + 1)/(1 − α) for i ≠ 0. All rates are positive and their inverse rates sum to one. For j = 2,3,4 the exact moments are αʲ + (1 − α)ʲ/(m + 1)ʲ⁻¹. With z = 1/(m + 1), the difference is −12[α² + (1 − α)²z]² + 18[α³ + (1 − α)³z²] − 6[α⁴ + (1 − α)⁴z³]/[α² + (1 − α)²z]. The denominator at zero is α² > 0. Continuity and z → 0 give f(α) = C. Every term is an actual value in the asserted set, so every lower bound of that set is at most C.

Dividing these rates by their common tail rate gives exceptional rate a = (1 − α)/(α(N − 1)) = A/(N − 1), with A = (5 + √17)/4. This finite family differs from the source’s A/N, and their ratio N/(N − 1) tends to one. The proof directly evaluates its own finite vectors and requires no finite-family identification.

For N = 2 the interpretation is two oriented, channel-labelled phase stages with one marked winding crossing; that physical identification is not explicitly supplied by the source notation. The theorem proves the rational inverse-rate expressions; it constructs no stochastic process, renewal identification or empirical correspondence. Physical N = 1, bidirectional, thermal and multicyclic networks and finite-time cumulants are outside its scope. The result claims neither strict finite inequality nor finite-N optimizer classification.

References