Queue Isomorphism and the Local Lipschitz Reduction for Selection Cost
We establish an exact algebraic isomorphism between the DP recursion in ZFCρ and the Lindley equation of queueing theory, and reduce the Local Lipschitz property ($\mathrm{Var}(A \mid \Omega = k) = O_k(1)$) to three explicit input conditions.
The core discovery (Theorem 1): the defect $D_n = M_n - \rho_E(n)$ satisfies exactly $D_{n+1} = \max(D_n + \Delta M_n^{\mathrm{glob}} - 1, 0)$, where $\Delta M_n^{\mathrm{glob}} = M_{n+1} - M_n$ is the one-step global gradient of the multiplicative target.
Building on this, we prove a conditional theorem (Theorem 3): if (0) the global Lindley process has bounded variance, (I) the sampled one-step global gradient has bounded variance on $\Omega$-shells, and (II) the predecessor defect has bounded variance under $\Omega(n) = k$, then $\mathrm{Var}(A \mid \Omega = k) = O_k(1)$. All three conditions are supported by overwhelming numerical evidence ($N = 10^7$, cross-validated across 11 shells).
The definitive contribution of this paper is the queue isomorphism itself (§2) and its corollaries (the memory-wipe mechanism), together with the resulting precise reduction framework.
Keywords: integer complexity, ρ-arithmetic, Lindley equation, queue isomorphism, Local Lipschitz, selection cost, defect, memory wipe
§1. Introduction
§1.1. Background
Paper 20 unconditionally closed the B-bound: $\mathrm{Var}(B_\rho \mid \Omega = k, n \leq N) = O_k(1)$, where $B_\rho(n) = \rho_E(n) - \rho_E(n/P^-(n)) - \rho_E(P^-(n)) - 2$ is the SPF defect defined in Paper 16. The next critical input for the $D(N) \to 1$ proof chain is the A-bound (Local Lipschitz property).
§1.2. Target
Conditional Theorem (Local Lipschitz Property). If input conditions (0), (I), (II) (see §5.1) hold, then for each fixed $k \geq 2$:
$$\mathrm{Var}(A \mid \Omega(n) = k,\; n \leq N) = O_k(1) \quad (N \to \infty)$$
§1.3. Paper Structure
- Proved: Exact algebraic isomorphism between the DP recursion and the Lindley equation (§2).
- Reduction framework: Conditional theorem deriving $\mathrm{Var}(A) = O_k(1)$ from three input conditions (§3–§5).
- Numerical support: Overwhelming numerical evidence for all three conditions (§3, §5, Appendices).
§1.4. Notation
Global gradient $\Delta M_n^{\mathrm{glob}} = M_{n+1} - M_n$: target change between consecutive integers. Used in the Lindley identity (§2).
Shell gradient $\Delta M_{k,i}^{\mathrm{shell}} = M_{n_{i+1}} - M_{n_i}$: target change between consecutive $\Omega = k$ shell elements. Reported in numerical analysis (§3).
Shell-sampled global gradient $\Delta M_{n-1}^{\mathrm{glob}} \mid \Omega(n) = k$: conditional distribution of the one-step global gradient at shell sampling points. Used in Condition (I) and the proof of Theorem 3.
§2. The Queue Isomorphism
§2.1. Definitions
- DP recursion: $\rho_E(n) = \min(\rho_E(n-1) + 1,\ M_n)$
- Multiplicative target: $M_n = \min_{ab=n,\,a,b\geq 2}(\rho_E(a) + \rho_E(b) + 2)$ (composites), $M_p = \rho_E(p)$ (prime extension)
- Jump gain: $G(n) = \rho_E(n-1) + 1 - M_n$
- Selection cost: $A(n) = j(n) - 1$, where $j(n) = \max(G(n), 0)$
- Defect: $D_n = M_n - \rho_E(n) \geq 0$
- SPF defect (Paper 16): $B_\rho(n) = \rho_E(n) - \rho_E(n/P^-(n)) - \rho_E(P^-(n)) - 2$
By definition, $M_n = \rho_E(n) + D_n = \rho_E(n/P^-) + \rho_E(P^-) + 2 + B_\rho(n) + D_n$.
§2.2. Theorem 1 (Lindley Identity)
Theorem 1. For all $n \geq 2$:
$$D_{n+1} = \max\!\left(D_n + \Delta M_n^{\mathrm{glob}} - 1,\; 0\right)$$
Proof. $D_n = \max(M_n - \rho_E(n-1) - 1, 0)$, so $\rho_E(n) = M_n - D_n$. Then:
$$D_{n+1} = \max(M_{n+1} - \rho_E(n) - 1,\; 0) = \max(D_n + \Delta M_n^{\mathrm{glob}} - 1,\; 0) \qquad \square$$
§2.3. Corollary 1 (Jump Gain via Queue)
$$G(n+1) = 1 - \Delta M_n^{\mathrm{glob}} - D_n$$
Proof. $G(n+1) = \rho_E(n) + 1 - M_{n+1} = (M_n - D_n) + 1 - M_{n+1} = 1 - \Delta M_n^{\mathrm{glob}} - D_n$. $\square$
§2.4. Corollary 2 (Memory Wipe at Jumps)
If $G(n) > 0$, then $D_n = 0$ and $G(n+1) = 1 - \Delta M_n^{\mathrm{glob}}$.
Proof. $G(n) > 0$ means $\rho_E(n-1) + 1 > M_n$, so $\rho_E(n) = M_n$ and $D_n = 0$. $\square$
§3. Numerical Support for Input Conditions
§3.1. Input Condition (I): Bounded Variance of Shell-Sampled Global Gradient
Theorem 3 requires: under $\Omega(n) = k$, the variance of $\Delta M_{n-1}^{\mathrm{glob}}$ is bounded.
Condition (I). For each fixed $k \geq 2$: $\mathrm{Var}(\Delta M_{n-1}^{\mathrm{glob}} \mid \Omega(n) = k,\; n \leq N) = O_k(1)$
Numerical evidence ($N = 10^7$):
| $k$ | $\mathrm{Var}(\Delta M_{n-1}^{\mathrm{glob}} \mid \Omega(n)=k)$ | $|\Delta M|_{\max}$ | Shell size |
|---|---|---|---|
| 2 | 2.113 | 9 | 1,904,324 |
| 3 | 1.425 | 7 | 2,444,359 |
| 4 | 1.335 | 8 | 2,050,696 |
| 5 | 1.372 | 7 | 1,349,779 |
| 6 | 1.394 | 7 | 774,078 |
| 7 | 1.418 | 7 | 409,849 |
| 8 | 1.454 | 7 | 207,207 |
| 9 | 1.485 | 8 | 101,787 |
| 10 | 1.490 | 7 | 49,163 |
| 11 | 1.529 | 8 | 23,448 |
| 12 | 1.556 | 7 | 11,068 |
All values < 2.2, stably bounded.
§3.2. Supplementary Shell Gradient Data
| $k$ | $\mathrm{Var}(\Delta M^{\mathrm{shell}})$ | $|\Delta M^{\mathrm{shell}}|_{\max}$ | $E[\mathrm{gap}]$ |
|---|---|---|---|
| 4 | 0.808 | 5 | 4.9 |
| 8 | 0.746 | 4 | 48.3 |
| 12 | 0.717 | 3 | 903 |
Shell gradient variance is smaller than the shell-conditioned global gradient variance — counterintuitively, since block sums typically have larger variance. This suggests systematic variance cancellation within shells.
§3.3. Component Decomposition
By §2.1, $M_n = \rho_E(n/P^-) + \rho_E(P^-) + 2 + B_\rho(n) + D_n$. For consecutive shell elements:
$$\Delta M^{\mathrm{shell}} = \Delta\alpha + \Delta\beta + \Delta B_\rho + \Delta D$$
where $\Delta\alpha = \Delta[\rho_E(n/P^-)]$, $\Delta\beta = \Delta[\rho_E(P^-)]$.
Key observations ($k = 8$ shell, $N = 10^7$):
- (a) $P(D = 0) = 99.88\%$, so $\Delta D \approx 0$.
- (b) $\mathrm{Var}(\Delta B_\rho^{\mathrm{shell}}) \leq 2\mathrm{Var}(B_\rho) = O_k(1)$ (directly from Paper 20 B-bound).
- (c) The difficult component is $\Delta\alpha$. $\mathrm{Var}(\Delta\alpha) \approx 1.18$, but $\mathrm{Cov}(\Delta\alpha, \Delta B_\rho) \approx -0.45$ compresses total variance from ~1.9 to $\mathrm{Var}(\Delta M^{\mathrm{shell}}) \approx 0.75$.
§3.4. Formalization Prospects
Complete analytic formalization of Condition (I) requires combining Paper 20's prime-power correction $\varepsilon$-control, the Turán–Kubilius framework on $\Omega = k$ shells, and $\rho_E$ growth-rate concentration (std of $\rho_E(n)/\ln n \approx 0.044$). The truth of Condition (I) is established beyond reasonable doubt at current numerical precision, but this paper does not claim it has been formally proved.
§4. Stability Framework
§4.1. Logarithmic Growth of $M_n$ (Proved)
Proposition B. There exists $C > 0$ such that $M_n \leq C \cdot \ln n + O(1)$ for all $n \geq 2$.
Proof. For composite $n$ with smallest prime factor $p = P^-(n)$: $M_n \leq \rho_E(p) + \rho_E(n/p) + 2 \leq C \cdot \ln n + 2$. For primes, $M_p = \rho_E(p) \leq C \cdot \ln p$. $\square$
Corollary (Negative drift). $(1/N) \sum_n \Delta M_n^{\mathrm{glob}} = O(\ln N / N) \to 0$.
§4.2. Classical Stability Theorem (Reference)
Remark. In ZFCρ, $\{\Delta M_n^{\mathrm{glob}}\}$ is a deterministic sequence. Applying Loynes' theorem requires constructing an appropriate probability space (e.g., via shift-dynamical embedding). This paper subsumes the legitimacy of such embedding within Condition (0).
§4.3. Defect Statistics (Numerical)
| $k$ | $E[D]$ | $\mathrm{Var}(D)$ | $P(D=0)$ |
|---|---|---|---|
| 2 | 0.732 | 1.018 | — |
| 3 | 0.186 | 0.228 | — |
| 4 | 0.054 | 0.062 | — |
| 5 | 0.018 | 0.020 | — |
| 6 | 0.007 | 0.007 | — |
| 7 | 0.003 | 0.003 | — |
| 8 | 0.001 | 0.001 | 99.88% |
| 9–12 | < 0.001 | < 0.001 | > 99.9% |
§4.4. Input Conditions (0) and (II)
Condition (0) (Global stability). Under an appropriate probabilistic framework, the global Lindley process $\{D_n\}$ has bounded variance: $\mathrm{Var}(D_n \mid n \leq N) = O(1)$
Numerical support. Global Cesàro mean $(1/N) \sum D_n \approx 0.199$ (stable), max $D = 8$, $P(D = 0) \approx 87\%$.
Condition (II) (Bounded predecessor defect variance on shells). Under $\Omega(n) = k$, the predecessor defect $D_{n-1}$ has bounded variance: $\mathrm{Var}(D_{n-1} \mid \Omega(n) = k,\; n \leq N) = O_k(1)$
Numerical support:
| $k$ | $E[D_{n-1} \mid \Omega(n)=k]$ | $\mathrm{Var}(D_{n-1} \mid \Omega(n)=k)$ |
|---|---|---|
| 2 | 0.139 | 0.228 |
| 4 | 0.232 | 0.405 |
| 6 | 0.250 | 0.407 |
| 8 | 0.224 | 0.344 |
| 10 | 0.200 | 0.297 |
| 12 | 0.196 | 0.299 |
All $\mathrm{Var}(D_{n-1} \mid \Omega(n) = k) < 0.43$, stably bounded. $\mathrm{Corr}(D_{n_i}, D_{n_{i+1}}) = -0.001$ on the $k = 8$ shell sequence. Mean inter-shell gap $\approx 48$ steps with drift $\approx -1$: $D$ resets to zero well before reaching the next shell point.
§5. Conditional Main Theorem
§5.1. Theorem 3
Theorem 3. Assume:
(0) Global stability: $\mathrm{Var}(D_n \mid n \leq N) = O(1)$;
(I) Bounded shell-sampled global gradient variance: $\mathrm{Var}(\Delta M_{n-1}^{\mathrm{glob}} \mid \Omega(n) = k, n \leq N) = O_k(1)$;
(II) Bounded predecessor defect variance on shells: $\mathrm{Var}(D_{n-1} \mid \Omega(n) = k, n \leq N) = O_k(1)$.
Then for each fixed $k \geq 2$: $\mathrm{Var}(A \mid \Omega(n) = k,\; n \leq N) = O_k(1)$
Proof.
Step 1. At $\Omega(n) = k$, by Corollary 1 (§2.3): $G(n) = 1 - \Delta M_{n-1}^{\mathrm{glob}} - D_{n-1}$. Taking conditional variance:
$$\mathrm{Var}(G \mid \Omega(n)=k) \leq 2\,\mathrm{Var}(\Delta M_{n-1}^{\mathrm{glob}} \mid \Omega(n)=k) + 2\,\mathrm{Var}(D_{n-1} \mid \Omega(n)=k)$$
Condition (I) controls the first term; Condition (II) controls the second. Hence $\mathrm{Var}(G \mid \Omega = k) = O_k(1)$.
Step 2. $A(n) = j(n) - 1$, where $j(n) = \max(G(n), 0) = (G(n))^+$. The map $x \mapsto x^+$ is 1-Lipschitz, so $\mathrm{Var}(j \mid \Omega = k) \leq \mathrm{Var}(G \mid \Omega = k) = O_k(1)$. Thus $\mathrm{Var}(A \mid \Omega = k) = \mathrm{Var}(j \mid \Omega = k) = O_k(1)$. $\square$
§5.2. Numerical Validation
| $k$ | $\mathrm{Var}(G|k)$ | $\mathrm{Var}(\Delta M|k)$ | $\mathrm{Var}(D_{n-1}|k)$ | $\mathrm{Cov}(\Delta M,D)$ | $\mathrm{Var}(\Delta M+D)$ | Check |
|---|---|---|---|---|---|---|
| 2 | 1.769 | 2.113 | 0.228 | −0.286 | 1.769 | ✓ |
| 4 | 0.900 | 1.335 | 0.405 | −0.420 | 0.900 | ✓ |
| 6 | 0.956 | 1.394 | 0.407 | −0.423 | 0.956 | ✓ |
| 8 | 1.101 | 1.454 | 0.344 | −0.349 | 1.101 | ✓ |
| 10 | 1.209 | 1.490 | 0.297 | −0.289 | 1.209 | ✓ |
| 12 | 1.298 | 1.556 | 0.299 | −0.278 | 1.298 | ✓ |
Consistency check. By Corollary 1, $G = 1 - \Delta M - D$, so $\mathrm{Var}(G) = \mathrm{Var}(\Delta M + D)$. The table confirms $\mathrm{Var}(G|k) = \mathrm{Var}(\Delta M+D)$ to four decimal places.
§6. Empirical Observations
§6.1. Approximate Independence of Jump Gains
By Corollary 2 (§2.4), at $D = 0$, $G(n+1) = 1 - \Delta M_n^{\mathrm{glob}}$, independent of prior queue state. Since $P(D = 0) = 99.88\%$ on the $k = 8$ shell, the distribution of the next $j$ value is nearly independent of the current $j$.
$j$ transition matrix ($k = 8$ shell, $N = 10^7$):
| $g \backslash g'$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| 0 | 0.014 | 0.107 | 0.328 | 0.358 | 0.156 | 0.034 | 0.004 |
| 1 | 0.014 | 0.115 | 0.327 | 0.358 | 0.157 | 0.028 | 0.002 |
| 2 | 0.015 | 0.121 | 0.330 | 0.349 | 0.154 | 0.029 | 0.002 |
| 3 | 0.019 | 0.134 | 0.333 | 0.341 | 0.145 | 0.027 | 0.002 |
| 4 | 0.020 | 0.141 | 0.336 | 0.337 | 0.138 | 0.026 | 0.002 |
| 5 | 0.026 | 0.145 | 0.319 | 0.341 | 0.141 | 0.025 | 0.002 |
| 6 | 0.034 | 0.164 | 0.294 | 0.335 | 0.133 | 0.038 | 0.002 |
Row-to-row variation is minimal, consistent with the algebraic memory-wipe mechanism of Corollary 2.
§6.2. $j$-tail Decay
$P(j \geq g)$ on the $k = 8$ shell decays faster than geometric (numerical observation); this paper does not claim superexponential decay as proved.
§7. Thermodynamic Interface
| ZFCρ (Queue framework) | Thermodynamic correspondence |
|---|---|
| $D_n$ (defect) | Local free energy excess |
| Drift $\approx -1$ | Dissipation |
| $\Delta M^{\mathrm{glob}}$ (target gradient) | External driving strength |
| $D = 0$ (queue empty) | Thermal equilibrium |
| $\mathrm{Var}(D) \to 0$ as $k$ increases | Fluctuation suppression in high dimensions |
§8. Updated Proof Landscape
| Input | Status |
|---|---|
| $\mathrm{Var}(A) = O_k(1)$ | Conditional — depends on (0)(I)(II), strong numerical support (this paper) |
| $\mathrm{Var}(B_\rho) = O_k(1)$ | Unconditionally closed — Paper 20 |
Unconditional closure of the A-bound requires:
- (0): Probabilistic embedding of the deterministic Lindley process
- (I): Analytic proof of bounded shell-sampled global gradient variance
- (II): Formalization of predecessor defect variance bound (derivable from (0) via Palm theory or sparse-sampling arguments)
References
- Asmussen, S. (2003). Applied Probability and Queues. Springer.
- Franken, P., König, D., Arndt, U. & Schmidt, V. (1982). Queues and Point Processes. Akademie-Verlag.
- Kingman, J.F.C. (1962). Some inequalities for the queue GI/G/1. Biometrika 49, 315–324.
- Lindley, D.V. (1952). The theory of queues with a single server. Proc. Cambridge Phil. Soc. 48, 277–289.
- Loynes, R.M. (1962). The stability of a queue with non-independent inter-arrival and service times. Proc. Cambridge Phil. Soc. 58, 497–520.
- H. Qin, ZFCρ Paper I. DOI: 10.5281/zenodo.18914682.
- H. Qin, ZFCρ Paper II. DOI: 10.5281/zenodo.18927658.
- H. Qin, ZFCρ Paper XVI. DOI: 10.5281/zenodo.19013602.
- H. Qin, ZFCρ Paper XIX. DOI: 10.5281/zenodo.19026991.
- H. Qin, ZFCρ Paper XX. DOI: 10.5281/zenodo.19027893.
The queue isomorphism (Theorem 1) emerged from four-AI parallel exploration. Gemini first identified the Lindley equation structure; Claude completed the gap analysis and discovered the Global→Shell transfer problem and the B-notation conflict; ChatGPT provided exact numerical verification, component decomposition, and rigorous review; Grok provided $k$-scaling data.
All computations use $N = 10^7$ with exact integer arithmetic (Python/NumPy). Three independent implementations cross-validated. $M_n$ is extended to primes via $M_p = \rho_E(p)$ (equivalently, $D_p = 0$). Shells: $\{n \leq N : \Omega(n) = k\}$, sorted by natural ordering.
C.1. Global $\Delta M^{\mathrm{glob}}$ Histogram
| $\Delta M$ | Count | Fraction |
|---|---|---|
| −8 | 20 | 0.0002% |
| −7 | 295 | 0.0030% |
| −6 | 3,199 | 0.0320% |
| −5 | 24,263 | 0.2426% |
| −4 | 114,622 | 1.1462% |
| −3 | 379,970 | 3.7997% |
| −2 | 1,000,473 | 10.0047% |
| −1 | 2,063,520 | 20.6352% |
| 0 | 2,641,566 | 26.4157% |
| 1 | 2,411,159 | 24.1116% |
| 2 | 877,084 | 8.7708% |
| 3 | 339,566 | 3.3957% |
| 4 | 108,348 | 1.0835% |
| 5 | 28,597 | 0.2860% |
| 6 | 6,120 | 0.0612% |
| 7 | 1,046 | 0.0105% |
| 8 | 135 | 0.0014% |
| 9 | 15 | 0.0002% |
| 10 | 1 | 0.0000% |
C.2. $k = 8$ Shell Statistics
$j$ distribution: $j=0$ (1.71%), $j=1$ (12.82%), $j=2$ (33.10%), $j=3$ (34.56%), $j=4$ (14.82%), $j=5$ (2.77%), $j\geq 6$ (0.22%).
$D$ distribution: $D=0$ (99.88%), $D=1$ (0.11%), $D=2$ (0.003%).
Gap statistics: $E[\mathrm{gap}] = 48.26$, Median = 32, Range = [1, 560]. Most frequent gaps: 16 (20,805), 32 (15,511), 24 (12,767).
本文建立 ZFCρ 框架中 DP 递推与排队论之间的精确代数同构，并将局部 Lipschitz 性（$\mathrm{Var}(A \mid \Omega = k) = O_k(1)$）归约为三个明确的输入条件。
核心发现（定理 1）：缺陷量 $D_n = M_n - \rho_E(n)$ 精确满足 Lindley 方程 $D_{n+1} = \max(D_n + \Delta M_n^{\mathrm{glob}} - 1, 0)$，其中 $\Delta M_n^{\mathrm{glob}} = M_{n+1} - M_n$ 是连续整数间的乘性目标梯度。
在此基础上，本文证明条件定理（定理 3）：若 (0) 全局 Lindley 过程具有有界方差，(I) 壳层采样点上的单步全局梯度方差有界，且 (II) $D_{n-1}$ 在 $\Omega(n) = k$ 条件下方差有界，则 $\mathrm{Var}(A \mid \Omega = k) = O_k(1)$。三个条件均有压倒性的数值支持（$N = 10^7$，11 个壳层交叉验证）。
本文的确定性贡献是排队同构本身（§2）及其推论（记忆擦除机制），以及由此产生的精确归约框架。
关键词：整数复杂度，ρ-算术，Lindley 方程，排队同构，局部 Lipschitz，选择代价，缺陷，记忆擦除
§1. 引言
§1.1. 背景
论文 20 无条件闭合了 B-bound：$\mathrm{Var}(B_\rho \mid \Omega = k, n \leq N) = O_k(1)$，其中 $B_\rho(n) = \rho_E(n) - \rho_E(n/P^-(n)) - \rho_E(P^-(n)) - 2$ 是论文 16 定义的 SPF 缺陷。$D(N) \to 1$ 证明链的下一个关键输入是 A-bound（局部 Lipschitz 性）。
§1.2. 目标
条件定理（局部 Lipschitz 性）. 若输入条件 (0), (I), (II)（见 §5.1）成立，则对每个固定 $k \geq 2$：
$$\mathrm{Var}(A \mid \Omega(n) = k,\; n \leq N) = O_k(1) \quad (N \to \infty)$$
§1.3. 论文结构
- 已证： DP 递推与 Lindley 方程的精确代数同构（§2）。
- 归约框架： 从三个输入条件推出 $\mathrm{Var}(A) = O_k(1)$ 的条件定理（§3–§5）。
- 数值支持： 三个条件的压倒性数值证据（§3, §5, 附录）。
§1.4. 记号约定
全局梯度 $\Delta M_n^{\mathrm{glob}} = M_{n+1} - M_n$：连续整数 $n, n+1$ 之间的目标变化。§2 的 Lindley 恒等式使用此量。
壳层梯度 $\Delta M_{k,i}^{\mathrm{shell}} = M_{n_{i+1}} - M_{n_i}$：$\Omega = k$ 壳层中连续元素之间的目标变化。§3 的数值分析报告此量。
壳层采样全局梯度 $\Delta M_{n-1}^{\mathrm{glob}} \mid \Omega(n) = k$：全局单步梯度在壳层采样点上的条件分布。条件 (I) 和定理 3 的证明使用此量。
§2. 排队同构
§2.1. 基本量
- DP 递推：$\rho_E(n) = \min(\rho_E(n-1) + 1,\ M_n)$
- 乘性目标：$M_n = \min_{ab=n,\,a,b\geq 2}(\rho_E(a) + \rho_E(b) + 2)$（合数），$M_p = \rho_E(p)$（素数延拓）
- 跳跃增益：$G(n) = \rho_E(n-1) + 1 - M_n$
- 选择代价：$A(n) = j(n) - 1$，其中 $j(n) = \max(G(n), 0)$
- 缺陷：$D_n = M_n - \rho_E(n) \geq 0$
- SPF 缺陷（论文 16）：$B_\rho(n) = \rho_E(n) - \rho_E(n/P^-(n)) - \rho_E(P^-(n)) - 2$
由定义，$M_n = \rho_E(n) + D_n = \rho_E(n/P^-) + \rho_E(P^-) + 2 + B_\rho(n) + D_n$。
§2.2. 定理 1（Lindley 恒等式）
定理 1. 对所有 $n \geq 2$：
$$D_{n+1} = \max\!\left(D_n + \Delta M_n^{\mathrm{glob}} - 1,\; 0\right)$$
证明. $D_n = \max(M_n - \rho_E(n-1) - 1, 0)$，故 $\rho_E(n) = M_n - D_n$。于是：
$$D_{n+1} = \max(M_{n+1} - \rho_E(n) - 1,\; 0) = \max(D_n + \Delta M_n^{\mathrm{glob}} - 1,\; 0) \qquad \square$$
§2.3. 推论 1（跳跃增益的排队表示）
$$G(n+1) = 1 - \Delta M_n^{\mathrm{glob}} - D_n$$
证明. $G(n+1) = \rho_E(n) + 1 - M_{n+1} = (M_n - D_n) + 1 - M_{n+1} = 1 - \Delta M_n^{\mathrm{glob}} - D_n$。$\square$
§2.4. 推论 2（跳跃处的记忆擦除）
若 $G(n) > 0$，则 $D_n = 0$，且 $G(n+1) = 1 - \Delta M_n^{\mathrm{glob}}$。
证明. $G(n) > 0$ 即 $\rho_E(n-1) + 1 > M_n$，故 $\rho_E(n) = M_n$，$D_n = 0$。$\square$
§3. 输入条件的数值支持
§3.1. 输入条件 (I)：壳层采样点上的全局梯度方差有界
条件 (I). 对每个固定 $k \geq 2$：$\mathrm{Var}(\Delta M_{n-1}^{\mathrm{glob}} \mid \Omega(n) = k,\; n \leq N) = O_k(1)$
数值证据（$N = 10^7$）：
| $k$ | $\mathrm{Var}(\Delta M_{n-1}^{\mathrm{glob}} \mid \Omega(n)=k)$ | $|\Delta M|_{\max}$ | 壳层大小 |
|---|---|---|---|
| 2 | 2.113 | 9 | 1,904,324 |
| 3 | 1.425 | 7 | 2,444,359 |
| 4 | 1.335 | 8 | 2,050,696 |
| 5 | 1.372 | 7 | 1,349,779 |
| 6 | 1.394 | 7 | 774,078 |
| 7 | 1.418 | 7 | 409,849 |
| 8 | 1.454 | 7 | 207,207 |
| 9 | 1.485 | 8 | 101,787 |
| 10 | 1.490 | 7 | 49,163 |
| 11 | 1.529 | 8 | 23,448 |
| 12 | 1.556 | 7 | 11,068 |
所有值 < 2.2，稳定有界。
§3.2. 壳层梯度的补充数据
| $k$ | $\mathrm{Var}(\Delta M^{\mathrm{shell}})$ | $|\Delta M^{\mathrm{shell}}|_{\max}$ | $E[\mathrm{gap}]$ |
|---|---|---|---|
| 4 | 0.808 | 5 | 4.9 |
| 8 | 0.746 | 4 | 48.3 |
| 12 | 0.717 | 3 | 903 |
壳层梯度方差小于全局梯度的壳层条件方差——这是反直觉的（块和的方差通常大于单步方差），暗示壳层内部存在系统性的方差抵消。
§3.3. 分量分解
由 §2.1，$\Delta M^{\mathrm{shell}} = \Delta\alpha + \Delta\beta + \Delta B_\rho + \Delta D$，其中 $\Delta\alpha = \Delta[\rho_E(n/P^-)]$，$\Delta\beta = \Delta[\rho_E(P^-)]$。
关键观察（$k = 8$ 壳层，$N = 10^7$）：
- (a) $P(D = 0) = 99.88\%$，故 $\Delta D \approx 0$。
- (b) $\mathrm{Var}(\Delta B_\rho^{\mathrm{shell}}) \leq 2\mathrm{Var}(B_\rho) = O_k(1)$（论文 20 B-bound 直接给出）。
- (c) 真正困难的分量是 $\Delta\alpha$。$\mathrm{Var}(\Delta\alpha) \approx 1.18$，但 $\mathrm{Cov}(\Delta\alpha, \Delta B_\rho) \approx -0.45$ 将总方差压缩至 $\mathrm{Var}(\Delta M^{\mathrm{shell}}) \approx 0.75$。
§3.4. 形式化前景
条件 (I) 的完整解析形式化需要论文 20 的素幂修正 $\varepsilon$ 控制、$\Omega = k$ 壳层上的 Turán–Kubilius 框架，以及 $\rho_E$ 增长率的集中性。条件 (I) 的真实性在当前数值精度下已超越合理怀疑，但本文不声称其已被形式证明。
§4. 稳定性框架
§4.1. $M_n$ 的对数增长（已证）
命题 B. 存在 $C > 0$，使得对所有 $n \geq 2$，$M_n \leq C \cdot \ln n + O(1)$。
证明. 对最小素因子为 $p = P^-(n)$ 的合数 $n$：$M_n \leq \rho_E(p) + \rho_E(n/p) + 2 \leq C \cdot \ln n + 2$。素数情况类似。$\square$
推论（负漂移）. $(1/N) \sum_n \Delta M_n^{\mathrm{glob}} = O(\ln N / N) \to 0$。
§4.2. 经典稳定性定理（参考）
注. 在 ZFCρ 中，$\{\Delta M_n^{\mathrm{glob}}\}$ 是确定性序列。应用 Loynes 定理需要构造适当的概率空间（例如通过移位动力嵌入）。本文将此合法性包含在条件 (0) 中。
§4.3. 缺陷统计（数值）
| $k$ | $E[D]$ | $\mathrm{Var}(D)$ | $P(D=0)$ |
|---|---|---|---|
| 2 | 0.732 | 1.018 | — |
| 3 | 0.186 | 0.228 | — |
| 4 | 0.054 | 0.062 | — |
| 5 | 0.018 | 0.020 | — |
| 6 | 0.007 | 0.007 | — |
| 7 | 0.003 | 0.003 | — |
| 8 | 0.001 | 0.001 | 99.88% |
| 9–12 | < 0.001 | < 0.001 | > 99.9% |
§4.4. 输入条件 (0) 与 (II)
条件 (0)（全局稳定性）. 在适当的概率框架下，全局 Lindley 过程 $\{D_n\}$ 具有有界方差：$\mathrm{Var}(D_n \mid n \leq N) = O(1)$
数值支持. 全局 Cesàro 均值 $(1/N) \sum D_n \approx 0.199$（稳定），$\max D = 8$，$P(D = 0) \approx 87\%$。
条件 (II)（壳层上前驱缺陷方差有界）. 在 $\Omega(n) = k$ 条件下，前驱缺陷 $D_{n-1}$ 具有有界方差：$\mathrm{Var}(D_{n-1} \mid \Omega(n) = k,\; n \leq N) = O_k(1)$
| $k$ | $E[D_{n-1} \mid \Omega(n)=k]$ | $\mathrm{Var}(D_{n-1} \mid \Omega(n)=k)$ |
|---|---|---|
| 2 | 0.139 | 0.228 |
| 4 | 0.232 | 0.405 |
| 6 | 0.250 | 0.407 |
| 8 | 0.224 | 0.344 |
| 10 | 0.200 | 0.297 |
| 12 | 0.196 | 0.299 |
所有 $\mathrm{Var}(D_{n-1} \mid \Omega(n) = k) < 0.43$，稳定有界。$k = 8$ 壳层序列上 $\mathrm{Corr}(D_{n_i}, D_{n_{i+1}}) = -0.001$。平均壳层间距约 48 步，漂移 $\approx -1$：$D$ 在到达下一个壳层点之前已充分重置为零。
§5. 条件主定理
§5.1. 定理 3
定理 3. 假设：
(0) 全局稳定性：$\mathrm{Var}(D_n \mid n \leq N) = O(1)$；
(I) 壳层采样点上全局梯度方差有界：$\mathrm{Var}(\Delta M_{n-1}^{\mathrm{glob}} \mid \Omega(n) = k, n \leq N) = O_k(1)$；
(II) 壳层上前驱缺陷方差有界：$\mathrm{Var}(D_{n-1} \mid \Omega(n) = k, n \leq N) = O_k(1)$。
则对每个固定 $k \geq 2$：$\mathrm{Var}(A \mid \Omega(n) = k,\; n \leq N) = O_k(1)$
证明.
步骤 1. 在 $\Omega(n) = k$ 处，由推论 1（§2.3）：$G(n) = 1 - \Delta M_{n-1}^{\mathrm{glob}} - D_{n-1}$。取条件方差：
$$\mathrm{Var}(G \mid \Omega(n)=k) \leq 2\,\mathrm{Var}(\Delta M_{n-1}^{\mathrm{glob}} \mid \Omega(n)=k) + 2\,\mathrm{Var}(D_{n-1} \mid \Omega(n)=k)$$
条件 (I) 控制第一项，条件 (II) 控制第二项。故 $\mathrm{Var}(G \mid \Omega = k) = O_k(1)$。
步骤 2. $A(n) = j(n) - 1$，$j(n) = \max(G(n), 0)$。映射 $x \mapsto x^+$ 是 1-Lipschitz 的，故 $\mathrm{Var}(j \mid \Omega = k) \leq \mathrm{Var}(G \mid \Omega = k) = O_k(1)$。从而 $\mathrm{Var}(A \mid \Omega = k) = O_k(1)$。$\square$
§5.2. 数值验证
| $k$ | $\mathrm{Var}(G|k)$ | $\mathrm{Var}(\Delta M|k)$ | $\mathrm{Var}(D_{n-1}|k)$ | $\mathrm{Cov}(\Delta M,D)$ | $\mathrm{Var}(\Delta M+D)$ | 验证 |
|---|---|---|---|---|---|---|
| 2 | 1.769 | 2.113 | 0.228 | −0.286 | 1.769 | ✓ |
| 4 | 0.900 | 1.335 | 0.405 | −0.420 | 0.900 | ✓ |
| 6 | 0.956 | 1.394 | 0.407 | −0.423 | 0.956 | ✓ |
| 8 | 1.101 | 1.454 | 0.344 | −0.349 | 1.101 | ✓ |
| 10 | 1.209 | 1.490 | 0.297 | −0.289 | 1.209 | ✓ |
| 12 | 1.298 | 1.556 | 0.299 | −0.278 | 1.298 | ✓ |
一致性验证. 由推论 1，$G = 1 - \Delta M - D$，故 $\mathrm{Var}(G) = \mathrm{Var}(\Delta M + D)$。表格精确验证了该代数恒等式（精度达四位小数）。
§6. 经验观察
§6.1. 跳跃增益的近似独立性
由推论 2（§2.4），在 $D = 0$ 时，$G(n+1) = 1 - \Delta M_n^{\mathrm{glob}}$，与此前队列状态无关。由于 $k = 8$ 壳层上 $P(D = 0) = 99.88\%$，下一个 $j$ 值的分布几乎独立于当前 $j$。
$j$ 转移矩阵（$k = 8$ 壳层，$N = 10^7$）：
| $g \backslash g'$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| 0 | 0.014 | 0.107 | 0.328 | 0.358 | 0.156 | 0.034 | 0.004 |
| 1 | 0.014 | 0.115 | 0.327 | 0.358 | 0.157 | 0.028 | 0.002 |
| 2 | 0.015 | 0.121 | 0.330 | 0.349 | 0.154 | 0.029 | 0.002 |
| 3 | 0.019 | 0.134 | 0.333 | 0.341 | 0.145 | 0.027 | 0.002 |
| 4 | 0.020 | 0.141 | 0.336 | 0.337 | 0.138 | 0.026 | 0.002 |
| 5 | 0.026 | 0.145 | 0.319 | 0.341 | 0.141 | 0.025 | 0.002 |
| 6 | 0.034 | 0.164 | 0.294 | 0.335 | 0.133 | 0.038 | 0.002 |
行间变化极小，与推论 2 的代数记忆擦除机制一致。
§6.2. $j$ 尾部衰减
$k = 8$ 壳层上 $P(j \geq g)$ 的衰减速度快于几何分布（数值观察）；本文不声称这已被证明为超指数衰减。
§7. 热力学对应
| ZFCρ（排队框架） | 热力学对应 |
|---|---|
| $D_n$（缺陷） | 局部自由能超额 |
| 漂移 $\approx -1$ | 耗散 |
| $\Delta M^{\mathrm{glob}}$（目标梯度） | 外部驱动强度 |
| $D = 0$（队列为空） | 热平衡 |
| $\mathrm{Var}(D) \to 0$（随 $k$ 增大） | 高维中涨落抑制 |
§8. 更新后的证明格局
| 输入 | 状态 |
|---|---|
| $\mathrm{Var}(A) = O_k(1)$ | 条件性成立 — 依赖 (0)(I)(II)，数值支持强（本文） |
| $\mathrm{Var}(B_\rho) = O_k(1)$ | 无条件闭合 — 论文 20 |
A-bound 的无条件闭合需要：
- (0)：确定性 Lindley 过程的概率嵌入
- (I)：壳层采样全局梯度方差有界的解析证明
- (II)：前驱缺陷方差界的形式化（可由 (0) 通过 Palm 概率理论或稀疏采样论证推出）
参考文献
- Asmussen, S. (2003). Applied Probability and Queues. Springer。
- Franken, P., König, D., Arndt, U. & Schmidt, V. (1982). Queues and Point Processes. Akademie-Verlag。
- Kingman, J.F.C. (1962). Some inequalities for the queue GI/G/1. Biometrika 49, 315–324。
- Lindley, D.V. (1952). The theory of queues with a single server. Proc. Cambridge Phil. Soc. 48, 277–289。
- Loynes, R.M. (1962). The stability of a queue with non-independent inter-arrival and service times. Proc. Cambridge Phil. Soc. 58, 497–520。
- H. Qin，ZFCρ 论文 I，DOI: 10.5281/zenodo.18914682。
- H. Qin，ZFCρ 论文 II，DOI: 10.5281/zenodo.18927658。
- H. Qin，ZFCρ 论文 XVI，DOI: 10.5281/zenodo.19013602。
- H. Qin，ZFCρ 论文 XIX，DOI: 10.5281/zenodo.19026991。
- H. Qin，ZFCρ 论文 XX，DOI: 10.5281/zenodo.19027893。
排队同构（定理 1）从四 AI 并行探索中涌现。Gemini 率先识别出 Lindley 方程结构；Claude 完成了缺口分析，发现了 Global→Shell 转移问题和 B-记号冲突；ChatGPT 提供了精确的数值验证、分量分解和严格的审阅；Grok 提供了 $k$-缩放数据。
所有计算使用 $N = 10^7$ 的精确整数算术（Python/NumPy）。三个独立实现交叉验证。$M_n$ 通过 $M_p = \rho_E(p)$（等价地，$D_p = 0$）延拓至素数。壳层：$\{n \leq N : \Omega(n) = k\}$，按自然序排列。
C.1. 全局 $\Delta M^{\mathrm{glob}}$ 直方图
| $\Delta M$ | 计数 | 比例 |
|---|---|---|
| −8 | 20 | 0.0002% |
| −7 | 295 | 0.0030% |
| −6 | 3,199 | 0.0320% |
| −5 | 24,263 | 0.2426% |
| −4 | 114,622 | 1.1462% |
| −3 | 379,970 | 3.7997% |
| −2 | 1,000,473 | 10.0047% |
| −1 | 2,063,520 | 20.6352% |
| 0 | 2,641,566 | 26.4157% |
| 1 | 2,411,159 | 24.1116% |
| 2 | 877,084 | 8.7708% |
| 3 | 339,566 | 3.3957% |
| 4 | 108,348 | 1.0835% |
| 5 | 28,597 | 0.2860% |
| 6 | 6,120 | 0.0612% |
| 7 | 1,046 | 0.0105% |
| 8 | 135 | 0.0014% |
| 9 | 15 | 0.0002% |
| 10 | 1 | 0.0000% |
C.2. $k = 8$ 壳层统计
$j$ 分布：$j=0$ (1.71%)，$j=1$ (12.82%)，$j=2$ (33.10%)，$j=3$ (34.56%)，$j=4$ (14.82%)，$j=5$ (2.77%)，$j\geq 6$ (0.22%)。
$D$ 分布：$D=0$ (99.88%)，$D=1$ (0.11%)，$D=2$ (0.003%)。
间距统计：$E[\mathrm{gap}] = 48.26$，中位数 = 32，范围 = [1, 560]。最频繁间距：16 (20,805)，32 (15,511)，24 (12,767)。