Dyadic Shell Constant and Positivity of the Predecessor Gap
DOI: 10.5281/zenodo.19054726This paper attacks a remaining technical bottleneck on the (L3) route of the D(N) → 1 proof chain — mean positivity of $R_2$ (Paper 24 reduction). We introduce the dyadic shell constant $c_{\mathrm{dyad}}(k; N)$, shifting the problem from the global $c^*$ lower bound (open since Selfridge 1953) to a shell-conditioned dyadic interval average — a strictly weaker proposition. The core tool is Dyadic Telescoping (Proposition 12, corollary of Paper XII):
$$\rho_E(2m-1) - \rho_E(m-1) = m - \sum_{n=m}^{2m-1} j(n)$$where $j(n) = \max(G(n), 0)$ is the jump function from Paper XII. The predecessor gap equals $m$ minus the jump function sum over the dyadic interval $[m, 2m-1]$. $R_2$ positivity reduces to: the conditional expectation of the jump function sum must be strictly less than the conditional expectation of $m$ minus the target gap threshold $\tau_k(N) \approx 2.498$.
Numerical findings ($N = 10^9$): E[pred gap] exceeds $\tau_k \approx 2.498$ on all representative shells (k = 5, 8, 10, 12) with pred gap margin 0.20–0.36. Jump rate is statistically uniform across dyadic intervals (variation < 0.002). Predecessor gap takes integer values with mode 3 (42%). In the finite sample up to $N = 10^9$, worst-case $\rho_E(n)/\ln n$ is driven entirely by smooth numbers; though the best smooth ratio is only 3.567 (< 3.61), the shell-conditioned dyadic Cesàro average already exceeds the threshold stably ($c_{\mathrm{dyad}}$ margin above $\tau_k/\ln 2 \approx 3.604$ is 0.29–0.52). This separation between "low pointwise extreme" and "high interval average" is the core argument for Route B over Route A.
§1. Introduction
§1.1 Remaining Bottleneck on the (L3) Route
Paper 24 isolated the remaining target-gap problem on the (L3) route as mean positivity of $R_2$. For fixed cutoff $N$:
$$E[R_2 \mid \Omega(m) = k, D_m = 0, m \leq N] > 0$$if and only if
$$E[\rho_E(2m-1) - \rho_E(m-1) \mid \Omega(m) = k, D_m = 0, m \leq N] > \tau_k(N)$$where $\tau_k(N) := E[M_{2m} - M_m \mid \Omega(m) = k, D_m = 0, m \leq N]$, numerically $\tau_k(N) \approx 2.498$.
A natural heuristic translates this as: $\rho_E(2m-1) - \rho_E(m-1) \approx c^* \ln 2$, requiring $c^* > 2.498/\ln 2 \approx 3.61$. But Paper XI's proved lower bound is only $c^* \geq 3/\ln 3 \approx 2.73$, and integer complexity lower bounds have progressed very slowly since Selfridge (1953).
Key observation: The $c^*$ lower bound is a strictly stronger condition than what is actually needed. We require not the global pointwise $\liminf \rho_E(n)/\ln n > 3.61$, but only shell-conditioned local average positivity over dyadic intervals — a strictly weaker proposition.
§1.2 Contributions
- Dyadic Shell Constant $c_{\mathrm{dyad}}$ (§2). Shifts the problem from global $c^*$ to a shell-conditioned dyadic average.
- Dyadic Telescoping (§3, corollary of Paper XII). pred gap = $m - \sum j(n)$, connecting directly to Paper XII's jump function telescoping.
- Formal reduction (§3.3). $R_2$ positivity reduces to $E[\sum j] < E[m] - \tau_k(N)$, all conditioned on $\Omega = k, D_m = 0$.
- Numerical structure (§4). On representative shells (k = 5, 8, 10, 12), pred gap exceeds $\tau_k$ ($N = 10^9$ data); jump rate is statistically uniform within dyadic intervals (variation < 0.002); worst-case $\rho_E/\ln n$ integers are all smooth numbers.
§1.3 Comparison of Three Routes
| Route | Target | Difficulty | Leverage |
|---|---|---|---|
| A ($c^*$ lower bound) | Prove $c^* > 3.61$ (global pointwise) | Very high (open 70 years) | High but overshoots |
| B (pred gap, this paper) | Shell-conditioned dyadic average > $\tau_k$ | Medium-high | Directly closes (L3) |
| C (smooth only) | Prove pred gap only for smooth numbers | Medium | Medium (still need non-smooth ≥ smooth) |
This paper executes Route B. Routes A and C are compared in Appendix A.
§2. Dyadic Shell Constant
§2.1 Definition
Definition. For fixed $k \geq 2$, the dyadic shell constant is:
$$c_{\mathrm{dyad}}(k; N) := \frac{E[\rho_E(2m-1) - \rho_E(m-1) \mid \Omega(m) = k, D_m = 0, m \leq N]}{\ln 2}$$$R_2$ mean positivity (Paper 24) is equivalent to:
$$c_{\mathrm{dyad}}(k; N) > \frac{\tau_k(N)}{\ln 2}$$where $\tau_k(N) := E[M_{2m} - M_m \mid \Omega(m) = k, D_m = 0, m \leq N]$. Numerically $\tau_k(N) \approx 2.498$ for all tested $k$, giving threshold $\tau_k/\ln 2 \approx 3.604$.
§2.2 Numerical Values ($N = 10^9$)
| k | E[pred gap] | $c_{\mathrm{dyad}}$ | margin above $\tau_k/\ln 2$ |
|---|---|---|---|
| 5 | 2.855 | 4.119 | +0.515 |
| 8 | 2.753 | 3.971 | +0.367 |
| 10 | 2.711 | 3.911 | +0.307 |
| 12 | 2.698 | 3.893 | +0.289 |
$c_{\mathrm{dyad}}$ decreases from 4.12 (k=5) to 3.89 (k=12), remaining consistently above 3.604.
§2.3 Relation to Global $c^*$
The global $c^*$ is $\liminf_{n \to \infty} \rho_E(n)/\ln n$ (pointwise). $c_{\mathrm{dyad}}$ is a shell-conditioned Cesàro average over dyadic intervals. The two differ:
- The finite-sample minimum of $\rho_E/\ln n$ depends on the starting threshold: min = 3.32 ($n \geq 10^3$), 3.44 ($n \geq 10^5$), 3.50 ($n \geq 10^8$), all achieved by smooth numbers ($2^a \cdot 3^b$ type)
- $c_{\mathrm{dyad}}$ is an interval average; individual low values from smooth numbers are "averaged away"
§3. Dyadic Telescoping (Corollary of Paper XII)
§3.1 Identity
Proposition 12 (Dyadic Telescoping). For any $m \geq 2$:
$$\rho_E(2m-1) - \rho_E(m-1) = m - \sum_{n=m}^{2m-1} j(n)$$where $j(n) = \max(G(n), 0) = \max(\rho_E(n-1) + 1 - M_n, 0)$ is the jump function from Paper XII.
Note. This is a direct corollary of Paper XII's $\delta(N) = \sum j(n)$ telescoping applied to the dyadic interval. Its value lies not in the proof itself (a one-line corollary), but in exposing the correct local object for Route B: the predecessor gap equals $m$ minus the jump function sum over the dyadic interval.
Proof. By the DP recursion $\rho_E(n) = \min(\rho_E(n-1) + 1, M_n)$, for each $n$:
$$\rho_E(n) = \rho_E(n-1) + 1 - j(n)$$(When $G(n) > 0$: $\rho_E(n) = M_n = \rho_E(n-1) + 1 - j(n)$; when $G(n) \leq 0$: $\rho_E(n) = \rho_E(n-1) + 1$ and $j(n) = 0$.)
Summing over $n = m, m+1, \ldots, 2m-1$ ($m$ terms of telescoping):
$$\rho_E(2m-1) - \rho_E(m-1) = m - \sum_{n=m}^{2m-1} j(n) \qquad \square$$§3.2 Equivalent Rewriting
Define $A(n) = j(n) - 1$ (Paper XVII). Then:
$$\rho_E(2m-1) - \rho_E(m-1) = -\sum_{n=m}^{2m-1} A(n)$$The predecessor gap equals the negative sum of $A(n)$ over the dyadic interval. Since $A(n) \leq 0$ when $G(n) \leq 0$ (then $j = 0, A = -1$) and $A(n) \geq 0$ when $G(n) > 0$ (then $A = j - 1 \geq 0$), the imbalance between positive and negative contributions determines the sign and magnitude of the predecessor gap.
§3.3 Reduction to Jump Density
Define the target gap threshold:
$$\tau_k(N) := E[M_{2m} - M_m \mid \Omega(m) = k, D_m = 0, m \leq N]$$Numerically $\tau_k(N) \approx 2.498$ for all tested $k$ and $N$ (see Paper 24).
$R_2$ mean positivity is equivalent to:
$$E\left[\sum_{n=m}^{2m-1} j(n) \;\middle|\; \Omega(m) = k, D_m = 0, m \leq N\right] < E[m \mid \Omega(m) = k, D_m = 0, m \leq N] - \tau_k(N)$$That is, the conditional expectation of the jump function sum over the dyadic interval $[m, 2m-1]$ must be strictly less than the conditional expectation of $m$ minus the target gap threshold.
§4. Numerical Structure
§4.1 Step/Jump Decomposition
By Proposition 12, for a representative $m$ from the k=8 shell ($m \approx 164{,}000$), within the interval $[m, 2m-1]$:
$$\text{pred gap} = m - \sum_{n=m}^{2m-1} j(n)$$Positions where $j(n) = 0$ (non-jump) contribute $+1$ to the pred gap; positions where $j(n) > 0$ contribute $1 - j(n) \leq 0$.
| Quantity | Value |
|---|---|
| span $m$ | 164,068 |
| non-jump positions ($j=0$) | 67,123 |
| jump positions ($j > 0$) | 96,945 |
| $\sum j(n)$ | 164,068 − 2.53 = 164,065.47 |
| pred gap | 2.53 |
(Data: average over 5,000 samples from the k=8 shell.) Two O(m)-scale quantities ($m$ and $\sum j$) nearly cancel, leaving an O(1) positive residual.
§4.2 Statistical Uniformity of Jump Rate
The jump rate ($P(j(n) > 0)$) across 10 equal-fraction positions within the $[m, 2m-1]$ interval for the representative k=8 shell:
| Position | jump rate |
|---|---|
| 0.0–0.1 | 0.5891 |
| 0.1–0.2 | 0.5889 |
| 0.2–0.3 | 0.5893 |
| 0.3–0.4 | 0.5890 |
| 0.4–0.5 | 0.5892 |
| 0.5–0.6 | 0.5895 |
| 0.6–0.7 | 0.5891 |
| 0.7–0.8 | 0.5897 |
| 0.8–0.9 | 0.5899 |
| 0.9–1.0 | 0.5901 |
Variation < 0.002 — statistically uniform. No edge effects within the interval.
§4.3 Worst-Case Integers: All Smooth Numbers
At $N = 10^9$, all integers with the lowest $\rho_E(n)/\ln n$ are smooth numbers of the form $2^a \cdot 3^b$. Representative examples:
| n | $\rho/\ln$ | factorization |
|---|---|---|
| 1,024 | 3.318 | $2^{10}$ |
| 4,096 | 3.366 | $2^{12}$ |
| 12,288 | 3.398 | $2^{12} \cdot 3$ |
| 65,536 | 3.426 | $2^{16}$ |
| 1,048,576 | 3.463 | $2^{20}$ |
| 16,777,216 | 3.487 | $2^{24}$ |
| 268,435,456 | 3.504 | $2^{28}$ |
Smooth numbers have the most splitting options, giving the smallest $M_n$, so $\rho_E$ grows most slowly. The finite-sample low-ratio tail is driven by smooth numbers, but the shell-averaged predecessor gap is unaffected.
Finite-sample checkpoint ($N = 10^9$): $\rho_E(10^9)/\ln(10^9) = 75/20.72 = 3.6191$ — the first checkpoint above 3.61. However, smooth numbers have not crossed 3.61: the best smooth $\rho/\ln = 3.567$. The pure $2^k$ sequence at $k = 29$ gives $\rho/(k \cdot \ln 2) = 3.532$, rising slowly. Smooth number fit ($N = 10^9$ data): $\rho/\ln \approx 3.344 + 0.0102 \cdot \ln(n)$ ($R^2 = 0.72$), predicting crossing of 3.61 at $n \approx 10^{11.4}$.
§4.4 Predecessor Gap Distribution
The predecessor gap takes integer values. Unconditional distribution (all $m \in [2, 5 \times 10^6]$):
| pred gap | fraction |
|---|---|
| ≤ 0 | 1.1% |
| 1 | 6.9% |
| 2 | 24.2% |
| 3 | 42.0% |
| 4 | 25.9% |
Mode is 3. E[pred gap] $\approx 2.85$ unconditionally. Only 8.0% of $m$ have pred gap $\leq 1$.
§5. Formalization Prospects
§5.1 Telescoping + Renewal Route
Proposition 12 gives pred gap = $m - \sum j(n)$. Paper XXI's Lindley isomorphism guarantees memory erasure of $j(n)$ at jump points ($D = 0$ implies queue clearance). This suggests a possible renewal-block argument:
Partition $[m, 2m-1]$ by positions where $D = 0$ into regeneration blocks. If the ratio of block length to within-block $\sum j$ can be controlled, positivity of the pred gap may follow. This direction requires developing Paper XXI's algebraic isomorphism further into a block decomposition theorem for regenerative processes, and is left to subsequent work.
§5.2 Why This Is Easier Than $c^*$
$c^* > 3.61$ requires that every sufficiently large $n$ satisfies $\rho_E(n)/\ln n > 3.61$ — a pointwise lower bound. But the finite-sample minimum of $\rho/\ln$ depends on the starting threshold: 3.32 ($n \geq 10^3$), 3.44 ($n \geq 10^5$), 3.50 ($n \geq 10^8$). Even at $N = 10^9$, smooth numbers' $\rho/\ln$ has not crossed 3.61 (best smooth = 3.567).
pred gap $> \tau_k$ only requires a dyadic interval Cesàro average to exceed $\tau_k$ — an average lower bound.
The latter permits individual values of $\rho_E(n)/\ln n$ to fall below 3.61 (as smooth numbers do), as long as these bad points are averaged away by good points. The $N = 10^9$ data confirms this: the best smooth $\rho/\ln$ is only 3.567, yet E[pred gap] exceeds $\tau_k$ on all representative shells (pred gap margin 0.20–0.36; equivalently $c_{\mathrm{dyad}}$ margin 0.29–0.52).
§5.3 Smooth Number Stress Test (Route C as Auxiliary)
If one can prove:
- (a) E[pred gap] > $\tau_k$ for smooth numbers
- (b) E[pred gap] for non-smooth numbers $\geq$ E[pred gap] for smooth numbers
then (L3) closes. Part (a) can be studied via exact DP on the two-dimensional $2^a \cdot 3^b$ lattice. Part (b) requires an argument that "large prime factors increase $M_n$." Both steps remain open but are more tractable than Route A.
§6. Updated Proof Landscape
| Input | Status | Source |
|---|---|---|
| B (Sathe-Selberg) | Known | Classical |
| A' ($p_\infty(k) \to 1$) | Conditionally closed | Paper 22 |
| $A_q^\sharp$ | Strong numerical support, awaiting formalization | Paper 23 |
| Trivial bound + Defect criterion | Proved | Paper 24 |
| (L1) $p=2$ layer | Exact reduction | Paper 24 |
| (L1) $p \geq 3$ layer | Open | Paper 24 |
| $c_{\mathrm{dyad}} > \tau_k/\ln 2$ | Strong numerical support ($N=10^9$, margin 0.29–0.52) | Paper 25 |
| Dyadic Telescoping | Corollary of Paper XII (Proposition 12) | Paper 25 |
| $R_2$ mean positivity | Reduced to $E[\sum j] < E[m] - \tau_k(N)$ | Paper 25 |
Precise characterization of remaining work: Under the shell condition $\Omega(m) = k, D_m = 0$, prove that the conditional expectation of the jump function sum over the dyadic interval $[m, 2m-1]$ is strictly less than $E[m] - \tau_k(N)$. This is a problem about the local Cesàro average of $j(n)$, independent of the global $c^*$.
§7. Thermodynamic Interface
| ZFCρ | Thermodynamics |
|---|---|
| pred gap = m − Σj | Free energy = capacity − total dissipation |
| Jump rate statistically uniform | Macroscopic projection of microscopic reversibility |
| smooth = worst case | Most "ordered" systems are hardest to self-organize (analogy to Third Law) |
| $c_{\mathrm{dyad}} > 3.604$ | Energy balance positive per dimensional upgrade — system far from equilibrium |
References
- Altman, H. (2016). Integer complexity, addition chains, and well-ordering. Ph.D. thesis.
- Selfridge, J. L. (1953). Cited in Guy, R. K. Unsolved Problems in Number Theory, Problem F26.
- Qin, H. (2025a). ZFCρ Paper XI. DOI: 10.5281/zenodo.18975756.
- Qin, H. (2025b). ZFCρ Paper XII. DOI: 10.5281/zenodo.18977948.
- Qin, H. (2025c). ZFCρ Paper XV. DOI: 10.5281/zenodo.19007312.
- Qin, H. (2025d). ZFCρ Paper XVII. DOI: 10.5281/zenodo.19016958.
- Qin, H. (2025e). ZFCρ Paper XXI. DOI: 10.5281/zenodo.19037934.
- Qin, H. (2025f). ZFCρ Paper XXII. DOI: 10.5281/zenodo.19039953.
- Qin, H. (2025g). ZFCρ Paper XXIII. DOI: 10.5281/zenodo.19041689.
- Qin, H. (2025h). ZFCρ Paper XXIV. DOI: 10.5281/zenodo.19044696.
Appendix A
Route Selection Basis
Route B (dyadic shell average) is favored over Route A (global $c^*$) based on the following evidence: at $N = 10^9$, the best smooth-number $\rho/\ln$ is only 3.567 (still far below 3.61), while E[pred gap] exceeds $\tau_k$ on all representative shells (pred gap margin 0.20–0.36; $c_{\mathrm{dyad}}$ margin 0.29–0.52). This paradox — smooth pointwise extreme below threshold but shell Cesàro average above threshold — becomes more pronounced at larger scales, directly supporting Route B's feasibility. Route C (smooth numbers only) is retained as an extremal stress-test line.
Appendix B
Complete Numerical Tables
B.1 $c_{\mathrm{dyad}}$ Representative Shells ($N = 10^9$)
(See §2.2 table. The text displays k = 5, 8, 10, 12 as representative shells. Complete shell data is reproducible from the accompanying scripts.)
B.2 Predecessor Gap Distribution
(See §4.4 table.)
B.3 $\rho_E(2^k)/(k \cdot \ln 2)$ ($N = 10^9$)
| k | $2^k$ | ρ_E | ρ/(k·ln2) |
|---|---|---|---|
| 10 | 1,024 | 23 | 3.318 |
| 16 | 65,536 | 38 | 3.426 |
| 20 | 1,048,576 | 48 | 3.463 |
| 24 | 16,777,216 | 58 | 3.487 |
| 28 | 268,435,456 | 68 | 3.504 |
| 29 | 536,870,912 | 71 | 3.532 |
Rising slowly. At $2^{28}$, still only 3.504, well below 3.61.
B.4 Global min(ρ/ln) by Threshold ($N = 10^9$)
| $n \geq$ | min(ρ/ln) | achieved at |
|---|---|---|
| $10^3$ | 3.318 | $2^{10}$ |
| $10^5$ | 3.444 | $2^{12} \cdot 3^3$ |
| $10^6$ | 3.463 | $2^{20}$ |
| $10^8$ | 3.504 | $2^{28}$ |
| $10^9$ | 3.619 | $10^9$ |
本文攻击 D(N) → 1 证明链中 (L3) 路线上的一个剩余技术瓶颈——$R_2$ 的均值正性（Paper 24 归约）。我们引入 dyadic shell constant $c_{\mathrm{dyad}}(k; N)$，将 predecessor gap 问题从全局 $c^*$ 下界（70 年未解）转移到壳层条件的 dyadic 区间平均。核心工具是 Dyadic Telescoping（命题 12，Paper XII 的推论）：
$$\rho_E(2m-1) - \rho_E(m-1) = m - \sum_{n=m}^{2m-1} j(n)$$其中 $j(n) = \max(G(n), 0)$ 是 Paper XII 的跳跃函数。predecessor gap 精确等于 $m$ 减去 dyadic 区间 $[m, 2m-1]$ 上的跳跃函数总和。$R_2$ 正性归约为：dyadic 区间上跳跃函数的条件期望总和严格小于条件期望下的 $m$ 减去 target gap 阈值 $\tau_k(N) \approx 2.498$。
数值发现（$N = 10^9$）：E[pred gap] 在所展示的代表壳层（k = 5, 8, 10, 12）上 > $\tau_k \approx 2.498$（pred gap margin 0.20–0.36），jump rate 在代表壳层的 dyadic 区间内统计均匀（波动 < 0.002），predecessor gap 取整数值，众数为 3（占 42%）。
在截至 $N = 10^9$ 的有限样本中，worst-case $\rho_E(n)/\ln n$ 仍由 smooth numbers 主导；尽管 smooth family 中的最佳比值仅 3.567（< 3.61），壳层条件的 dyadic Cesàro 平均已经稳定高于阈值（$c_{\mathrm{dyad}}$ 超过 $\tau_k/\ln 2 \approx 3.604$ 的 margin 为 0.29–0.52）。正是这种"点态极值低、区间平均高"的分离，使 Route B 比 Route A 更贴题。
§1. 引言
§1.1 (L3) 的剩余瓶颈
Paper 24 在 (L3) 路线上把剩余 target-gap 问题隔离为 $R_2$ 的均值正性。对固定 cutoff $N$：
$$E[R_2 \mid \Omega(m) = k, D_m = 0, m \leq N] > 0$$等价于
$$E[\rho_E(2m-1) - \rho_E(m-1) \mid \Omega(m) = k, D_m = 0, m \leq N] > \tau_k(N)$$其中 $\tau_k(N) := E[M_{2m} - M_m \mid \Omega(m) = k, D_m = 0, m \leq N]$，数值上 $\tau_k(N) \approx 2.498$。
一个自然的 heuristic 是：$\rho_E(2m-1) - \rho_E(m-1) \approx c^* \ln 2$，所以需要 $c^* > 2.498/\ln 2 \approx 3.61$。但 Paper XI 的已证下界只有 $c^* \geq 3/\ln 3 \approx 2.73$，且整数复杂度下界自 Selfridge (1953) 以来进展极慢。
本文的关键观察： $c^*$ 下界是一个比实际需要更强的条件。我们需要的不是全局点态 $\liminf \rho_E(n)/\ln n > 3.61$，而是壳层条件下 dyadic 区间的局部平均正性——一个严格更弱的命题。
§1.2 本文贡献
- Dyadic Shell Constant $c_{\mathrm{dyad}}$（§2，定义 + 数值）。 将问题从全局 $c^*$ 转移到壳层条件的 dyadic 平均。
- Dyadic Telescoping（§3，Paper XII 的推论）。 pred gap = $m - \sum j(n)$——直接连回 Paper XII 的跳跃函数 telescoping。
- Formal reduction（§3.3）。 $R_2$ 正性归约为 $E[\sum j] < E[m] - \tau_k(N)$，条件均在 $\Omega = k, D_m = 0$ 下。
- 数值结构（§4）。 在代表壳层（k = 5, 8, 10, 12）上 pred gap > 2.50（$N = 10^9$ 数据）；jump rate 在代表壳层 k=8 的 dyadic 区间内统计均匀（波动 < 0.002）；worst-case $\rho_E/\ln n$ 全部出现在 smooth numbers 上。
§1.3 三条路线的比较
| 路线 | 目标 | 难度 | 杠杆 |
|---|---|---|---|
| A（$c^*$ 下界） | 证 $c^* > 3.61$（全局点态） | 极高（70 年未解） | 高但过强 |
| B（pred gap，本文主攻） | 证壳层条件 dyadic 平均 > 2.50 | 中高 | 直接闭合 (L3) |
| C（smooth only） | 只证 smooth 数的 pred gap | 中 | 中（仍需非 smooth ≥ smooth） |
本文执行 Route B。Route A 和 C 的比较见附录 A。
§2. Dyadic Shell Constant
§2.1 定义
定义. 对固定 $k \geq 2$，定义 dyadic shell constant：
$$c_{\mathrm{dyad}}(k; N) := \frac{E[\rho_E(2m-1) - \rho_E(m-1) \mid \Omega(m) = k, D_m = 0, m \leq N]}{\ln 2}$$$R_2$ 均值正性（Paper 24）等价于：
$$c_{\mathrm{dyad}}(k; N) > \frac{\tau_k(N)}{\ln 2}$$其中 $\tau_k(N) := E[M_{2m} - M_m \mid \Omega(m) = k, D_m = 0, m \leq N]$。数值上 $\tau_k(N) \approx 2.498$（对所有测试 k），因此阈值 $\tau_k/\ln 2 \approx 3.604$。
§2.2 数值（$N = 10^9$）
| k | E[pred gap] | $c_{\mathrm{dyad}}$ | margin above $\tau_k/\ln 2$ |
|---|---|---|---|
| 5 | 2.855 | 4.119 | +0.515 |
| 8 | 2.753 | 3.971 | +0.367 |
| 10 | 2.711 | 3.911 | +0.307 |
| 12 | 2.698 | 3.893 | +0.289 |
$c_{\mathrm{dyad}}$ 从 4.12（k=5）缓降到 3.89（k=12），始终 > 3.604。
§2.3 与全局 $c^*$ 的关系
全局 $c^*$ 是 $\liminf_{n \to \infty} \rho_E(n)/\ln n$（点态）。$c_{\mathrm{dyad}}$ 是壳层条件下 dyadic 区间上的 Cesàro 平均。两者不同：
- 全局 min(ρ/ln) 依赖于起始阈值：min = 3.32（n ≥ 1,000），3.44（n ≥ 100,000），3.49（n ≥ 5×10⁶），均由 smooth numbers（2^a · 3^b 型）达到
- $c_{\mathrm{dyad}}$ 是区间平均，smooth number 的个别低值被壳层内其他整数"平均掉"
§3. Dyadic Telescoping（Paper XII 的推论）
§3.1 恒等式
命题 12（Dyadic Telescoping）. 对任何 $m \geq 2$：
$$\rho_E(2m-1) - \rho_E(m-1) = m - \sum_{n=m}^{2m-1} j(n)$$其中 $j(n) = \max(G(n), 0) = \max(\rho_E(n-1) + 1 - M_n, 0)$ 是 Paper XII 的跳跃函数。
注. 这是 Paper XII 的 $\delta(N) = \sum j(n)$ telescoping 在 dyadic 区间上的直接推论。它的价值不在于证明本身（一行 corollary），而在于暴露了 Route B 的正确局部对象：predecessor gap 精确等于 $m$ 减去 dyadic 区间上的跳跃函数总和。
证明. 由 DP 递推 $\rho_E(n) = \min(\rho_E(n-1) + 1, M_n)$，对每个 $n$：
$$\rho_E(n) = \rho_E(n-1) + 1 - j(n)$$（当 $G(n) > 0$ 时 $\rho_E(n) = M_n = \rho_E(n-1) + 1 - j(n)$；当 $G(n) \leq 0$ 时 $\rho_E(n) = \rho_E(n-1) + 1$ 且 $j(n) = 0$。）
对 $n = m, m+1, \ldots, 2m-1$ 求和（共 $m$ 项 telescoping）：
$$\rho_E(2m-1) - \rho_E(m-1) = m - \sum_{n=m}^{2m-1} j(n) \qquad \square$$§3.2 等价重写
定义 $A(n) = j(n) - 1$（Paper XVII），则：
$$\rho_E(2m-1) - \rho_E(m-1) = -\sum_{n=m}^{2m-1} A(n)$$predecessor gap 精确等于 dyadic 区间上 $A(n)$ 的负总和。由于 $A(n) \leq 0$ 当 $G(n) \leq 0$（此时 $j = 0, A = -1$），$A(n) \geq 0$ 当 $G(n) > 0$（此时 $A = j - 1 \geq 0$），正负贡献的不平衡决定了 predecessor gap 的符号和大小。
§3.3 归约为跳跃密度
定义 target gap 阈值：
$$\tau_k(N) := E[M_{2m} - M_m \mid \Omega(m) = k, D_m = 0, m \leq N]$$数值上 τ_k(N) ≈ 2.498（对所有测试的 k 和 N，见 Paper 24）。
$R_2$ 均值正性等价于：
$$E\left[\sum_{n=m}^{2m-1} j(n) \;\middle|\; \Omega(m) = k, D_m = 0, m \leq N\right] < E[m \mid \Omega(m) = k, D_m = 0, m \leq N] - \tau_k(N)$$即 dyadic 区间 $[m, 2m-1]$ 上的跳跃函数条件总和的期望必须严格小于条件期望下的 $m$ 减去 target gap 阈值。
§4. 数值结构
§4.1 Step/Jump 分解
由命题 12，对 k=8 壳层的代表 $m$（$m \approx 164{,}000$），区间 $[m, 2m-1]$ 内：
$$\text{pred gap} = m - \sum_{n=m}^{2m-1} j(n)$$其中 $j(n) = 0$ 的位置（non-jump）贡献 $+1$ 到 pred gap，$j(n) > 0$ 的位置贡献 $1 - j(n) \leq 0$。
| 量 | 值 |
|---|---|
| span $m$ | 164,068 |
| non-jump 位置 ($j=0$) | 67,123 |
| jump 位置 ($j > 0$) | 96,945 |
| $\sum j(n)$ | 164,068 − 2.53 = 164,065.47 |
| pred gap | 2.53 |
（数据为 k=8 壳层 5,000 个样本的平均值。）两个 O(m) 级的量（$m$ 和 $\sum j$）几乎精确抵消，留下 O(1) 的正残差。
§4.2 Jump Rate 统计均匀性
Jump rate（即 $P(j(n) > 0)$）在代表壳层 k=8 的 $[m, 2m-1]$ 区间内 10 个等分位置上：
| 位置 | jump rate |
|---|---|
| 0.0–0.1 | 0.5891 |
| 0.1–0.2 | 0.5889 |
| 0.2–0.3 | 0.5893 |
| 0.3–0.4 | 0.5890 |
| 0.4–0.5 | 0.5892 |
| 0.5–0.6 | 0.5895 |
| 0.6–0.7 | 0.5891 |
| 0.7–0.8 | 0.5897 |
| 0.8–0.9 | 0.5899 |
| 0.9–1.0 | 0.5901 |
波动 < 0.002——统计均匀。区间内无边缘效应。
§4.3 Worst-Case 整数：全是 Smooth Numbers
在 $N = 10^9$ 下，$\rho_E(n)/\ln n$ 最小的整数全部是 $2^a \cdot 3^b$ 型 smooth numbers。代表例：
| n | $\rho/\ln$ | 分解 |
|---|---|---|
| 1,024 | 3.318 | $2^{10}$ |
| 4,096 | 3.366 | $2^{12}$ |
| 12,288 | 3.398 | $2^{12} \cdot 3$ |
| 65,536 | 3.426 | $2^{16}$ |
| 1,048,576 | 3.463 | $2^{20}$ |
| 16,777,216 | 3.487 | $2^{24}$ |
| 268,435,456 | 3.504 | $2^{28}$ |
Smooth numbers 拥有最多分裂选择 → $M_n$ 最小 → $\rho_E$ 增长最慢。有限样本中 $\rho_E/\ln n$ 的低值尾部由 smooth numbers 驱动，但 predecessor gap 的壳层平均不受此限制。
Finite-sample checkpoint（$N = 10^9$）： $\rho_E(10^9)/\ln(10^9) = 75/20.72 = 3.6191$——首次 checkpoint 过 3.61。但 smooth numbers 仍未过 3.61：最佳 smooth ρ/ln = 3.567。纯 $2^k$ 序列在 $k = 29$ 时 ρ/(k·ln2) = 3.532，上升缓慢。Smooth number 拟合（$N = 10^9$ 数据）：$\rho/\ln \approx 3.344 + 0.0102 \cdot \ln(n)$（$R^2 = 0.72$），预测在 $n \approx 10^{11.4}$ 处过 3.61。
§4.4 Predecessor Gap 分布
predecessor gap 取整数值。无条件分布（所有 $m \in [2, 5 \times 10^6]$）：
| pred gap | 比例 |
|---|---|
| ≤ 0 | 1.1% |
| 1 | 6.9% |
| 2 | 24.2% |
| 3 | 42.0% |
| 4 | 25.9% |
众数为 3。E[pred gap] ≈ 2.85 无条件。仅 8.0% 的 $m$ 有 pred gap ≤ 1。
§5. 形式化前景
§5.1 Telescoping + Renewal 路线
命题 12 给出 pred gap = $m - \sum j(n)$。Paper XXI 的 Lindley 同构保证了 $j(n)$ 在跳跃处的记忆擦除（$D = 0$ 时队列清空）。这暗示了一种可能的 renewal-block 论证方向：
将 $[m, 2m-1]$ 按 $D = 0$ 的位置分成再生块。如果块长和块内 $\sum j$ 的比率可以控制，pred gap 的正性有望跟出。这一方向需要将 Paper XXI 的代数同构进一步发展为再生过程的分块定理，留待后续工作。
§5.2 为什么这比 $c^*$ 更容易
$c^* > 3.61$ 要求每一个大 $n$ 的 $\rho_E(n)/\ln n > 3.61$——点态下界。但全局 min(ρ/ln) 依赖于起始阈值：3.32（$n \geq 10^3$），3.44（$n \geq 10^5$），3.50（$n \geq 10^8$）。即使在 $N = 10^9$ 下，smooth numbers 的 ρ/ln 仍未突破 3.61（最佳 smooth = 3.567）。
pred gap > $\tau_k$ 只要求 dyadic 区间上的 Cesàro 平均 > $\tau_k$——平均下界。
后者允许个别 $n$ 的 $\rho_E(n)/\ln n$ 低于 3.61（如 smooth numbers），只要这些坏点被好点平均掉。$N = 10^9$ 的数据确认了这一点：smooth numbers 的 ρ/ln 最佳仅 3.567，但 E[pred gap] 在代表壳层上 > $\tau_k$（pred gap margin 0.20–0.36；等价地 $c_{\mathrm{dyad}}$ margin 0.29–0.52）。
§5.3 Smooth Number 压力测试（Route C 作为副线）
如果能证明：
- (a) smooth numbers 上的 E[pred gap] > 2.50
- (b) 非 smooth 数的 E[pred gap] ≥ smooth 数的 E[pred gap]
则 (L3) 闭合。(a) 可以在 $2^a \cdot 3^b$ 的二维网格上用精确 DP 研究。(b) 需要"大素因子使 $M_n$ 增大"的论证。两步都是 open 的，但比 Route A 更可控。
§6. 证明格局更新
| 输入 | 状态 | 来源 |
|---|---|---|
| B（Sathe-Selberg） | 已知 | 经典 |
| A'（$p_\infty(k) \to 1$） | 条件性闭合 | Paper 22 |
| $A_q^\sharp$ | 数值强支持，待形式化 | Paper 23 |
| Trivial bound + Defect criterion | 已证 | Paper 24 |
| (L1) $p=2$ 层 | 精确归约 | Paper 24 |
| (L1) $p \geq 3$ 层 | 开放 | Paper 24 |
| $c_{\mathrm{dyad}} > \tau_k/\ln 2$ | 数值强支持（$N=10^9$，margin 0.29–0.52） | Paper 25 |
| Dyadic Telescoping | Paper XII 的推论（命题 12） | Paper 25 |
| $R_2$ 均值正性 | 归约为 $E[\sum j] < E[m] - \tau_k(N)$ | Paper 25 |
剩余工作的精确刻画：在壳层条件 $\Omega(m) = k, D_m = 0$ 下，证明 dyadic 区间 $[m, 2m-1]$ 上跳跃函数条件总和的期望严格小于 $E[m] - \tau_k(N)$。这是一个关于 $j(n)$ 局部 Cesàro 平均的问题，不依赖全局 $c^*$。
§7. 热力学接口
| ZFCρ | 热力学 |
|---|---|
| pred gap = m − Σj | 自由能 = 容量 − 耗散总量 |
| jump rate 统计均匀 | 微观可逆性的宏观投影 |
| smooth = worst case | 最"有序"的系统最难自组织（热力学第三定律的类比） |
| $c_{\mathrm{dyad}}$ > 3.604 | 每次维度提升的能量收支为正——系统远离平衡 |
参考文献
- Altman, H. (2016). Integer complexity, addition chains, and well-ordering. Ph.D. thesis.
- Selfridge, J. L. (1953). Cited in Guy, R. K. Unsolved Problems in Number Theory, Problem F26.
- Qin, H. (2025a). ZFCρ Paper XI. DOI: 10.5281/zenodo.18975756.
- Qin, H. (2025b). ZFCρ Paper XII. DOI: 10.5281/zenodo.18977948.
- Qin, H. (2025c). ZFCρ Paper XV. DOI: 10.5281/zenodo.19007312.
- Qin, H. (2025d). ZFCρ Paper XVII. DOI: 10.5281/zenodo.19016958.
- Qin, H. (2025e). ZFCρ Paper XXI. DOI: 10.5281/zenodo.19037934.
- Qin, H. (2025f). ZFCρ Paper XXII. DOI: 10.5281/zenodo.19039953.
- Qin, H. (2025g). ZFCρ Paper XXIII. DOI: 10.5281/zenodo.19041689.
- Qin, H. (2025h). ZFCρ Paper XXIV. DOI: 10.5281/zenodo.19044696.
附录 A
路线选择的依据
Route B（dyadic shell 平均）优于 Route A（全局 $c^*$）的数据依据：$N = 10^9$ 下，smooth numbers 的 ρ/ln 最佳仅 3.567（仍远低于 3.61），而 E[pred gap] 在代表壳层上 > $\tau_k$（pred gap margin 0.20–0.36；$c_{\mathrm{dyad}}$ margin 0.29–0.52）。这一悖论（smooth 点态极值 < 阈值但壳层 Cesàro 平均 > 阈值）在更大尺度下更加确定，直接支持 Route B 的可行性。Route C（smooth numbers only）作为极值压力测试线保留。
附录 B
完整数值表
B.1 $c_{\mathrm{dyad}}$ 代表壳层（$N = 10^9$）
（见 §2.2 表格。正文展示 k = 5, 8, 10, 12 四个代表壳层。完整壳层数据可由附带脚本复现。）
B.2 Predecessor gap 分布
（见 §4.4 表格）
B.3 $\rho_E(2^k)/(k \cdot \ln 2)$（$N = 10^9$）
| k | $2^k$ | ρ_E | ρ/(k·ln2) |
|---|---|---|---|
| 10 | 1,024 | 23 | 3.318 |
| 16 | 65,536 | 38 | 3.426 |
| 20 | 1,048,576 | 48 | 3.463 |
| 24 | 16,777,216 | 58 | 3.487 |
| 28 | 268,435,456 | 68 | 3.504 |
| 29 | 536,870,912 | 71 | 3.532 |
上升缓慢。$2^{28}$ 时仍为 3.504，远低于 3.61。
B.4 Global min(ρ/ln) by threshold（$N = 10^9$）
| $n \geq$ | min(ρ/ln) | 达到于 |
|---|---|---|
| $10^3$ | 3.318 | $2^{10}$ |
| $10^5$ | 3.444 | $2^{12} \cdot 3^3$ |
| $10^6$ | 3.463 | $2^{20}$ |
| $10^8$ | 3.504 | $2^{28}$ |
| $10^9$ | 3.619 | $10^9$ |