AI 中文总结
研究指数序列自引用首位数，推导收缩目标准则等，对\(c\geq2\)得出相关结论，给出特定情况下计数公式等，还给出插值连分数定位器复杂度及\((2,10)\)的认证实例，其无穷性待解。
AI 中文摘要
对于\(c > 1\)和整数基数\(b \geq 2\)，我们研究满足\(mb^k \leq c^m < (m + 1)b^k\)（其中\(k \geq 0\)）的正整数\(m\)；对于整数\(c\)，这是自前缀首位数条件。我们推导出精确的收缩目标准则；对于\(c \geq 2\)，精确的符号差异恒等式确定了无穷性和推测的对数计数。对于具有非整数对数斜率的\(c \geq 2\)，兰伯特\(W_{-1}\)反演产生一个具有最终双间隙定律和精确计数公式的候选序列；对于\((c, b) = (2, 10)\)，所有连续候选间隙为\(3\)或\(4\)。对于具有无理数\(\log_b c\)的代数\(c\)，兰伯特根相位在严格低于临界尺度的显式非平凡幂范围内满足确定性移动目标渐近性。对于无理数对数斜率，实际命中服从固定差异和算术链刚性；对于乘法独立的整数参数，在地板共振中心的相干端点命中迫使每个中间项。最后，设\(\rho = \{ \log_b c \}\)。对于固定的乘法独立整数\(c, b\)，插值连分数定位器对于每个\(\nu > \mu(\rho)\)具有位复杂度\(O(N^{1 - 1 / \nu} \text{polylog} N)\)。我们给出了\((2, 10)\)的一个显式认证实例，其无穷性仍然未知。
英文摘要
For $c>1$ and an integer radix $b\ge2$, we study the positive integers $m$ for which $mb^k\le c^m<(m+1)b^k$ for some $k\ge0$; for integer $c$, this is the self-prefix leading-digit condition. We derive an exact shrinking-target criterion; for $c\ge2$, an exact signed-discrepancy identity isolates both infinitude and the conjectural logarithmic count. For $c\ge2$ with nonintegral logarithmic slope, Lambert $W_{-1}$ inversion produces a candidate sequence with an eventual two-gap law and an exact counting formula; for $(c,b)=(2,10)$ all consecutive candidate gaps are $3$ or $4$. For algebraic $c$ with irrational $\log_b c$, the Lambert-root phases satisfy deterministic moving-target asymptotics in an explicit nontrivial power range strictly below the critical scale. For irrational logarithmic slope, actual hits obey fixed-difference and arithmetic-chain rigidity; for multiplicatively independent integer parameters, coherent endpoint hits at floor resonance centers force every intermediate term. Finally, set $ρ=\{\log_b c\}$. For fixed multiplicatively independent integers $c,b$, an interpolated continued-fraction locator has bit complexity $O(N^{1-1/ν}\operatorname{polylog}N)$ for every $ν>μ(ρ)$. We give an explicit certified instance for $(2,10)$, whose infinitude remains open.
Comments49 pages
