Self-Referential Leading Digits of Exponential Sequences: Arithmetic Structure and Certified Search

Abstract

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.

Disclosure

“clares no competing interests. CRediT authorship contribution statement Zihang Fang: Conceptualization, Formal analysis, Investigation, Methodology, Software, Validation, Writing–original draft, Writing–review and editing. Declaration of generative AI and AI-assisted technologies in the manuscript preparation process During the preparation of this work, the author used ChatGPT and Codex (OpenAI) to assist with literature searches and proof-checking. The author independently checked all”

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Proof ideas or individual proof-step assistance

Multiplier

8

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2026-08-23T19:08:52.178664Z

Structural counts