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

Zihang Fang

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

T0 review · reviewed 2026-07-30 · grok-4.5

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Pith's one-line read Self-prefix powers reduce to one signed discrepancy and a two-gap candidate law, with a certified sublinear search whose infinitude for 2^m still open.

desk verdict Solid structural and algorithmic package on self-prefix digits that honestly stops short of infinitude for 2^m. read the letter →

arxiv 2607.23662 v1 pith:LBMJY6TE submitted 2026-07-26 math.NT

Zihang Fang This is my paper · ORCID

classification math.NT MSC 11A6311J7011J7111Y16

keywords leadingdigitsshrinkingtargetsLambertWfunctiondiscrepancyresonancecontinuedfractionscertifiedsearchself-prefix

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies when c^m begins with the digits of m itself (or the analogous inequality for real c>1). It rewrites the event as a one-sided shrinking target of width about 1/m and proves an exact identity that turns the hit count into the signed discrepancy of a single fixed interval for the nonlinear sequence nα−log_b(n+1). For c≥2, Lambert W_{-1} inversion produces a rigid candidate sequence with an eventual two-gap law and an exact count; for base-10 powers of 2 the gaps are only 3 or 4. Actual hits obey fixed-difference shells and arithmetic-chain rigidity, and at floor resonance centers endpoint hits force every intermediate term. For multiplicatively independent integer pairs the paper builds an interpolated continued-fraction locator whose bit cost is O(N^{1−1/ν} polylog N) whenever the irrationality exponent of {log_b c} is finite, with an explicit machine-checkable safe threshold for (2,10). Infinitude of the classic sequence and the conjectured logarithmic growth remain open; the work isolates exactly what must be proved at the critical scale.

What carries the argument

The exact signed-discrepancy identity (Theorem 4.1): A_{c,b}(N)=log_b(N+1)+B_{α,b}(N)−ρN+{y_N}. It converts the original count into discrepancy of one fixed interval and isolates both open questions; Lambert W_{-1} layer endpoints and the interpolated convergent-block locator carry the geometry and the algorithm.

What would settle it

Decide the sign of d_+(x_j)−(z_j−x_j) for the explicit Lambert roots x_j,z_j of the (2,10) layer equations: infinitely many negative signs prove S_{2,10} infinite; a persistently positive liminf at second order (or higher) would prove it finite.

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Extended reading notes

Core claim

For c≥2 the self-prefix count equals log_b(N+1) plus the signed discrepancy of one fixed arc for y_n=nα−log_b(n+1), so infinitude and the logarithmic law are equivalent to that discrepancy being unbounded or o(log N). Lambert inversion supplies an exact candidate geometry (eventual gaps in {⌊1/ρ⌋,⌊1/ρ⌋+1}, closed counting formula), while resonance and continued-fraction packing give arithmetic rigidity and a certified sublinear search.

Load-bearing premise

The power-saving and sublinear-complexity claims rest on finite Diophantine type for log_b c coming from lower bounds on linear forms in logarithms and a discrepancy theorem for nα+β log n; if that type control fails, those exponents collapse while the exact identities can still stand.

Editorial extensions

If this is right

Infinitude of S_{c,b} is exactly equivalent to log_b(N+1)+D_{α,b}(N) being unbounded.
The logarithmic law A∼log_b N is exactly equivalent to D_{α,b}(N)=o(log N).
For (2,10) every large enough block of length ≤44 699 994 contains at most one index that needs exact verification.
Multiplicatively dependent integer pairs are completely classified and already obey A=log_b X+O(1).
Fixed differences and long arithmetic progressions of hits are confined to explicit resonance shells and force continued-fraction convergents.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same discrepancy identity suggests that any future one-sided approximation theorem for the Lambert roots at scale 1/j would settle both open problems at once.
The certified locator can be reused as a black-box filter for other self-referential or moving-target digit problems once a finite-type bound is known.
Because candidate gaps are only 3 or 4 for (2,10), exhaustive verification of candidates up to very large M is limited by the critical phase test, not by candidate density.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The useful core is not a resolution of van de Lune’s problem, but a clean reorganization of it. Theorem 4.1 turns the count into an exact signed discrepancy for one nonlinear sequence, so infinitude and the log law become precise statements about D rather than folklore. Alongside that, the Lambert W_{-1} layer geometry gives a genuine candidate sequence with an exact count and a global 3–4 gap law for (2,10), and the paper never pretends candidates are hits.

What is new is the problem-specific package: coupled inverse-root widths and complete monotonicity, fixed-difference resonance shells, floor-center nesting with endpoint filling and a coherent-skeleton size bound, and an interpolated continued-fraction locator with bit complexity O(N^{1-1/ν} polylog N) under finite type, plus a shipped interval-arithmetic certificate for a safe singleton threshold past about 1.9×10^{15}. Table 1 and §7 keep the logical boundaries honest; that is real discipline, not padding.

Soft spots are proportional and mostly expected. Quantitative sparsity, subcritical moving targets, and the complexity exponent lean on Matveev and Tichy–Turnwald; for fixed algebraic/integer parameters that is standard, not a hidden fit, but if you only care about unconditional critical-scale recurrence you will leave empty-handed. The manuscript is long and unformalized, so ordinary verification risk remains, including the Arb certificate. Significance is moderate: organizational and algorithmic progress inside a classical open OEIS-type question, not a breakthrough on the open problems themselves.

This is for people who work on leading digits, shrinking targets, or certified Diophantine search. Citations look appropriate; no circularity burden on the exact identities. I would send it to referees. Worth engaging if that is your neighborhood; skip if you only wanted a proof that S_{2,10} is infinite.

Circularity Check

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No significant circularity: exact identities, candidate geometry, and rigidity are self-contained reformulations or one-way implications; quantitative exponents are conditional on external Diophantine inputs.

full rationale

The paper’s load-bearing claims do not reduce to their inputs by construction. Theorem 4.1 is a telescoping identity from the prefix indicator and fractional-part arithmetic; the equivalences for infinitude and A∼log_b N are exact restatements of that identity, not smuggled conclusions. Lambert W_{-1} layer geometry (Theorems 3.1–3.5) inverts Φ and Ψ and counts candidates m_j=⌈x_j⌉ with an eventual two-gap law; the text repeatedly separates candidates from actual hits S_{c,b}. Resonance and nesting results (Theorems 5.13, 5.19, 5.23, 5.26) are one-way implications from assumed hits to shells, chain bounds, and endpoint filling—they do not posit hits from Diophantine data. Subcritical discrepancy and complexity exponents invoke Matveev and Tichy–Turnwald as external finite-type inputs and state μ(ρ)<∞ as a hypothesis (Theorems 3.10, 5.18, 6.6). The (2,10) safe-threshold certificate is a machine-checkable computation, not a fit renamed as prediction. The sole author self-citation is the code/data release for that certificate, which is independent reproducibility support rather than a load-bearing uniqueness or ansatz step. No self-definitional loop, fitted-input-as-prediction, or renaming of a known empirical law as a derivation was found.

Assumptions & free parameters 2 free parameters · 6 assumptions · 2 invented entities

Core results are definitional plus standard real/complex and Diophantine tools. Load-bearing external axioms are Matveev lower bounds, Tichy–Turnwald finite-type discrepancy, classical continued-fraction separation/Legendre, and Gelfond–Schneider for the rational/transcendental dichotomy. No data-fitted constants drive the main theorems; ν>μ(ρ) is a free complexity parameter, not a fit. Constructed objects (Lambert layers, shells, coherent skeletons) are explicit definitions, not physical postulates.

free parameters (2)

ν > μ(ρ) in complexity/sparsity exponents = arbitrary ν>μ(ρ); for bounded partial quotients effective exponent 1/2
Any number strictly larger than the irrationality exponent of ρ={log_b c} may be chosen; it parametrizes the proven power in O(N^{1-1/ν} polylog N) and related sieves, not a fit to hit data.
Subcritical exponents (τ,σ,κ) in Theorem 3.12
Free choices in 0<τ<σ<1/(η+1) and κ>0 controlling the moving-target count below critical scale; not fitted to S_{c,b}.

assumptions (6)

standard math Matveev-type effective lower bounds for nonzero linear forms in logarithms of algebraic numbers (used for type of 1/ρ and certification of signs).
Invoked in Theorems 3.10, 5.18, Lemma 6.5; supplies η and polylog separation for floors/endpoints.
standard math Tichy–Turnwald discrepancy bounds for sequences a n + β log n at finite Diophantine type.
Lemma 3.9 / Theorem 3.10 transfer ordinary and log-weighted discrepancy to Lambert roots below critical scale.
standard math Classical continued-fraction best-approximation, three-distance/packing separation, and Legendre’s convergent criterion.
Block packing (Thm 6.2), long hit-chain ⇒ convergent (Cor 5.20), interpolated locator (Thm 6.6).
standard math Gelfond–Schneider / Baker framework: for integer c,b≥2, log_b c rational iff multiplicatively dependent; otherwise transcendental.
Proposition 5.1 splits dependent classification from independent rigidity.
standard math Real Lambert W_{-1} branch inversion and analytic Lagrange–Bürmann expansion on the large-root layer.
Section 3 defines candidate endpoints x_j, z_j and width asymptotics.
domain assumption Deterministic multiprecision ball arithmetic correctly decides signs once radius is below Matveev separation (computational model §6).
Needed for Corollary 6.8 certificate and bit-complexity claims; pinned to Arb/python-flint in the supplement.

invented entities (2)

Lambert candidate sequence m_j = ⌈x_j⌉ from coupled W_{-1} layer roots independent evidence
purpose: Enumerate all integers that can possibly be prefix hits in each logarithmic layer and prove two-gap/exact-count geometry independent of actual hits.
Defined by exact inversion Φ(x)=j, Ψ(z)=j; not assumed to equal S_{c,b}.
Fixed-difference resonance shells I_{h,k} and floor-resonance coherent skeletons K_j independent evidence
purpose: Localize double hits and force intermediate hits when endpoints of nested windows are hits.
Derived containment/sieve objects from the prefix inequalities; potential skeletons need not contain hits (Remark 5.32).

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Pith. "Pith review of Self-Referential Leading Digits of Exponential Sequences: Arithmetic Structure and Certified Search." pith.science (2026). https://pith.science/paper/LBMJY6TE

@misc{pith2026260723662,
  author       = {Pith},
  title        = {Pith review of: Self-Referential Leading Digits of Exponential Sequences: Arithmetic Structure and Certified Search},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LBMJY6TE}},
  note         = {Machine review of arXiv:2607.23662}
}

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 $\rho=\{\log_b c\}$. For fixed multiplicatively independent integers $c,b$, an interpolated continued-fraction locator has bit complexity $O(N^{1-1/\nu}\operatorname{polylog}N)$ for every $\nu>\mu(\rho)$. We give an explicit certified instance for $(2,10)$, whose infinitude remains open.

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Pith tools

Reviewed July 30, 2026 · model on record in the stance chip above.

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Bach and J

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A. Berger and T. P. Hill.An Introduction to Benford’s Law. Princeton, NJ: Princeton University Press, 2015. doi:10.1515/9781400866588

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doi:10.1007/bf02124750 1996

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2026