arXiv:2604.17136v2

On the normality of the concatenated Fibonacci constant

José Ricardo G. Mendonça

math.NTmath.PRmath.ST11K1611B3911J71

Paper metadata

Original submission: April 18, 2026
Latest revision: July 14, 2026 · version 2
Claimed MSC: 11K16, 11B39, 11J71
arXiv categories: math.NT, math.PR, math.ST
Paper license: CC BY 4.0 ↗
Comments: AMSart style, 21 pages, 8 tables, no figures, 26 references

Abstract

We study the concatenated Fibonacci constant $\mathcal{F} := 0.F_{1}F_{2}F_{3}\cdots = 0.11235813\cdots$ , obtained by concatenating the Fibonacci numbers in the fractional part, and ask whether it is normal. We show that several classical sufficient conditions for normality by concatenation do not apply to the Fibonacci sequence because of its exponential growth, while a criterion of Pollack and Vandehey implies that the normality of $\mathcal{F}$ in base $10$ would follow if almost all Fibonacci numbers were $(\varepsilon,k)$ -normal in base $10$ . The Benford bias of leading digits and the Pisano periodicity of trailing digits are shown to contribute asymptotically negligible fractions of the total digits, isolating the distribution of the deep digits of large Fibonacci numbers as the remaining obstruction. Large-scale numerical experiments on the first $500{,}000$ Fibonacci numbers in bases $10$ and $2$ indicate that global single-digit counts and $k$ -block statistics for $k = 2, 3, 4$ are compatible with i.i.d.-like fluctuations at the scales tested, and that a positional decomposition concentrates the visible structured deviation at the boundaries between consecutive Fibonacci numbers, while pooled interior blocks remain close to uniform. Our computations suggest that any obstruction to normality lies in the asymptotic behavior of the deep digits of $F_{n}$ .

AI-generated audit

Audit summary

Audited against arXiv v2

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 15, 2026

01Statements3 reported findingsCorrect

The exact counting formula, the sufficient normality criterion as applied to Fibonacci numbers, and the leading-, trailing-, and row-versus-column structural conclusions are correct. The paper does not claim to prove normality; its finite computations are explicitly presented only as diagnostics.

Proposition 1Correct

Exact counting function for Fibonacci numbers

Pages 2–3 · Proposition 1 and equations (2)–(3) · arXiv:2604.17136v2

The nearest-integer form of Binet's formula gives $F_n\leq N$ exactly when $n<\log_\phi(\sqrt5(N+1/2))$ . The endpoint cannot be integral: if $\phi^m=\sqrt5(N+1/2)$ , then $2\phi^m=(2N+1)\sqrt5$ , whereas $2\phi^m=F_m+2F_{m-1}+F_m\sqrt5$ has a strictly positive rational part. Counting the positive integers below the resulting nonintegral endpoint gives the printed floor formula, including the multiplicity of $F_1=F_2=1$ .

Full paper, version 2 ↗

Theorem 3, Proposition 4, and Corollary 5Correct

Almost-all internal digit normality is a sufficient route

Pages 4–5 · Theorem 3, Proposition 4, and Corollary 5 · arXiv:2604.17136v2

The total length of the first $m$ Fibonacci strings is $\frac12m^2\log_{10}\phi+O(m)$ , while the largest individual length is $m\log_{10}\phi+O(1)$ . Thus boundary-crossing blocks contribute $o(1)$ of all blocks and an $o(m)$ exceptional set contributes $o(1)$ of the total digit mass. On the remaining strings, $(\varepsilon,k)$ -normality gives the required weighted block frequencies; first letting $m\to\infty$ and then $\varepsilon\to0$ proves the stated sufficient implication.

Pollack–Vandehey source cited for the concatenation method ↗

Sections 4.1–4.4 and Example 6Correct

Boundary structure is negligible and row balance is insufficient

Pages 5–11 · Sections 4.1–4.4 and Example 6 · arXiv:2604.17136v2

Any fixed number of leading or trailing digits contributes $O(m)$ positions among $D(m)\asymp m^2$ , so the Benford and Pisano boundary layers have zero limiting density. The exact digit-extraction function at depth $j$ has variation $\asymp10^j$ , and the universal lower bound $D_M^*\geq1/(2M)$ makes the Koksma–Hlawka estimate vacuous once $j-\log_{10}m\to\infty$ ; hence that particular discrepancy route cannot reach the typical depths $j\asymp m$ . Example 6 is also valid: every row is exactly period- $10$ balanced, while constant columns give limiting mass $1/10$ to each diagonal two-block and zero to every off-diagonal two-block.

Full paper, version 2 ↗

02Proofs3 reported findingsCorrect

The proofs of the exact count, the sufficient normality implication, the boundary-density estimates, the discrepancy-method limitation, and the row-versus-column counterexample are correct and complete. The large numerical tables are not used as proof inputs and are expressly qualified as finite heuristic diagnostics.

Proposition 1 and Proposition 4Correct and complete

Counting and length asymptotics

Pages 2–5 · Propositions 1 and 4 · arXiv:2604.17136v2

The strict endpoint in the Binet calculation is handled explicitly, and summing $|\sigma_{10}(F_n)|=n\log_{10}\phi+O(1)$ verifies both length hypotheses with the correct orders $m^2$ and $m$ .

Section 4.3Correct and complete

Fixed-depth digit extraction and the discrepancy horizon

Pages 6–8 · Equations (8)–(9) and Section 4.3 · arXiv:2604.17136v2

The identity $F_n=10^{L_n-1+\{\log_{10}F_n\}}$ gives the displayed digit extractor. It is an indicator of $9\cdot10^{j-2}$ intervals and therefore has total variation $18\cdot10^{j-2}+O(1)$ . The exponentially small Binet perturbation changes star discrepancy only by its displacement plus the negligible initial proportion. The lower discrepancy bound alone then proves the claimed logarithmic horizon for this method.

Example 6 and positional countingCorrect and complete

The ragged-array counterexample and vanishing boundary proportion

Pages 9–11 and 14–15 · Example 6, equations (11)–(17), and Section 5.5 · arXiv:2604.17136v2

Weighted sums over each residue class modulo $10$ equal one tenth of $\sum_{n\leq N}n$ up to $O(N)$ , proving simple normality of the constructed concatenation. Internal equal-symbol pairs have limiting frequency $1/10$ , whereas unequal pairs occur only at $O(N)$ boundaries against total length $\asymp N^2$ . The same elementary count gives $O(kN)/D(N)=O(k/N)$ for the leading, trailing, and boundary categories of the Fibonacci concatenation.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

Detailed audit reportFull reasoning, manuscript locations, and references.

Open report PDF ↗

Author response

Challenge an audit finding

Local workflow preview

A listed author may submit formal evidence that an audit is inaccurate. The response would be considered in a fresh AI re-evaluation; it would not edit the audit automatically.

Paper: arXiv:2604.17136v2
Authors listed: José Ricardo G. Mendonça
Audit date: August 15, 2026

01Establish identityMatch an authenticated scholarly identity to this paper.
02Submit evidenceIdentify the finding and give a formal mathematical response.
03Re-evaluateA separate agent checks the response and records a disposition.

Email fallback for papers without a linked ORCID

A production fallback could send a one-time link only when the submitted address matches an independently maintained author-contact allowlist for this paper. MathAudit must return the same message for every address so the form cannot reveal which contacts are on that list.

Structured response preview

This form remains unavailable until production identity verification succeeds. Nothing entered here is submitted.

Audit finding challengedWhy the evaluation is inaccurateFormal evidence or repairVerifiable sources, if anyI am submitting evidence, not instructions for the reviewing agent.

This panel never establishes authorship in the local prototype. A production result should be described narrowly as an authenticated ORCID match or control of a separately allowlisted author-contact mailbox.

Audit controls

Status: Report ready
Source: arXiv v2
Prompt: Standard prompt · locked

The report is available below and in the public summary.

Review prompt

You are a mathematical auditor. Evaluate only the correctness of the paper's central statements, the correctness and completeness of its proofs, and evidence-backed non-novelty. Do not make editorial comments or evaluate importance, exposition, style, or suitability for publication. Keep the report neutral, precise, and professional. PART 1 — CORRECTNESS OF THE MAIN STATEMENTS Identify the paper's central statements and assess each one separately using exactly one of these outcomes: - Correct: the statement has a correct, verifiable proof; or, despite a defect in the printed proof, you can give a fully verified repair or independent proof. - Incorrect: you can give strong formal evidence that the statement as written is false, preferably a counterexample or a complete contradiction. - Not able to verify: the supplied proof does not establish the statement, and you can neither supply a fully verified proof nor disprove it. For every adverse or unverified outcome, restate the exact claim and hypotheses, identify its location in the reviewed version, and give the decisive reasoning. Do not infer that a theorem is false merely because its printed proof is defective. Assign the overall statements status by the following priority: 1. Contains wrong statements, if at least one central statement is Incorrect. 2. Contains unsupported statements, if no central statement is Incorrect but at least one is Not able to verify. 3. Correct, if every central statement is Correct, apart from any typographical or minor formal findings allowed by Part 4. PART 2 — CORRECTNESS AND COMPLETENESS OF THE PROOFS Audit the proofs of the central statements and every material internal or cited input on which they depend. For each proof or proof segment, use the narrowest supported outcome: - Correct and complete: every substantive step is verifiable and no case or crucial nontrivial step is missing. - Incorrect as written: a displayed calculation, inference, lemma, or cited application is formally false or inapplicable. - Incomplete as written: a necessary substantive step or case is missing, even though no explicit contradiction has been proved. - Not able to verify: the available paper and sources are insufficient to decide whether the step is valid. Do not flag omitted routine details, compressed arguments, or “analogously” steps when the omitted reasoning is recoverable and correct. For every defect, identify its smallest exact location, explain its downstream dependency, and classify any proposed repair as one of: - Verified repair: supply enough reasoning to verify that the repair works throughout every affected use. - Plausible repair only: describe the possible repair but do not treat it as established. - No repair supplied. Assign the overall proofs status by the following priority: 1. Contains incorrect or incomplete proofs, if at least one proof is Incorrect as written or Incomplete as written. 2. Contains unverified proofs, if there is no proved incorrectness or incompleteness but at least one material proof is Not able to verify. 3. Correct, if all audited proofs are Correct and complete, apart from any typographical or minor formal findings allowed by Part 4. PART 3 — NOVELTY OF THE MAIN STATEMENTS This is a one-sided, evidence-only check. Evaluate only central results. Report a result only when a verifiable primary source, publicly available before the earliest public version of the paper that contains the result, proves the same statement up to a change of notation, or when the result follows directly and immediately from such an earlier theorem. For each non-novelty finding, provide the earlier source, its public date, the exact theorem or proposition, and an explicit hypothesis-and-conclusion mapping. A later source cannot establish non-novelty. Do not flag a result merely because an easier proof exists or because it follows through a nontrivial argument. Do not flag a result that the paper itself explicitly presents as known, a survey item, or an alternative proof. Mention only statements that meet this non-novelty threshold. Say nothing about statements for which no such evidence was found: do not certify novelty, list unsuccessful searches, or add statement-level “no prior source found” entries. If the list is empty, the section may say only “No non-novelty findings.” PART 4 — TECHNICAL AND EVIDENTIARY RULES 1. Criticism must never be overstated. Call a statement incorrect, a proof incorrect or incomplete, or a result non-new only when strong formal evidence supports that exact judgment. Otherwise use Not able to verify where that outcome exists, or remain silent in the novelty section. 2. Review a specified paper version. Give exact page, theorem, equation, or proof locations; restate relevant quantifiers and hypotheses; and check domains, endpoints, parameter ranges, constants, and dependencies explicitly. Apply a defect only to the regimes and downstream results that actually depend on it. 3. Immediate-consequence closure: when deciding whether a conclusion is supported, treat the argument as including every immediate formal consequence of facts established earlier in the paper and available at that point. An immediate formal consequence is a local, mechanically checkable inference using only definitions already in force, elementary logic, direct substitution or renaming, weakening of an inequality or set inclusion, or a monotonicity or symmetry property explicit in a definition or already proved. 4. Such an inference must introduce no new lemma, estimate, construction, case analysis, limiting or compactness argument, external result, or new issue of uniformity or parameter dependence. Check the direction of monotonicity and the dependence of every threshold or constant explicitly. Do not exchange pointwise and uniform conclusions, change exceptional sets, interchange quantifiers or limits, or extend an endpoint under this rule unless the required fact was already established. 5. If the claimed conclusion follows after immediate-consequence closure, it is fully supported. Do not report a gap, unsupported statement, minor formal correction, or recommendation to add an explanatory sentence, and do not mention the suppressed item in the report. This suppression does not apply to a genuine typographical error as defined below, because a typo is a literal notation defect that still requires a textual correction. For example, if

P(delta)

is proved for every sufficiently small

delta>0

and

P

is immediately monotone in the required direction, then the claim that

P(delta)

holds for every

delta>0

requires no correction; the omitted monotonicity step is exposition rather than an audit finding. 6. Immediate-consequence closure permits reconstruction of an omitted inference; it does not permit silent alteration of the printed domain, hypotheses, or quantifiers of an intermediate statement later invoked outside its stated scope. If formal validity requires changing the text of that statement or its stated range, report the appropriate finding even when the repair is easy. Thus a result stated only for

t>0

and subsequently applied at

t=0

remains a formal scope mismatch unless the endpoint case was separately established before the application. 7. Apply this mandatory gate before recording any adverse, unverified, typographical, or minor formal finding: (a) identify the exact assertion or inference alleged to be defective; (b) reconstruct the strongest conclusion following from the preceding argument after all admissible immediate formal consequences are added; (c) identify the specific defect that remains — a formally demonstrated falsehood, a concrete nontrivial unresolved proof obligation, or a genuine domain, hypothesis, quantifier, or notation mismatch requiring a textual correction; and (d) state the evidence and minimal repair. If no specific defect remains, or if the only possible addition would be explanatory, suppress the finding entirely. Use Not able to verify only when a precise, nontrivial unresolved obligation can be identified. 8. A typographical error is an incorrect symbol, index, exponent, sign, variable name, or notation mismatch whose unique intended correction is mechanically determined from the surrounding text and does not alter the mathematical argument. Report it in yellow as Typo, with the printed text, correction, exact location, and reason it is harmless. It does not lower an overall status. 9. A minor formal correction is a boundary, endpoint, or displayed parameter-range defect that can be repaired consistently in every affected statement by a local change, without a new argument, without invalidating any related result or application, and without creating another substantive defect. Report it in yellow as Minor formal correction, with the same details. It does not lower an overall status. 10. Do not use either minor category if more than one materially different repair is possible; if the repair changes the substance or scope of a central result; if an excluded case is used later; if the correction fails to propagate consistently; or if a new mathematical idea is required. In such cases use the strongest substantive classification actually supported by evidence. 11. Write every mathematical expression as valid, compilable LaTeX. Use

...

for inline mathematics and

...

for displayed mathematics, with balanced delimiters and braces. Do not leave raw LaTeX commands outside math delimiters. 12. Separate theorem truth from the correctness of the printed proof. A theorem can be Correct in Part 1 because of a fully verified independent proof while its printed proof is adverse in Part 2. Conversely, a defective proof alone never licenses an Incorrect theorem classification. Organize the report under Parts 1–3 and apply Part 4 throughout. Do not add editorial commentary, scores, rankings, recommendations, or a list of suppressed observations.

This panel is visibly marked as owner-only for the concept prototype. Authentication is not enabled locally.

On the normality of the concatenated Fibonacci constant

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Exact counting function for Fibonacci numbers

Almost-all internal digit normality is a sufficient route

Boundary structure is negligible and row balance is insufficient

Counting and length asymptotics

Fixed-depth digit extraction and the discrepancy horizon

The ragged-array counterexample and vanishing boundary proportion

Challenge an audit finding

Authenticate with ORCID, then require an exact arXiv match