arXiv:2606.17303v1

Tail Criteria, No-Go Audits, and Apéry-Type Certificate Obstructions for the Irrationality of e+π

Runlong Yu

math.GM11J8111J7211Y60

Paper metadata

Original submission: June 15, 2026
Claimed MSC: 11J81, 11J72, 11Y60
arXiv categories: math.GM
Paper license: CC BY 4.0 ↗

Abstract

The irrationality of e+pi remains open, despite the separate transcendence of e and pi. This paper studies the problem from the viewpoint of finite irrationality certificates and gives a bounded no-go audit for low-complexity Apéry-type proof mechanisms. First, we prove exact equivalences between the hypothesis e+pi in Q and eventual factorial-arithmetic phenomena: a ceiling recurrence, a factorial-Cantor digit condition, and a divisibility criterion. These criteria identify what rationality would force, while showing why tail conditions are not finite obstructions. Second, we formulate an Apéry-type certificate framework based on integer linear forms L_n = A_n(e+pi)+B_n with A_n,B_n in Z, L_n nonzero, and |L_n| tending to zero. A mixed integration-by-parts identity produces such forms from integer polynomials. We then audit several low-complexity constructions, including mixed Padé approximation, crossed separate approximations to e and pi, simple J-fractions, holonomic ansatzes, Rodrigues-type families, and an integer kernel-lattice search. The main contribution is a rigid boundary probe: no-go filters marking a tested zone where analytic smallness is destroyed by denominator clearing, coefficient growth, primitive reduction, or continued-fraction shadows. In the final kernel-lattice audit, 145 raw candidates reduce to 133 primitive records; the best signals are dominated by continued-fraction shadows, while non-CF candidates do not form a degree-continuing family. Thus, within the tested low-complexity families, no non-circular Apéry-type mechanism for e+pi is found.

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Audit summary

Audited against arXiv v1

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 findingsContains unsupported statements

The exact ceiling-recurrence, factorial-digit, divisibility, certificate, and mixed-kernel statements are correct. The paper's other central contribution is a bounded computational no-go audit, but the information supplied does not permit independent verification of its reported search outcome. No mathematical theorem was found to be false.

Theorems 3.1, 3.3, and 3.4Correct

The three factorial-tail criteria are exact

Pages 4–7 · Theorems 3.1, 3.3, and 3.4 · arXiv:2606.17303v1

The recurrence $\rho_{N+1}=(N+1)\rho_N-1$ for the factorial tail of $e$ makes eventual rationality equivalent to the ceiling recurrence for $A_N=\lceil N!\pi\rceil$ . The factorial-Cantor digit identity $d_N(\pi)=\lfloor N!\pi\rfloor-N\lfloor(N-1)!\pi\rfloor$ converts this recurrence exactly into $d_N(\pi)=N-2$ . Finally, $Q_N-NQ_{N-1}=d_N(\pi)-(N-2)$ lies in $[2-N,1]$ , so eventual divisibility by $N$ forces it to vanish. Both directions of all three equivalences are thereby established.

Full paper, version 1 ↗

Lemma 2.2 and Proposition 4.2Correct

The certificate criterion and mixed-kernel identity are valid

Pages 3 and 7–8 · Lemma 2.2 and Proposition 4.2 · arXiv:2606.17303v1

If $e+\pi=a/b$ and $L_n=A_n(e+\pi)+B_n$ is a nonzero integer linear form, then $bL_n$ is a nonzero integer, so $|L_n|\geq1/b$ ; hence a sequence tending to zero proves irrationality. Repeated integration by parts gives $\int_0^1P(x)e^x\,dx=A(P)e-S_P(0)$ , and adding $4A(P)\int_0^1(1+x^2)^{-1}\,dx=A(P)\pi$ yields the displayed integer linear form in $e+\pi$ .

Full paper, version 1 ↗

Computational finding 7.2Not able to verify

The final bounded-search outcome is not independently verifiable

Pages 12–16 · Computational finding 7.2 and Appendices B–C · arXiv:2606.17303v1

The abstract and Computational finding 7.2 report that 145 raw candidates reduce to 133 classified primitive records and that no non-CF degree-continuing family appears. Appendix B gives only a generic protocol: it does not identify the actual structured polynomial bases, degree ranges, sampling grid, lattice construction and objective, reduction parameters, candidate records, or a machine-checkable output certificate. Appendix C also lists 32 convergent shadows, 5 semiconvergent shadows, and 95 non-CF records, which sum to 132 rather than the asserted 133. The available material therefore neither reproduces the bounded search nor determines which count should be corrected. This is an evidentiary limitation, not evidence that the reported conclusion is false.

Full paper, version 1 ↗

02Proofs3 reported findingsContains unverified proofs

All formal theorem and lemma proofs are correct and complete. The computational finding cannot be audited from the generic protocol and summary counts supplied, so the bounded-search component remains unverified rather than disproved.

Sections 2–5Correct and complete

The formal certificate, tail, and kernel arguments are complete

Pages 3–9 · Lemma 2.2, Section 3, Proposition 4.2, and Lemma 5.1 · arXiv:2606.17303v1

The contradiction in the certificate lemma, the bounded-solution argument for the ceiling recurrence, factorial-digit extraction, divisibility reduction, integration-by-parts identity, and denominator-clearing filter all follow exactly from the displayed equations. Endpoint ambiguities in factorial expansions are excluded by the irrationality of $\pi$ , and no missing direction or case was found.

Appendices B–CNot able to verify

The computational audit lacks a reproducible verification record

Pages 15–16 · Appendices B–C · arXiv:2606.17303v1

To verify the finite-search conclusion one must reconstruct the exact search space and check every retained record, its primitive reduction, its continued-fraction classification, and the claimed absence of a degree-continuing family. The paper supplies an outline of these steps but not the instantiated bases, bounds, lattice inputs, outputs, or certificate. The category totals in Appendix C also leave one of the asserted 133 records unaccounted for. A fixed data file and executable specification, together with exact or interval-certified residuals and a complete classification table, would resolve this proof obligation.

Appendix DCorrect and complete

The displayed degree-14 continued-fraction shadow is correctly certified

Page 16 · Appendix D · arXiv:2606.17303v1

The displayed continued-fraction prefix gives consecutive convergents $59759/10198$ and $159664/27247$ , whose mediant is $219423/37445$ . The certified inequalities place $e+\pi$ between $159664/27247$ and that mediant, exactly the corresponding continued-fraction cylinder. Thus $159664/27247$ is the stated convergent, and the strict inequalities also certify that its integer linear form is nonzero. This verifies the showcased candidate, but not the omitted full search record.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2606.17303v1
Authors listed: Runlong Yu
Audit date: August 15, 2026

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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. 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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

...

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...

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Tail Criteria, No-Go Audits, and Apéry-Type Certificate Obstructions for the Irrationality of e+π

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The three factorial-tail criteria are exact

The certificate criterion and mixed-kernel identity are valid

The final bounded-search outcome is not independently verifiable

The formal certificate, tail, and kernel arguments are complete

The computational audit lacks a reproducible verification record

The displayed degree-14 continued-fraction shadow is correctly certified

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