{"id":"32aae2cd-931b-4e24-86cb-8992b9f96564","arxiv_id":"2607.03439","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Finite integral limits turn a non-Borel-summable double-well series into an absolutely convergent series that recovers the exact answer after L\to∞, including when expanded about a single minimum.","lead":"A simple double-well integral that produces a non-Borel-summable asymptotic series becomes an absolutely convergent power series when the integral is cut off at finite limits ±L. Summing that series and then sending L to infinity recovers the exact closed form, even when the expansion is performed around only one of the two minima.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The mathematics that supports the strongest claim is transparent, free of fitting parameters, and recovers an independently known closed form. The only soft spot is the motivational leap from this one-dimensional integral to realistic non-Borel systems; that leap is already identified by the reader and correctly keeps the verdict CONDITIONAL rather than ACCEPT. No further internal inconsistency or calculational error is present, so the reader’s assessment stands.","tokens_in":12178,"tokens_out":372,"duration_ms":3484,"concrete_test":"Independently recompute the first four coefficients of Iseries(L) in (17) and (33) by direct numerical quadrature of the finite-L integrals for a fixed pair (a=1, L=2) and several λ; confirm that they match the analytic incomplete-gamma / hypergeometric expressions to machine precision and that the partial sums approach Ianalytic as L is increased.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper’s strongest claim is fully internal and self-contained: for the concrete integral (1), both finite-L expansions (about x=0 and about one minimum) produce absolutely convergent power series in λ whose L\to∞ limits recover the known closed form (2). The derivations (incomplete-gamma representations, ratio-test limits (20) and (35), and the subsequent summations that yield the modified-Bessel expression) are elementary and can be verified by hand; no hidden assumption or algebraic gap appears. The reader’s weakest_assumption correctly flags that the toy model may not capture every mechanism of non-Borel summability in QM or QCD, but that is a question of future scope, not a flaw in the claim that is actually proved.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies the elementary integral I=∫_{-∞}^∞ exp((a/2)x^{2}-λx^{4})dx (a>0, λ>0), whose integrand has a double-well shape. The standard expansion about one minimum produces a non-Borel-summable asymptotic series in λ (singularity of the Borel transform on the positive real axis). Under finite limits ±L the author constructs two power series in λ—one by expanding about the local maximum at x=0 and one by expanding about a single minimum—proves both are absolutely convergent for every λ>0 by the ratio test, sums them in closed form, and shows that lim_{L\to∞} recovers exactly the known analytic expression involving modified Bessel functions I_{±1/4}. The result is presented as a first illustration that finite path-integral limits remain effective when the series is not Borel summable.","tokens_in":12322,"tokens_out":884,"duration_ms":18141,"significance":"If the explicit constructions hold (they do), the work supplies a clean, fully checkable demonstration that absolute convergence and exact recovery of the non-perturbative answer are possible even for a non-Borel-summable series generated by a degenerate vacuum. Strengths include the complete algebraic transparency of every step (binomial expansions, incomplete-gamma and hypergeometric representations, ratio-test limits (20) and (35), term-by-term summation, and the final L\to∞ limits that match the independent closed form (2)), the fact that the one-minimum expansion already captures both wells, and the absence of any fitted parameters. The paper correctly frames itself as a toy-model first step toward the QM double-well energy series and IR renormalons; that limited scope does not diminish the internal result.","major_comments":[],"minor_comments":[{"comment":"Several typographical errors should be corrected before publication: “Aknowledgments” (p. 13), “expnasions” in the title of Ref. [8], “the the” (p. 12, line 3 of the penultimate paragraph), and the inconsistent capitalization of “Series” versus “series” in the notation I_series(L).","section":null},{"comment":"In Sec. 3.2 the finite-L integral is written over y∈[-L,L] after the shift x=x_{+}+y. While the subsequent L\to∞ limit is exact, a short remark clarifying that the finite-L domain is not symmetric about the original origin (and therefore does not yet “see” both wells equally for moderate L) would help readers who wish to evaluate the truncated series numerically.","section":null},{"comment":"The incomplete-gamma and _{2}F_{2} representations (18), (34) are correct but become cumbersome for practical high-order evaluation. A brief note on the numerical stability of these special functions for large L and large order would improve usability.","section":null},{"comment":"References [8] and [9] are the author’s own preceding works; a sentence or two situating the present calculation more explicitly against the Borel-summable anharmonic-oscillator case treated in [9] would sharpen the logical progression for the reader.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is a short, self-contained note that correctly solves a well-chosen toy problem. Its main value is pedagogical and conceptual rather than a breakthrough for realistic systems; that is acceptable for a hep-th letter-style paper, but the editor may wish to confirm that the journal’s current standards for toy-model illustrations are met. No originality or citation concerns beyond the usual self-citation pattern of a continuing series."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The punchline is simple: for the elementary integral of exp(a x^{2}/2 − \\lambda x^{4}) with a>0, \tau>0, the usual expansion about one minimum produces a non-Borel asymptotic series, but the same expansion (and also the expansion about the local max at zero) under finite limits \\pm L yields absolutely convergent power series in \tau whose L\to∞ limits both recover the known closed form in modified Bessel functions. That is new relative to the author’s earlier finite-limit papers, which treated Borel-summable cases.\n\nWhat the paper does well is write every algebraic step out: binomial expansion, incomplete-gamma representation of the truncated moments, ratio-test proofs of absolute convergence for any finite L and any \tau>0, term-by-term summation via the series for the incomplete gamma or the hypergeometric, and the final L\to∞ limit that matches the textbook expression. No parameters are fitted; the target closed form is independent. The observation that an expansion about only one of the two minima still captures the full non-perturbative answer is the most interesting conceptual point.\n\nThe soft spot is exactly the one the author flags: this is still a one-dimensional integral. Whether the same finite-limit trick will tame the energy series of the quantum double well or the IR renormalons of QCD is left as future work. That does not undercut the claim that is actually proved, but it does keep the immediate significance modest.\n\nThe math and citation pattern look solid; the paper is honest about its scope. It is for people who already care about Borel summability, large-order perturbation theory, and non-perturbative vacuum structure. A serious referee should see it. I would accept it for peer review.","headline":"Clean, fully explicit toy-model proof that finite cutoffs convert a non-Borel double-well integral into absolutely convergent series that recover the exact answer, even from a single-minimum expansion.","tokens_in":12909,"tokens_out":462,"would_cite":false,"duration_ms":8284,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Finite integral limits turn a non-Borel-summable double-well series into an absolutely convergent one that recovers the exact answer, even when expanded about only one minimum.","keywords":["finite path integral limits","non-Borel summable series","double-well potential","perturbative expansion","strong coupling","asymptotic series","vacuum structure","modified Bessel functions"],"falsifier":"Sum the finite-L series numerically to high order for a large but finite L and a chosen strong coupling, then verify that the result approaches the known closed-form Bessel expression as L is increased further; any residual systematic deviation that survives the L\to∞ limit would falsify the claim.","tokens_in":13074,"feed_emoji":"∞","tokens_out":911,"duration_ms":18796,"temperature":0.7,"pith_summary":"Ordinary power-series expansions of a basic integral whose exponent has a double-well shape produce an asymptotic series that cannot be Borel-resummed, because the two minima force every term to share the same sign. Replacing the infinite integration limits by finite cutoffs ±L before the expansion is performed yields an absolutely convergent series for every positive coupling, whether one expands about the local maximum at the origin or about a single minimum. After the series is summed analytically, the infinite-L limit exactly reproduces the known closed-form expression involving modified Bessel functions. A reader cares because the same finite-limit device already worked for the Borel-summable anharmonic oscillator; success on this non-Borel toy model suggests a practical route to strong-coupling results in systems whose vacuum structure normally blocks Borel resummation, without multi-instanton or other non-perturbative machinery.","feed_headline":"Finite cutoffs turn non-Borel series into exact answers","feed_subtitle":"Even expansion about one double-well minimum recovers the full closed form once the cutoff is removed.","key_machinery":"Finite path-integral cutoffs ±L: the replacement of infinite integration limits by a finite L before the power series in the coupling is formed, followed by exact summation and only then the limit L\to∞.","core_discovery":"Under finite integral limits from −L to L the two perturbative expansions (about the origin and about one minimum) of the double-well integral are absolutely convergent for every positive coupling. Their sums, once the limit L\to∞ is taken, equal the exact analytic value of the original integral, so an expansion performed about a single minimum already encodes the contribution of both wells.","pith_inferences":["The same finite-cutoff device may supply a practical way to resolve infrared-renormalon ambiguities in QCD without explicitly introducing a condensate vacuum.","Energy eigenvalues of the quantum double-well potential could be extracted from a convergent finite-L series rather than multi-instanton calculus.","Absolute convergence for every coupling suggests that the radius of convergence is controlled only by the cutoff size, independent of vacuum topology.","The method could be checked on other non-Borel systems whose exact answers are known, such as certain sine-Gordon observables."],"forward_implications":["Absolutely convergent strong-coupling series exist for non-Borel-summable models with degenerate vacua once the integration limits are kept finite.","A perturbative expansion performed about only one vacuum can still reproduce the full non-perturbative content of a double well after the cutoff is removed.","The finite-limit procedure previously applied to the Borel-summable anharmonic oscillator extends without change to a non-Borel case.","Analytic summation of the finite-L series followed by L\to∞ yields the exact closed form without any Borel transform or resummation step."],"fun_headline_variants":["Finite limits turn non-Borel series into exact answers","One-min expansion recovers full double-well value via finite L","Finite path limits make non-Borel series absolutely convergent","Expansion about single min captures both wells under finite cutoffs","Finite cutoffs convert non-Borel series to exact integral results"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The elementary one-dimensional double-well integral already contains the essential mechanism that makes realistic systems with non-trivial vacua produce non-Borel-summable series, so success on this toy model is informative for those harder cases.","fun_headline_variants_meta":{"raw":{"variants":["Finite limits turn non-Borel series into exact answers","One-min expansion recovers full double-well value via finite L","Finite path limits make non-Borel series absolutely convergent","Expansion about single min captures both wells under finite cutoffs","Finite cutoffs convert non-Borel series to exact integral results"]},"model":"grok-4.5","effort":"low","cost_usd":0.010312,"raw_usage":{"total_tokens":2366,"prompt_tokens":875,"num_sources_used":0,"completion_tokens":87,"cost_in_usd_ticks":103120000,"prompt_tokens_details":{"text_tokens":875,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1404,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":875,"tokens_out":87,"duration_ms":9930,"temperature":1.0,"reasoning_tokens":1404,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T02:26:50.731636+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Sum the finite-L series numerically to high order for a large but finite L and a chosen strong coupling, then verify that the result approaches the known closed-form Bessel expression as L is increased further; any residual systematic deviation that survives the L\to∞ limit would falsify the claim.","supporting_citations":[],"review_version":1}