{"id":"76fca936-32fb-4b03-baea-7c4dba105c97","arxiv_id":"2502.07935","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"PrecisionLauricella is a Mathematica package that computes epsilon-expansions of Lauricella F_A, F_B, and F_D functions for n up to 3 using Frobenius-series analytic continuation.","lead":"A Mathematica package computes high-precision series expansions of Lauricella hypergeometric functions whose parameters depend on a small parameter epsilon. It uses one-variable Frobenius series and analytic continuation in the complex plane instead of multidimensional sums. A generalist reader might care because such functions appear in Feynman diagram calculations, and a fast accurate tool could speed up multi-loop computations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Accuracy claim is unsupported: no numerical validation or error-bound derivation appears in the paper, leaving the package's central promise of high-precision epsilon-expansions unsubstantiated.","rationale":"The reader's verdict of CONDITIONAL is based on the same fundamental weakness: the analytic-continuation engine is assumed correct but is neither proved nor tested in this paper. My stress-test pass confirms that this is the most load-bearing assumption behind the central claim. I considered whether a more specific technical objection could be raised, such as the geometric reachability of the path condition Eq. (13) with non-negative real kappa_i, but further analysis shows that with kappa_i allowed to be zero, axis-aligned polygonal segments can reach any complex target point, so that particular concern does not land. The missing numerical validation, however, is decisive: the paper's own error estimate is unverified, no comparison data are shown, and the correctness hinges on nontrivial analytic continuation with logarithmic terms and branch cuts. A concrete benchmark against independent high-precision methods would settle the issue. Since the reader already identified this weakness and assigned CONDITIONAL, my analysis does not change the verdict; it reinforces it. The package may well be correct and useful, but the paper as written does not provide the evidence needed to accept the accuracy claim unconditionally.","tokens_in":12076,"tokens_out":13298,"duration_ms":128047,"concrete_test":"Run NExpandHypergeometry on a benchmark set covering F_A, F_B, and F_D for n<=3, including points outside the original convergence regions and with complex or negative arguments, at requested precisions of 20, 100, and 200 digits. Compare each output against an independent high-precision evaluation: for arguments inside the convergence domain, use direct numerical summation or numerical integration of the Euler integral; for outside-domain points, use a high-precision numerical solution of the Pfaffian system via an adaptive ODE integrator (e.g., Mathematica NDSolve) along the same path, or an independent Mellin-Barnes implementation. Report the number of agreeing digits for each case; the accuracy claim is supported only if the agreement reaches the requested precision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that PrecisionLauricella delivers high-precision Laurent expansions about epsilon=0 for arbitrary nonsingular argument values (Section 1). For this to hold, the Frobenius-based analytic-continuation engine must correctly compute the true analytic continuation along the chosen path, the matching of fundamental solution matrices across overlapping regions must be accurate, branch cuts and logarithmic terms must be handled correctly, and the epsilon-reconstruction step must recover the true Laurent coefficients. None of these are demonstrated in the paper. Section 5 asserts that the package was validated against existing tools, but no numerical tables, comparison figures, or error measurements are provided. The error estimate in Eq. (17) is stated without derivation or numerical verification, and the Frobenius truncation error Delta_Frob is not defined or quantified. The timing figures (Figs. 5 and 6) demonstrate only runtime scaling, not accuracy. Since the algorithm's correctness depends on several nontrivial analytic and numerical steps, and the companion paper [40] is not included, the reader cannot verify that the implementation actually computes the claimed values. If any step fails silently, the outputs would be incorrect while appearing plausible, making the missing validation the most load-bearing weakness.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces PrecisionLauricella, a Wolfram Mathematica package for high-precision numerical evaluation of the Lauricella functions F_A, F_B, and F_D with n ≤ 3 and parameters depending linearly on a small parameter ε. The package computes Laurent expansions in ε about ε = 0 at user-specified argument values and precision. The method relies on analytic continuation via Frobenius generalized power series solutions to the Pfaffian systems satisfied by these functions, along piecewise-linear paths in the argument space, followed by reconstruction of the ε-dependence from evaluations on a lattice of ε values. The manuscript describes the algorithm and its flowchart, the package interface and options, and reports timing measurements for several examples. The central claim is that the package provides an accurate and efficient alternative to multidimensional series or Mellin–Barnes representations for these ε-expansions.","tokens_in":12293,"tokens_out":6107,"duration_ms":53648,"significance":"If the implementation is correct, the package would be a useful tool for physics applications, particularly Feynman integral computations, where high-precision ε-expansions of Lauricella functions are frequently needed. The use of one-dimensional Frobenius series rather than multidimensional sums, and the ε-lattice reconstruction with parallelization, are attractive and potentially more efficient than existing approaches. The public repository and supplementary notebook are strengths that support reproducibility. However, the paper does not present the numerical validation necessary to support the claimed accuracy, so the significance is conditional on the authors supplying the missing evidence in a revision.","major_comments":[{"comment":"The paper asserts: \"We validated the accuracy and efficiency of PrecisionLauricella by comparing its results, where applicable, with existing tools ... These comparisons demonstrated the robustness and reliability of the presented approach.\" However, no numerical comparison, error table, or convergence test appears anywhere in the manuscript. Figures 5 and 6 show only runtime scaling. Without concrete validation data, the reader cannot verify the central claim of high-precision evaluation. I request that the authors include a dedicated validation section with tables comparing against known analytic values (e.g., cases reducible to 2F1 or polylogarithms), against direct series summation inside the convergence region, and against independent numerical tools such as those cited in Refs. [48, 62]. Each comparison should report the requested precision, the achieved error, and the runtime.","section":"Section 5, first paragraph"},{"comment":"The error estimate max(h^{2n−2⌊k/2⌋}, Δ_Frob) is stated without derivation, and Δ_Frob is neither defined nor quantified. Since the options FrobeniusNumberTerms=\"Auto\" and InternalPrecision=\"Auto\" presumably rely on this estimate to achieve the requested accuracy, the paper must explain how Δ_Frob is estimated or bounded, and should numerically verify the formula by comparing predicted versus actual errors on test cases. As written, the claimed accuracy control is not testable.","section":"Section 3, Eq. (17)"},{"comment":"The condition that all κ_i in Eq. (13) be non-negative real numbers is stated to be necessary and the justification is deferred to Ref. [40]; the claim that the search \"never more than four times\" is likewise unsupported in this paper. The existence of such a path is load-bearing for the analytic continuation engine and hence for the package's promise of evaluating at arbitrary argument values. Please either summarize the argument for path existence, state the precise conditions under which it holds, and describe the failure mode if no path is found; or qualify the \"arbitrary argument values\" claim to the class of points for which such a path is guaranteed.","section":"Section 3, path selection and Eq. (13)"},{"comment":"The automatic determination of the number of Frobenius series terms and of internal precision is not documented. The text says \"Auto\" determines these \"based on the desired accuracy,\" but no criterion is given. Without a definition of the truncation error Δ_Frob or an empirical convergence test, there is no evidence that the default settings actually deliver the requested accuracy. Please document the automatic truncation strategy and provide a test showing that the achieved error tracks the requested precision across a range of parameter values and expansion orders.","section":"Section 3 (Frobenius truncation) and Section 4 (Auto options)"}],"minor_comments":[{"comment":"\"staring point\" should be \"starting point\" in all three captions.","section":"Captions of Figs. 2, 3, 4"},{"comment":"The notation \"2 n\" is ambiguous; it should be clarified whether this means 2n (two times n) and all variables (h, n, k) should be defined in one place near the equation.","section":"Section 3, Eq. (17)"},{"comment":"The phrase \"arbitrary argument values\" is too strong given the restriction n ≤ 3 and the dependence on path existence. It would be more precise to state \"for a wide class of nonsingular argument values\" and to note explicitly that singular points and possibly other excluded configurations are not covered.","section":"Section 1, Introduction"},{"comment":"The companion paper is cited as [1] in the Program Summary but as [40] in the main text; please harmonize the numbering or add a cross-reference so readers are not confused.","section":"Program Summary and References"},{"comment":"A brief usage example with actual numerical output in the text would help readers; the supplementary notebook is mentioned but not visible in the arXiv listing, so the paper should be self-contained enough to demonstrate the package interface.","section":"Section 4"}],"recommendation":"major_revision","confidential_remarks":"The missing numerical validation is the main obstacle. The paper's own validation list includes Ref. [39] by the same authors, and the companion method paper [40] is also by the same group; an independent cross-check, for example against direct series evaluation or another public code, would substantially increase confidence. I would also suggest that the editor obtain the supplementary notebook and, if possible, the companion paper during the review process, since key mathematical assumptions (path existence, Frobenius truncation error) are deferred to [40]."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I read the PrecisionLauricella paper. The concrete artifact is a Mathematica package that evaluates Lauricella F_A, F_B, F_D for n<=3 with indices linear in epsilon, by continuing the defining series along a path via Frobenius generalized power series and then reconstructing the epsilon-expansion from evaluations on a lattice using Lagrange interpolation. The package is downloadable, GPL, comes with a notebook, and the paper describes the options and the continuation graph clearly. The parallelization approach is sensible: evaluating the epsilon lattice independently is a good use of cores.\n\nWhat is genuinely new is the implementation and the automation, not the mathematics. The Frobenius/Pfaffian method is in the companion paper [40], and this paper is honest about that. The epsilon-reconstruction and the option set (SimpleAnalyticContinuation, DeltaPrescription, parallelization) are new relative to the cited tools.\n\nThe soft spot is exactly what the stress-test says: the accuracy claim is not demonstrated in this paper. Section 5 asserts comparisons with Refs. [62,48,38,39] but gives no numerical table, no error plot, no convergence test. Eq. (17) is an error bound stated without derivation, and Delta_Frob is never defined. The correctness of the whole engine rests on nontrivial steps: the existence of a path with non-negative kappa_i, the matching of fundamental solution matrices, branch-cut handling, and resonance treatment. If any of those fails silently, the package returns plausible but wrong values. The paper currently gives the reader no way to check.\n\nThe 'arbitrary argument values' phrase is overbroad: the implementation is for n<=3, and for exceptional parameter values (resonances) the treatment is described only in the companion paper. That is a minor wording issue.\n\nOn balance, this is a legitimate software paper with a real gap. It deserves a serious referee, not a desk reject. The referee should ask for a validation section with actual numbers, a derivation or at least a precise statement of the error bound, and a definition of Delta_Frob. If the companion paper supplies the missing proofs, this can become a solid contribution.","headline":"A real Mathematica package for epsilon-expansions of Lauricella functions, but the paper's accuracy claim is asserted rather than shown.","tokens_in":12836,"tokens_out":2487,"would_cite":false,"duration_ms":22268,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["33C65","65D20","33C70"],"pacs":[],"model":"deepseek-v4-flash","headline":"PrecisionLauricella, a Mathematica package, computes high-precision Laurent expansions in a parameter ε for Lauricella functions FA, FB, and FD in up to three variables at arbitrary complex argument values, using analytic continuation via…","keywords":["Lauricella functions","epsilon expansion","Frobenius method","analytic continuation","Pfaffian system","high-precision computation","Mathematica package"],"falsifier":"Pick a target point where the package's analytic continuation requires several overlapping regions and compare its output at 200 digits with an independent high-precision evaluation from a Mellin–Barnes representation or a known polylogarithmic expression; any disagreement larger than the paper's stated error bound would refute the claim of arbitrary-argument correctness. A second check targets the path-existence assumption: attempt a point on a branch cut with both DeltaPrescription → +I and → −I and verify the two results are the appropriate branches of the same analytic function.","tokens_in":11885,"feed_emoji":"📐","tokens_out":10385,"duration_ms":80844,"temperature":0.7,"pith_summary":"PrecisionLauricella is a Mathematica package for numerically expanding Lauricella functions $F_A^{(n)}$, $F_B^{(n)}$, and $F_D^{(n)}$ with $n \\leq 3$ in Laurent series in a small parameter $\\varepsilon$, given that all function indices depend linearly on $\\varepsilon$. The paper's central claim is that Frobenius generalized power series, continued analytically along paths through the argument space, produce these expansions to user-selected precision at arbitrary argument values, including points outside the series' convergence domains. This makes the package an alternative to multidimensional series summation and Mellin–Barnes integral representations, which become expensive when many digits are required. If the claim is correct, the package offers a practical tool for high-precision $\\varepsilon$-expansions, with runtime growing linearly in the number of expansion terms.","feed_headline":"PrecisionLauricella computes Lauricella ε-expansions to high precision","feed_subtitle":"One-dimensional Frobenius series replace costly multidimensional sums and Mellin–Barnes integrals.","key_machinery":"The mechanical core is the Pfaffian system, a first-order system $dJ = (M_x dx + M_y dy) J$ for a vector $J$ of $\\theta$-derivatives of the Lauricella function, together with the path ansatz $x_i = \\kappa_i t$, which reduces it to a single-variable system $dJ/dt = M(t)J$. Solutions are written as Frobenius series $U = \\sum_{\\lambda\\in S} t^\\lambda \\sum_{n=0}^\\infty \\sum_{k=0}^{m_\\lambda} c_n(\\lambda,k) t^n \\log^k t$, with exponents determined by eigenvalues of the leading matrix $A_0 = \\lim_{t\\to 0} tM(t)$ and logarithmic terms appearing when eigenvalues differ by integers. The continuation is organized by an intersection graph of circular convergence regions, with branch cuts handled by treating regions on opposite sides as disjoint; the package finds a path through the graph and multiplies the fundamental solution matrices along it. Finally, the $\\varepsilon$-lattice approach with Lagrange interpolation reconstructs the Laurent series coefficients in $\\varepsilon$ from independent evaluations.","core_discovery":"The central discovery is that a single package can automate the entire pipeline: construct a Pfaffian system for the chosen Lauricella function, restrict it to a line $x_i = \\kappa_i t$ through the origin, solve the resulting one-dimensional system by Frobenius generalized power series in overlapping circular regions, and chain those solutions along a path from the origin to the target point. The $\\varepsilon$-dependence is handled by evaluating the whole chain on a lattice of numerical $\\varepsilon$ values and reconstructing the Laurent coefficients by Lagrange interpolation, which is why the computation parallelizes and why the runtime grows linearly with the number of terms. The paper demonstrates the approach on $F_A^{(n)}$, $F_B^{(n)}$, and $F_D^{(n)}$ for $n \\le 3$, reporting accuracy at 20, 100, and 200 digits and comparing where possible with existing tools.","pith_inferences":["If the same Frobenius-path machinery extends to higher $n$ or to other Horn-type hypergeometric families, it could replace Mellin–Barnes integration for a broader class of Feynman-integral master integrals; the package currently stops at $n=3$.","The $\\varepsilon$-lattice reconstruction implies the method's accuracy at fixed $\\varepsilon$-order is limited by the Lagrange interpolation step; a natural test is to compare the reconstructed Laurent coefficients against a direct $\\varepsilon$-expansion of the Pfaffian matrices, which the paper does not report.","The path-existence condition on $\\kappa_i$ is the least tested input; a systematic scan over random target points would show whether 'arbitrary argument values' holds in practice or only for points reachable by the graph-search heuristic.","The reported linear runtime in $\\varepsilon$-order suggests that the Frobenius truncation order, not the number of $\\varepsilon$ terms, will dominate wall-clock time at very high digit counts; users may need to raise InternalPrecision to keep the final digits stable."],"forward_implications":["Users can request Laurent expansions of $F_A$, $F_B$, and $F_D$ for $n\\le 3$ at arbitrary complex argument values to a specified number of digits without implementing multi-dimensional summation or Mellin–Barnes integration themselves.","The cost of adding more terms in the $\\varepsilon$-expansion grows linearly, so deep expansions (for example ten terms or more) remain practical on a laptop, as the paper's benchmark timings show.","Because each $\\varepsilon$ lattice point is evaluated independently, the computation parallelizes across available cores; the paper reports timings with 8 and 16 kernels.","The package's error estimate $\\max\\{h^{2n-2\\lfloor k/2\\rfloor}, \\Delta_{\\mathrm{Frob}}\\}$ gives a user-controllable trade-off between lattice step size, expansion depth, and achievable precision.","Where independent tools exist (Appell F2, Appell F1/F3, Lauricella FD(3)), the paper reports agreement, so users can reuse the package as a cross-check for other hypergeometric computations."],"supporting_citations":[{"why":"Supplies the mathematical framework of analytic continuation via Frobenius generalized power series that the package implements, including the requirement that continuation paths have non-negative real κ_i.","marker":"[40]"},{"why":"Provides the pre-derived Pfaffian systems for the implemented Lauricella functions that the package solves along the continuation path.","marker":"[39]"},{"why":"Used as an accuracy baseline in the paper's validation of Appell F2 evaluations.","marker":"[62]"},{"why":"Another comparison tool for analytic continuations and numerical evaluations of Appell F1, F3 and Lauricella FD(3) in the validation section.","marker":"[48]"},{"why":"A Mathematica package for expanding multivariate hypergeometric functions in multiple polylogarithms, used as a further comparison in the conclusion.","marker":"[38]"}],"fun_headline_variants":["PrecisionLauricella: high-precision Lauricella with ε parameter","Frobenius series method powers PrecisionLauricella package","Mathematica package for ε-dependent Lauricella functions","One-dimensional path to high-precision Lauricella evaluations","New package simplifies Lauricella ε-expansions to high precision"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method assumes that the one-dimensional Frobenius-series solutions really continue the Lauricella function to the requested target point, and specifically that a continuation path with all direction coefficients $\\kappa_i$ real and non-negative exists; the paper takes this existence condition from its companion work and does not prove or test it.","fun_headline_variants_meta":{"raw":{"variants":["PrecisionLauricella: high-precision Lauricella with ε parameter","Frobenius series method powers PrecisionLauricella package","Mathematica package for ε-dependent Lauricella functions","One-dimensional path to high-precision Lauricella evaluations","New package simplifies Lauricella ε-expansions to high precision"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000603,"raw_usage":{"total_tokens":2778,"prompt_tokens":873,"completion_tokens":1905,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":489,"completion_tokens_details":{"reasoning_tokens":1816}},"tokens_in":489,"tokens_out":1905,"duration_ms":13214,"temperature":1.0,"reasoning_tokens":1816,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T11:22:17.517011+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Pick a target point where the package's analytic continuation requires several overlapping regions and compare its output at 200 digits with an independent high-precision evaluation from a Mellin–Barnes representation or a known polylogarithmic expression; any disagreement larger than the paper's stated error bound would refute the claim of arbitrary-argument correctness. A second check targets the path-existence assumption: attempt a point on a branch cut with both DeltaPrescription → +I and → −I and verify the two results are the appropriate branches of the same analytic function.","supporting_citations":[],"review_version":1}