{"id":"1bde1b0f-225b-4c78-bc93-666faeeda8ba","arxiv_id":"2505.13540","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"AmpRed now evaluates complex Feynman integrals at near-real-integral speed by choosing an auxiliary scale whose monomial coefficients have definite sign.","lead":"The AmpRed package is updated with a rule for choosing an auxiliary scale that fixes the analytic branch of complex Feynman integrals, speeding up their evaluation. Benchmarks show large speedups over AMFlow on several three- and four-loop integrals, though no direct comparison with the previous AmpRed is given.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Branch-fixing rule after Eq. (5) is unproved, and the 'always possible' justification covers only the U coefficient, leaving the y^0 coefficient of F_y potentially indefinite.","rationale":"The reader correctly identified the branch-fixing observation as the weakest point. My pass sharpens it: the flaw is not merely 'unproved' but the paper's own sufficiency argument appears incomplete, because a polynomial linear in x_{n+1} has two coefficient functions, and the cited positive-definite U only controls the y^1 term. The y^0 term (the second Symanzik polynomial F) can have mixed signs outside a Euclidean region, and even inside it the paper only says F has 'more positive terms', not that it is sign-definite. Since the entire efficiency gain rests on replacing the general branch-fixing method of ref. [5] sec. 3.4 by this sign-definiteness shortcut, a failure here invalidates the numerical results, not just the speed claim. The concrete test is deliberately a single notebook-level check: it verifies the actual polynomial condition for the benchmark kinematics and probes the branch choice by comparing a violating pair against AMFlow. I do not see evidence of fraud; this is a normal software-note gap. The previous-version benchmark is also missing, but that is secondary to branch correctness. Therefore I keep the CONDITIONAL verdict.","tokens_in":2960,"tokens_out":10320,"duration_ms":118713,"concrete_test":"Run the example notebook 'MasterIntegrals.nb' for one benchmark integral (e.g., Fig. 1b) at the Euclidean point used in Table I; print F_y after choosing i = n+1 and check whether every coefficient of y^a is sign-definite, including the y^0 coefficient F. Then repeat the same integral with an auxiliary pair for which the y^0 coefficient is explicitly not sign-definite, and compare both numerical results with AMFlow at the stated 20-digit precision goal. If the two AmpRed results agree with each other and with AMFlow, the branch rule is robust; if they differ, the missing sign-definiteness is load-bearing and the 'always possible' claim needs qualification.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The new method's correctness hinges on the observation after Eq. (5) that the branch of I_y is fixed by the -i0+ prescription whenever the coefficients of the monomials y^a in F_y are positive (or negative) definite. The paper gives no proof of this observation and, more importantly, the accompanying 'always possible' statement is not established: writing F = F + U x_{n+1}, the choice i = n+1 produces F_y with a y^1 coefficient U x_j (positive definite), but the y^0 coefficient is the second Symanzik polynomial F, which is not guaranteed to be sign-definite. The text's reason—'the coefficient of x_{n+1} ... is always positive definite'—therefore addresses only one of the required coefficients. If the benchmark integrals are evaluated in a Euclidean region where F is indeed positive, this gap may be harmless, but the paper does not state or verify that the condition holds for the examples, and the claim that the choice is 'always possible for loop integrals' is logically unsupported. Should the branch rule fail, the computed integrals would sit on the wrong Riemann sheet and the timing comparisons in Table I would not establish the claimed improvement.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This short paper reports an update to the AmpRed package for evaluating multi-loop Feynman master integrals in the Feynman-parameter representation. The proposed improvement targets the analytic continuation of complex integrals: by choosing the auxiliary scale pair {x_i, x_j} in the insertion E^(0) = x_i/x_j so that the coefficients of the monomials y^a in F_y have definite signs, the author claims that the branch of I_y is fixed by the -i0+ prescription of F. This would remove the need for the slower branch-fixing procedure of the previous version (ref. [5], sec. 3.4) and make complex integrals almost as efficient as real ones. The paper presents benchmarks for four integrals, from two to four loops, comparing AmpRed with AMFlow, and reports that AmpRed is faster for the three- and four-loop examples while AMFlow fails on the Wilson-line integral due to memory constraints. The main methodological claim, however, is stated as an observation without proof, and the paper does not provide a direct comparison with the previous AmpRed version or explicitly report numerical agreement with AMFlow.","tokens_in":3246,"tokens_out":6052,"duration_ms":61974,"significance":"If the branch-fixing observation is correct, the method offers a practically valuable improvement for evaluating master integrals at complex kinematic points, a common need in multi-loop calculations. The paper provides a concrete algorithmic change, a reproducible example notebook, and benchmarks against an independent package, which are all positives. The central claim is falsifiable and can be tested on the provided examples. However, the unproved branch-fixing rule is load-bearing: if it fails, the computed integrals lie on the wrong Riemann sheet and the timing comparisons are meaningless. The absence of a direct comparison with the previous AmpRed implementation and the lack of an explicit statement of numerical agreement with AMFlow further weaken the paper's headline claims.","major_comments":[{"comment":"The branch-fixing rule is stated as an 'observation' without proof or reference. The claim that the branch of I_y is fixed by the -i0+ prescription of F whenever the coefficients of the monomials y^a in F_y are positive (or negative) definite is the load-bearing step of the paper; if the rule fails, the computed integrals sit on the wrong Riemann sheet and the benchmark timings do not establish the method's correctness. Please provide a proof or a rigorous analytic-continuation argument, or give a precise reference to a derivation.","section":"Paragraph after Eq. (5)"},{"comment":"The statement that such a choice of {x_i,x_j} is 'always possible for loop integrals' is not supported by the given justification. Choosing i = n+1 makes the coefficient of the linear monomial x_{n+1} in F equal to the first Symanzik polynomial U, which is positive definite, but after the substitution x_{n+1} = y x_j the y^0 coefficient of F_y is the second Symanzik polynomial F, which is not sign-definite in general. The later comment that the benchmarks are performed in the Euclidean region, where F has more positive terms, does not establish strict sign-definiteness of all monomial coefficients in F_y. Please state the precise condition needed and verify that it holds for each benchmark integral.","section":"Paragraph after Eq. (5)"},{"comment":"The abstract claims that the new implementation 'significantly enhances computational efficiency' compared to the previous version of AmpRed, but Table I compares AmpRed only against AMFlow, not against the previous AmpRed method (e.g., the sec. 3.4 algorithm of ref. [5]). Without a direct timing comparison to the old implementation, the abstract's comparative claim is unsupported. Please add such a comparison or revise the abstract to claim only the efficiency relative to AMFlow.","section":"Abstract and Table I"},{"comment":"The paper does not state whether the numerical values produced by AmpRed and AMFlow agree. Since the method's central purpose is to obtain complex master integrals on the correct branch, the timing comparison is meaningful only if both packages yield the same results to the stated precision goal. Please report the numerical agreement explicitly or provide the computed values, e.g., as supplementary material.","section":"Table I and surrounding text"},{"comment":"The statement that 'the efficiency is thus close to that of real integrals' is not benchmarked. No comparison is shown between AmpRed's run time for a complex kinematic point and its run time for a real kinematic point on the same integral. Either provide such a comparison or qualify the claim so that it refers only to what Table I actually demonstrates.","section":"Text after Table I"}],"minor_comments":[{"comment":"The notation 'Fλ0 y' is ambiguous; it should be written as F_y^{λ0} to clearly denote the power of F_y.","section":"Eq. (5)"},{"comment":"The transformation of the integration measure from dΠ(n+1) to dΠ(n) after integrating out the δ-function is not explained; a short comment would help the reader follow the derivation.","section":"Eq. (4)"},{"comment":"The line 'Precision goal: 20' is a fragment; please state it as a full sentence, e.g., 'All integrals were computed to 20 significant digits' or 'The precision goal was 20 digits for both packages.'","section":"Table I"},{"comment":"Reference [5] omits the publication year and arXiv identifier; please complete the citation.","section":"References"},{"comment":"For reproducibility, the caption or text should state the specific Euclidean kinematic point (e.g., values of Mandelstam variables and masses) used in the benchmarks, rather than referring only to the example notebook.","section":"Fig. 1 caption"}],"recommendation":"major_revision","confidential_remarks":"The central idea is interesting and the benchmark setup is appropriate, but the manuscript is very brief for the weight of the central claim. The branch-fixing observation needs a proof or a rigorous justification, and the abstract's 'compared to the previous version' claim should be backed by a direct comparison. I would be willing to reconsider after a major revision that addresses these points."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a useful, honest software note. It adds a practical rule to AmpRed for choosing the auxiliary scale pair so that the branch of the continued integral is fixed by the i0+ prescription, avoiding the expensive branch-tracking of the previous method. The benchmarks up to four loops, including a Wilson-line integral, show real speedups over AMFlow, and the package ships with example notebooks, so the results are reproducible. That is genuine progress for multi-loop practitioners.\n\nThe new observation is genuinely new: the specific criterion that sign-definite coefficients of y^a in F_y fix the branch does not appear in the earlier papers, and the four-loop data are new. The rule is a modest extension of the authors' previous work, not a conceptual breakthrough, but for a software update that is fine.\n\nThe soft spots are real but minor. The statement that the choice is 'always possible for loop integrals' is not supported as written. Choosing i=n+1 makes the y^1 coefficient of F_y equal to U x_j, which is positive, but the y^0 coefficient is the second Symanzik polynomial F, which is not sign-definite in general. The paper later says it calculates in the Euclidean region because F has 'more positive terms' there, and for the tested massless integrals that probably makes F positive, so the examples are likely fine. But the general claim is logically incomplete, and the paper does not explicitly verify that F is sign-definite for each benchmark. This should be fixed in a revision.\n\nThe other soft spot is that the abstract claims improvement over the previous AmpRed version, but the table only compares against AMFlow. A head-to-head with the old branch-tracking method would support the claim directly. Also, the paper never states that the AmpRed and AMFlow numerical results agree; I assume they do, but a sentence saying so would make the timing comparison meaningful.\n\nThe citation pattern is clean. The relevant prior work is cited, and the benchmark is against an external package, so there is no circularity.\n\nOverall, this is a solid methods note with a narrow but useful contribution. It deserves peer review at a software-oriented journal, provided the author tightens the branch-fixing justification and either adds the previous-version comparison or softens the abstract claim. I would send it to referees.","headline":"A modest but real software improvement for evaluating complex multi-loop integrals; the branch-fixing rule is plausible but the 'always possible' claim needs qualification.","tokens_in":3634,"tokens_out":3661,"would_cite":false,"duration_ms":37102,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Choosing an auxiliary scale with sign-definite coefficients lets AmpRed evaluate complex integrals as fast as real ones, according to benchmarks up to four loops.","keywords":["analytic continuation","complex integrals","Feynman parameter representation","master integrals","multi-loop integrals","AmpRed","Wilson lines"],"falsifier":"Take a multi-scale integral at a complex kinematic point where the physical sheet is known independently, for example the two-loop example in Fig. 1a, and evaluate it both with a sign-definite auxiliary choice and with AMFlow's independent method. If the values disagree or land on different sheets for any sign-definite choice, the claim that sign-definiteness fixes the branch is falsified.","tokens_in":2764,"feed_emoji":"⚛️","tokens_out":10549,"duration_ms":95129,"temperature":0.7,"pith_summary":"AmpRed's update targets the bottleneck of multi-loop precision calculations: evaluating complex master integrals in the Feynman-parameter representation. The paper claims that by choosing the auxiliary scale $y=x_i/x_j$ so that the coefficients of monomials in the deformed integrand $F_y$ all have the same sign, the $-i0^+$ prescription fixes the physical branch and the integral can be evaluated exactly like a real one. Benchmarks on integrals up to four loops, including an integral with Wilson lines, show large speedups over AMFlow in three- and four-loop cases, and one case where the comparison package runs out of memory. The takeaway for a sympathetic reader is that branch selection in analytic continuation can be replaced by an algebraic sign condition, at least on the tested examples.","feed_headline":"A sign trick makes complex Feynman integrals as fast as real ones","feed_subtitle":"AmpRed's sign-definite auxiliary scale fixes the physical branch, and four-loop benchmarks show the speed gain.","key_machinery":"The central object is the auxiliary scale insertion $y=x_i/x_j$ through $\\delta(y-E^{(0)}(x))$, which rewrites the integral as $\\int dy\\, y^{\\lambda_i} I_y$ with $F_y=F|_{x_i=y x_j}$. The mechanism that makes complex evaluation cheap is sign-definiteness: if the coefficients of the monomials in $F_y$ are all positive (or all negative), the $-i0^+$ prescription is claimed to fix the branch of $I_y$, so the subsequent differential-equation computation runs as for real kinematics. The paper notes that this condition is always satisfiable for loop integrals because the coefficient of $x_{n+1}$ in $F$ equals the positive-definite first Symanzik polynomial $U$.","core_discovery":"The central claim is that the physical Riemann sheet of the auxiliary integral $I_y$ is fixed by the $-i0^+$ prescription whenever all coefficients of the monomials in $F_y$ have the same sign. For ordinary loop integrals such a pair $\\{x_i,x_j\\}$ always exists, because the coefficient of $x_{n+1}$ in $F$ is the first Symanzik polynomial $U$, which is positive definite. The paper therefore asserts that complex master integrals can be computed with the same algorithm and nearly the same cost as real integrals, without the expensive branch-point search of the earlier method. The claim is supported by timing benchmarks rather than by a proof of the branch-fixing observation.","pith_inferences":["If the branch-fixing observation is correct, the same sign-definiteness criterion should apply to cut integrals and to auxiliary variables other than $x_i/x_j$; the paper explicitly leaves cut integrals out, so testing those cases would extend the method.","The reported speedups depend on the hardware, the IBP solver, the precision goal, and the Euclidean-region choice; the qualitative advantage is likely to persist, but the exact factors are setup-specific.","A rigorous proof of the branch-fixing statement would upgrade the empirical benchmarks into a theorem; a targeted search for a counterexample in higher-loop multi-scale integrals would test where the condition starts to fail."],"forward_implications":["AmpRed can now evaluate complex master integrals at essentially the same cost as real ones, removing the main practical penalty of complex kinematics.","On the three- and four-loop benchmarks, the new implementation is about three to four times faster than AMFlow, and it handled the Wilson-line integral that made AMFlow run out of memory, using about 200 GB of RAM.","The method applies to integrals with linear propagators (Wilson lines) as well as ordinary loop integrals, because the sign-definite auxiliary choice relies only on the positive definiteness of the first Symanzik polynomial.","For multi-scale integrals, the paper recommends using differential equations with the boundary condition chosen at a singular point, together with the scaling relation $I(-s)=e^{-i\\pi(\\lambda_0+\\lambda_{n+1}+1)}I(s)^*$ for real $s$."],"supporting_citations":[{"why":"Defines the AmpRed package and the Feynman-parameter framework in which the update is implemented.","marker":"[1]"},{"why":"Supplies the iterative master-integral algorithm and the earlier branch-choice solution that this paper improves.","marker":"[5]"},{"why":"The differential-equation methods are used to evaluate the auxiliary integral $I_y$ for real and complex cases.","marker":"[9–11]"},{"why":"AMFlow is the comparison package whose computation times and memory usage are the benchmark baseline.","marker":"[12]"},{"why":"KIRA solves the integration-by-parts identities in both the AmpRed and AMFlow runs.","marker":"[13–15]"}],"fun_headline_variants":["AmpRed speeds up complex integrals via sign trick","Feynman integrals now compute as fast as real ones","Sign trick fixes branch, makes complex integrals fast","AmpRed's sign-definite trick speeds up complex integrals","Complex integrals speed up with AmpRed's sign trick"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire speedup rests on an unproved observation: when the auxiliary variable is chosen so that all coefficients in the integrand's expansion have the same sign, the integration automatically picks the physically correct branch. If that observation fails for some integral, the method would return a value on the wrong Riemann sheet.","fun_headline_variants_meta":{"raw":{"variants":["AmpRed speeds up complex integrals via sign trick","Feynman integrals now compute as fast as real ones","Sign trick fixes branch, makes complex integrals fast","AmpRed's sign-definite trick speeds up complex integrals","Complex integrals speed up with AmpRed's sign trick"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000214,"raw_usage":{"total_tokens":1300,"prompt_tokens":698,"completion_tokens":602,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":314,"completion_tokens_details":{"reasoning_tokens":524}},"tokens_in":314,"tokens_out":602,"duration_ms":5778,"temperature":1.0,"reasoning_tokens":524,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:30:56.937970+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a multi-scale integral at a complex kinematic point where the physical sheet is known independently, for example the two-loop example in Fig. 1a, and evaluate it both with a sign-definite auxiliary choice and with AMFlow's independent method. If the values disagree or land on different sheets for any sign-definite choice, the claim that sign-definiteness fixes the branch is falsified.","supporting_citations":[],"review_version":1}