{"id":"c1842810-d1e3-45fe-8192-510c40aa4b7a","arxiv_id":"2412.17137","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Complex Langevin results in one- and two-dimensional toy models match a linear combination of integration cycles when boundary terms vanish, and the kernel choice controls which cycles contribute.","lead":"This preprint tests a theorem about when complex Langevin simulations, a technique for quantum systems with complex probabilities, go wrong: with no boundary terms, the output is a mix of 'integration cycles', not always the physical one. Using simple toy models, the authors show that a tunable kernel selects which cycles contribute, and they present first numerical hints that the theorem extends to two dimensions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"2D confirmation rests on an unproven extension of Eq. (10) and on an asserted cycle count; both need direct numerical checks.","rationale":"The reader's weakest-assumption analysis identifies the same central gap: Eq. (10) is proven only in one dimension, and the 2D fits are interpreted under a conjecture. My stress-test agrees and sharpens the issue. The 2D numerical evidence is underpowered in two respects: the observable set is limited to low-degree monomials, and the cycle count at a=2 depends on an asserted singularity that is not derived. Both are directly testable, which is why a conditional verdict is appropriate rather than a rejection. The paper itself flags both limitations, so there is no hidden flaw or overclaim; however, the 2D claims should not be read as having the same status as the 1D confirmation. The proposed extended-observable fit would settle whether the apparent linear structure is an artifact of low-degree moments, and the explicit cutoff integration for the a=2 product cycles would settle the cycle-count question. Until those checks are run, the central claim is best regarded as a promising conjecture with partial numerical support.","tokens_in":33394,"tokens_out":9397,"duration_ms":91463,"concrete_test":"Use the a=0.5 model (34) and re-run the fit of Eq. (10) with the same Nγ=9 basis but an extended observable set that includes high-degree monomials (z1^6, z2^6, z1^4 z2^2, z1^2 z2^4) and at least one exponential observable such as e^{α z_i^k} with α in C. Require the whitened residuals to pass both normality and zero-mean/unit-variance checks. If the fitted ai change by more than the statistical errors or the residual tests fail, the current 2D fits do not confirm the conjectured extension. Separately, for a=2, numerically integrate all nine product-cycle partition functions with a large-R cutoff and verify that the seven non-basis cycles diverge; this would settle the Appendix A singularity claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The one-dimensional part is solid: Eq. (10) is proven in [1], and the fits are backed by multiple consistency checks (the sum rule (11), Shapiro-Wilk tests, alternative observable sets). The load-bearing gap is the two-dimensional extension. For d=2, Eq. (10) is a conjecture, so the fits in Sec. V B cannot by themselves confirm it; they demonstrate compatibility with one particular linear model. The compatibility is checked only with monomials of total degree at most 4 (14 observables, 28 real components). If the true CL state is not a cycle combination but happens to match the first few moments of one, the fit would still pass. Additionally, the Nγ=2 count at a=2 relies on the Appendix A assertion of a non-integrable singularity on r1=r2; this is not derived, and the graphical cuboid algorithm is presented informally. A wrong count would change the design matrix X in Eq. (23) and invalidate the a=2 fits. Finally, the Appendix B Shapiro-Wilk criterion tests normality of whitened residuals, not zero mean; a constant offset in the residuals would escape rejection. The paper is transparent about the conjecture, so this is not an overclaim, but the 2D evidence is not at the same level as the 1D evidence.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies complex Langevin (CL) simulations and the role of integration cycles. It numerically tests the Salcedo-Seiler theorem (Eq. (10)) that, in the absence of boundary terms, CL expectation values are a linear combination of integration-cycle averages with observable-independent coefficients. In one dimension (action S = λ z^4/4, kernel K = e^{iπm/24}), the authors measure monomial observables and boundary terms, and fit the coefficients a_i via covariance-weighted least squares to cycle averages computed by numerical integration. They find the theorem confirmed whenever boundary terms vanish, with the sum rule (11) satisfied; different kernels rotate the cycle-coefficient vector, and the 'perfect' kernel K = λ^{-1/2} selects the real cycle. In two dimensions (action S = λ/4 (z1^4 + z2^4 + a z1^2 z2^2)), they study a = 2 (O(2)-symmetric, Nγ = 2) and a < 2 (Nγ = 9). For a = 2 they find a1 ≈ 1 and a2 ≈ 0 for all λ with vanishing boundary terms; for a = 0.5 and suitable kernels they find four contributing cycles. They conclude that Eq. (10) holds in all their simulations, providing a first hint of its validity beyond one dimension.","tokens_in":33603,"tokens_out":11172,"duration_ms":93175,"significance":"If correct, the paper provides a useful numerical confirmation of the integration-cycle picture and, more importantly, demonstrates that kernels can be used to control which cycles are sampled, with practical implications for kernel searches in complex Langevin simulations. The one-dimensional results are convincing: the model is nontrivial, the fits are covariance-weighted, the sum rule is checked, alternative observable sets give consistent results, and the data and analysis scripts are publicly available. The two-dimensional results are weaker because Eq. (10) is a conjecture there and the cycle count at a = 2 relies on an asserted singularity that is not derived. Nevertheless, the paper is transparent about these limitations, and the two-dimensional evidence is a plausible first step toward a generalized theorem.","major_comments":[{"comment":"The count Nγ = 2 for |a| ≥ 2 is load-bearing for the two-dimensional fits, but the argument rests on the unproven assertion in Appendix A that for a = 2 and ϕ = π/4 (the line r1 = r2) 'a non-integrable singularity arises in the integral over certain cycles.' No derivation, estimate, or reference is given. Since the number of columns of the design matrix X in Eq. (23) and hence the a = 2 fits in Sec. V B 1 (Figs. 8 and 11) depend on this count, the authors should provide the missing derivation or an explicit numerical check (e.g., demonstrate divergence of candidate cycle integrals, or show rank deficiency of the matrix Mij = ⟨Oi⟩γj for nine candidate cycles). Without this, the a = 2 fits cannot be taken as a confirmation of Eq. (10).","section":"Appendix A, Eq. (36)"},{"comment":"For the a < 2 fits of Sec. V B 2, the nine-cycle basis (40) is used at a = 0.5, but the validity of representatives involving the one-dimensional γ3 for a ≠ 0 is only asserted in Appendix A. The numerical contours used to compute ⟨O⟩γi for the design matrix X are not specified. Please specify the contours and provide a convergence check for all nine cycle integrals at the couplings used (at least a = 0.5), or restrict the basis to cycles whose validity is demonstrated. Otherwise the small fitted values of a5...a9 in Fig. 15 may be artifacts of non-convergent integrals.","section":"Sec. V B 2 and Eq. (40)"},{"comment":"The Shapiro-Wilk test used in Appendix B as the goodness-of-fit criterion tests whether the whitened residuals are Gaussian, not whether their mean is zero. A model that is systematically offset by a constant would produce shifted Gaussian residuals and could still pass the p > 0.05 threshold. Since the statement 'we find the fit to be good if and only if it produces this unique answer' (Sec. V A 1) underpins the one- and two-dimensional conclusions, add an explicit zero-mean test on the whitened residuals (e.g., a t-test, or equivalently a χ² per degree of freedom after whitening) and report its outcome for all fits shown.","section":"Appendix B and Sec. V A 1"},{"comment":"The abstract states that the paper 'confirm[s] numerically' Eq. (10), but for d = 2 the equation is a conjecture, as acknowledged in Secs. III A and VI. Because the two-dimensional fits assume the linear model (10) as the regression model, they test compatibility with one particular linear combination rather than the validity of the decomposition. To strengthen the two-dimensional confirmation, perform a hold-out test: fit the coefficients a_i on a subset of observables and verify that the remaining observables are reproduced within uncertainties for the points in Figs. 8 and 11. This would mitigate the concern that a non-cycle distribution matching the first few monomial moments could pass the current test.","section":"Abstract and Sec. V B"}],"minor_comments":[{"comment":"The word 'Heavyside' should be 'Heaviside'.","section":"Sec. II B"},{"comment":"The phrase 'it has do be done online' should read 'it has to be done online'.","section":"Sec. IV D 2"},{"comment":"The covariance matrix Σ is said to be 'normalized by 1/√Nmeas' in the fit, while Appendix B uses the same symbol for the unnormalized covariance; please clarify the two uses.","section":"Sec. IV D 3"},{"comment":"The sentence 'The results are rounded to the first significant digit of the respective statistical uncertainties' is clear, but consider stating explicitly that the imaginary parts in the l = -1 and l = 1 rows are consistent with zero within the quoted errors.","section":"Sec. V A 1, Table I"},{"comment":"Reference [31] has an incomplete author entry ('T¨or¨ok' without initials); please complete the citation.","section":"References"},{"comment":"The FAIR principles are invoked, but the simulation code is only 'available upon request'; posting the code alongside the data would be more consistent with the stated reproducibility goals.","section":"Sec. IV and Data Availability"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a solid numerical study for the one-dimensional case, and the kernel-dependence of the cycle coefficients is a useful result. The main risk is the two-dimensional section, where the cycle count and the goodness-of-fit criterion need strengthening. I believe the issues are fixable within the scope of a revision. The paper is appropriate for a hep-lat journal; the fact that one author is also an author of the theorem being tested is not, in itself, a concern because the theorem is published and the fits use independently computed cycle integrals."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a careful, honest numerical study. The one-dimensional test of the Salcedo–Seiler decomposition is solid and stands on its own. The two-dimensional extension is genuinely new but conditional, and the authors say so in the text. The paper deserves a serious referee, and the public data means the referee can check the weak points directly.\n\nWhat is actually new: first numerical evidence for Eq. (10) in two dimensions, where it is a conjecture rather than a theorem; a systematic map of how the kernel phase selects which integration cycles are sampled; and the cuboid-counting algorithm in Appendix A, a concrete tool for counting cycles in polynomial models. The paper also ships data and analysis scripts on Zenodo, so the fits are reproducible. That counts for a lot.\n\nSoft spots: the 2D fits do not confirm the conjecture — they show compatibility with one particular linear model, and the observable set (monomials of total degree at most 4) is not wide enough to rule out a state whose low moments accidentally match a cycle combination. The Nγ = 2 count for a ≥ 2 depends on an asserted non-integrable singularity on r1 = r2 in Appendix A, and that count is load-bearing for the a ≥ 2 fits. Also, the Shapiro–Wilk test in Appendix B checks normality of whitened residuals, not zero mean, so a constant offset could in principle pass unnoticed. These are real limitations, but the authors explicitly flag the conjecture and the singularity, so this is not an overclaim. The 1D section is clean and independent.\n\nMinor: the a = 2 singularity could be probed directly by numerical integration, and a sentence on that would tighten the paper. The dismissal of reduced chi-square is a bit quick; the whitening approach is fine, but a line explaining that the issue is Gaussianity of residuals would help.\n\nWho should read this: anyone working on complex Langevin, wrong convergence, or kernel searches. The kernel–cycle connection is relevant for future kernel optimization, and the counting algorithm may transfer to other polynomial models. The paper does not resolve the 2D question, but it sets it up sharply.\n\nRecommendation: send it to peer review. The numerics are reproducible, the claims are calibrated to the evidence, and the 2D conjecture is the natural target for follow-up work. I would cite the data and the counting algorithm.","headline":"Solid 1D confirmation of the cycle decomposition, a genuinely new but explicitly conditional 2D test, and a useful cycle-counting algorithm; worth a serious referee.","tokens_in":34158,"tokens_out":3388,"would_cite":true,"duration_ms":28918,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["02.50.Ey","05.10.Gg"],"model":"deepseek-v4-flash","headline":"In quartic toy models, boundary-term-free complex Langevin averages are numerically confirmed to be observable-independent linear combinations of integration-cycle averages, with the kernel phase controlling which cycles contribute.","keywords":["complex Langevin","sign problem","integration cycles","kernel","wrong convergence","boundary terms","Dyson-Schwinger equations","two-dimensional toy model"],"falsifier":"Re-measure the even monomial observables in the model (34) at $a=2$ with kernels near $m=10$ and $m=34$, fit the two-cycle coefficients on subsets of observables of increasing order, and check whether the residuals drift with observable order; any order-dependent drift would falsify the claim that one coefficient set describes all observables. Independently, evaluate the integral of $\\operatorname{Re} S$ along the ray $r_1 = r_2$ for $a=2$: if no non-integrable divergence appears, the predicted reduction to $N_\\gamma = 2$ would need another explanation.","tokens_in":33171,"feed_emoji":"🧮","tokens_out":9724,"duration_ms":83492,"temperature":0.7,"pith_summary":"Complex Langevin simulations are a candidate cure for the sign problem, but they can converge to wrong answers even when the standard boundary-term test looks clean. This paper's central claim is that in such clean cases the simulation average is still a linear combination of integration-cycle averages, with the same coefficients for every observable, and that the kernel in the Langevin equation decides which combination is sampled. The authors verify the decomposition numerically in a one-dimensional quartic model, where it is a theorem, and in a two-dimensional extension, where it is only a conjecture. The practical punchline is that vanishing boundary terms are necessary but not sufficient for correctness, and that kernels used to cure the sign problem can silently redirect the simulation onto the wrong cycles.","feed_headline":"Kernel chooses which integration cycles get sampled","feed_subtitle":"Zero boundary terms are not enough: quartic-model tests find a cycle decomposition, and 2D suggests the same law.","key_machinery":"The central object is the integration cycle: an equivalence class of integration contours in the complexified field space along which $e^{-S}$ vanishes at infinity; for polynomial actions, cycles connect different angular good regions where the Boltzmann factor decays. The load-bearing identity is the decomposition (10) of the paper, which says every boundary-term-free complex Langevin average is a linear combination of cycle averages with observable-independent coefficients. The second piece of machinery is the kernel $K$ in the Langevin equation $dz = -K S'(z)\\,d\\tau + \\sqrt{K}\\,dw$; the argument of $K$ rotates the sampled distribution in the complex plane and thereby selects which cycle or mixture of cycles is sampled. To extract the coefficients, the paper fits measured monomial observables against numerically computed cycle averages by generalized least squares, using whitened residuals to judge whether the fit is trustworthy. A geometric counting algorithm based on decomposing cuboids in the angular torus supplies the number $N_\\gamma$ of independent cycles, including the drop from nine to two cycles at $|a|=2$.","core_discovery":"The central claim, carried through every simulation, is that in the absence of boundary terms a complex Langevin expectation value obeys the decomposition $\\langle O\\rangle_{\\mathrm{CL}} = \\sum_{i=1}^{N_\\gamma} a_i \\langle O\\rangle_{\\gamma_i}$, where the $\\gamma_i$ are the independent integration cycles of the complexified theory and the complex coefficients $a_i$ are the same for all observables, with $\\sum_i a_i = 1$. In one dimension the paper reproduces the known cases and exposes a strong example: for $\\lambda = e^{5i\\pi/6}$ with trivial kernel the boundary terms vanish yet the result is a 50/50 mixture of the real and imaginary cycles, not the desired real-cycle average. Scanning kernel phases $K = e^{im\\pi/24}$, the coefficients move through a series of plateaus: only the real cycle near $m=10$, only the imaginary cycle near $m=34$, and mixed combinations on smaller plateaus near $m=22$ and $46$. In the two-dimensional model $S_a = \\frac{\\lambda}{4}(z_1^4+z_2^4+a z_1^2 z_2^2)$, the same decomposition is found numerically, including the strong-coupling regime $a \\ge 2$ in which an angular counting algorithm predicts only two independent cycles and the fits correspondingly use $N_\\gamma = 2$. The authors therefore conclude that the one-dimensional theorem plausibly extends to arbitrary dimension.","pith_inferences":["If the two-dimensional conjecture is right, integration-cycle content should be part of the interpretation of every boundary-term-free complex Langevin simulation, not just one-dimensional models.","Automated kernel searches (for example by machine learning) could lock onto a harmless-looking kernel on a wrong-convergence plateau; using a family of observables, especially high powers, rather than a single one, would make such plateaus detectable.","The counting algorithm's prediction that strong coupling reduces cycle number suggests that some strongly interacting theories might have tractably few cycles, and that the danger of cycle contamination may be largest at weak coupling.","The asserted singularity at $r_1=r_2$ for $a=2$ is testable in isolation: a direct numerical integration along that ray would either substantiate or weaken the $N_\\gamma=2$ counting."],"forward_implications":["Vanishing boundary terms are confirmed insufficient for correctness: simulations can converge to wrong averages composed of several integration cycles while all measured boundary terms stay consistent with zero.","The kernel acts as a dial on cycle coefficients, at least in the quartic model: changing its phase moves the simulation from the real cycle to the imaginary cycle, with intermediate mixtures, and large plateaus of the kernel phase give correct results.","Kernel searches must track cycle content, not only boundary terms: there are large kernel regions with clean boundary terms but wrong averages.","In the two-dimensional model the number of independent cycles drops from nine to two for strong coupling $|a| \\ge 2$, and numerical fits there support the two-cycle decomposition.","Because the decomposition held in every boundary-term-free run, the authors expect a proof of the decomposition for arbitrary dimensions to be within reach."],"supporting_citations":[{"why":"States the decomposition theorem (10) that this paper sets out to test numerically in one and two dimensions.","marker":"[1]"},{"why":"Provides the formal proof that complex Langevin reproduces correct expectation values when boundary terms decay; its assumption is what the boundary-term measurements probe.","marker":"[36]"},{"why":"Introduces the observable-based boundary term $B_O$ and the plateau test used throughout the paper to decide whether boundary terms vanish.","marker":"[39]"},{"why":"Extends the boundary-term criterion, supporting the detection method the simulations rely on.","marker":"[40]"},{"why":"Earlier companion study of the quartic model that already connects the kernel to the decomposition; the present paper extends its observables and kernel scan.","marker":"[52]"},{"why":"Original observation that a kernel close to $\\lambda^{-1/2}$ restores correct results in the quartic model, the plateau effect this paper maps to cycle coefficients.","marker":"[62]"},{"why":"Adaptive step-size algorithm used in all runs to control runaway trajectories and keep the stochastic evolution stable.","marker":"[26]"},{"why":"Defines integration cycles as relative homology classes, the topological language the paper borrows for cycle counting.","marker":"[42]"},{"why":"Example of a $z$-dependent kernel restoring correct results, cited as the natural next test for the cycle-decomposition picture.","marker":"[66]"}],"fun_headline_variants":["Integration cycles, not kernels, decide wrong convergence","Kernel selects integration cycles in Complex Langevin","Complex Langevin mixes cycles when boundary terms vanish","Cycle decomposition explains wrong convergence in Complex Langevin"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes the integration-cycle decomposition holds in two dimensions, a conjecture the authors explicitly flag as unproved, and counts $N_\\gamma = 2$ for $a \\ge 2$ on the strength of an asserted but not derived non-integrable singularity on the line $r_1 = r_2$.","fun_headline_variants_meta":{"raw":{"variants":["Integration cycles, not kernels, decide wrong convergence","Kernel selects integration cycles in Complex Langevin","Complex Langevin mixes cycles when boundary terms vanish","Cycle decomposition explains wrong convergence in Complex Langevin"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000246,"raw_usage":{"total_tokens":1601,"prompt_tokens":1071,"completion_tokens":530,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":687,"completion_tokens_details":{"reasoning_tokens":471}},"tokens_in":687,"tokens_out":530,"duration_ms":5726,"temperature":1.0,"reasoning_tokens":471,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T05:46:02.519438+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Re-measure the even monomial observables in the model (34) at $a=2$ with kernels near $m=10$ and $m=34$, fit the two-cycle coefficients on subsets of observables of increasing order, and check whether the residuals drift with observable order; any order-dependent drift would falsify the claim that one coefficient set describes all observables. Independently, evaluate the integral of $\\operatorname{Re} S$ along the ray $r_1 = r_2$ for $a=2$: if no non-integrable divergence appears, the predicted reduction to $N_\\gamma = 2$ would need another explanation.","supporting_citations":[{"cited_title":"In order to define the two linearly independent integration cycles for this model, we introduce two variables ξ1 and ξ2 in analogy to (30)","cited_arxiv_id":null,"evidence_quote":"States the decomposition theorem (10) that this paper sets out to test numerically in one and two dimensions."},{"cited_title":"L’Ecuyer, R","cited_arxiv_id":null,"evidence_quote":"Original observation that a kernel close to $\\lambda^{-1/2}$ restores correct results in the quartic model, the plateau effect this paper maps to cycle coefficients."},{"cited_title":"Fujimura, K","cited_arxiv_id":null,"evidence_quote":"Adaptive step-size algorithm used in all runs to control runaway trajectories and keep the stochastic evolution stable."}],"review_version":1}