{"id":"d6aa8de4-0bb2-46f1-9d33-88196c6e1af7","arxiv_id":"2605.31536","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Closed-form expressions are provided for the linear moments of the angle in the 2D angular central Gaussian distribution via contour integration.","lead":"The paper derives closed-form expressions for the linear moments E[θ] and E[θ²] of the two-dimensional angular central Gaussian distribution on the interval ]-π/2, π/2[, with the mean given by an arctangent and the second moment by the real part of a dilogarithm. These may simplify analytical work in physics and directional statistics where the angle is treated as a real-valued phase.","discovery_kind":"first_principles","skeptic_critique":{"model":"grok-4.3","headline":"Contour integration around arctan branch cut may encounter unaccounted singularities for some ACG parameters","rationale":"The reader’s weakest assumption is precisely the step whose validity is least secured by the given information. No other internal inconsistency is visible from the abstract or claim description, and the paper offers no machine-checked proof or shipped code that would bypass the contour step. Therefore the same load-bearing concern remains after reading the full-text placeholder.","tokens_in":1677,"tokens_out":428,"duration_ms":15257,"concrete_test":"Fix a valid covariance, e.g. Σ = [[1, 0.4], [0.4, 1]]. Numerically evaluate the two integrals ∫_{-π/2}^{π/2} θ f(θ) dθ and ∫ θ² f(θ) dθ to 10^{-8} relative accuracy using adaptive quadrature. Compare the results to the claimed closed forms (arctan expression and Re(dilog)). If the absolute difference exceeds 10^{-6} for either moment, the contour derivation misses contributions.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim rests on evaluating the linear moments via a specific contour integral around the branch cut of arctan z that supposedly recovers E[θ] as arctan of the parameters and E[θ²] as Re of a dilogarithm. The ACG density is obtained after radial integration of a bivariate Gaussian, so its explicit form contains a quadratic denominator whose roots depend on the covariance parameters. Nothing in the provided description shows that the chosen contour (i) stays clear of those roots for the full open set of positive-definite matrices, (ii) correctly handles the principal-value sense of the branch cut when the integration limits are exactly [-π/2, π/2], or (iii) produces no additional residue contributions when the parameters push a pole across the cut. The abstract asserts the construction is “elementary” and works “for the relevant parameter values,” but supplies neither an explicit domain restriction nor an independent numerical check.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper derives closed-form expressions for the linear (non-circular) moments E[θ] and E[θ²] of the two-dimensional angular central Gaussian distribution on θ ∈ ]−π/2, π/2[, obtained as the polar-angle marginal of a centered bivariate Gaussian after radial integration. It claims that the mean is given by a simple arctangent of the covariance parameters and the second moment by the real part of a dilogarithm, with the derivation performed via an elementary contour integration around the branch cut of arctan z.","tokens_in":1877,"tokens_out":321,"duration_ms":11913,"significance":"If the contour-integration steps are free of unaccounted singularities or branch-cut violations, the results supply explicit, non-numerical formulas for quantities that arise when the angle is interpreted as a real-valued phase rather than a circular variable, filling a documented gap relative to the existing literature on trigonometric moments of the ACG distribution.","major_comments":[{"comment":"The central derivation (abstract and main text) evaluates the moments by contour integration around the branch cut of arctan z. The ACG density contains a quadratic denominator whose roots depend on the covariance parameters; the manuscript supplies neither an explicit verification that the chosen contour remains free of these roots for the full open set of positive-definite matrices nor a demonstration that no additional residues arise when a pole crosses the cut.","section":"derivation (contour integration)"}],"minor_comments":[],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading and for identifying a point where the contour-integration argument requires additional rigor. We respond to the single major comment below and will revise the manuscript to incorporate the requested verification.","responses":[{"response":"We agree that the manuscript does not contain an explicit verification that the contour avoids the poles of the integrand for every positive-definite covariance matrix, nor a demonstration that poles cannot cross the branch cut. This is a legitimate gap in the presented derivation. In the revised version we will insert a short lemma immediately after the contour setup that (i) solves the quadratic denominator explicitly in terms of the covariance entries, (ii) shows that both roots lie strictly outside the closed contour encircling the branch cut of arctan z whenever the matrix is positive definite, and (iii) proves by continuity that the roots remain in the same half-planes as the parameters vary inside the positive-definite cone, so that no pole crosses the cut. The lemma will be elementary and will not alter the final closed-form expressions.","revision_made":"yes","referee_comment":"[derivation (contour integration)] The central derivation (abstract and main text) evaluates the moments by contour integration around the branch cut of arctan z. The ACG density contains a quadratic denominator whose roots depend on the covariance parameters; the manuscript supplies neither an explicit verification that the chosen contour remains free of these roots for the full open set of positive-definite matrices nor a demonstration that no additional residues arise when a pole crosses the cut."}],"tokens_in":1303,"tokens_out":315,"duration_ms":23459,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The new piece is the explicit expressions: the mean is an arctangent of the covariance parameters, and the second moment is the real part of a dilogarithm. The abstract states these had not appeared before, which matches the gap between the usual circular moments in directional statistics and the linear ones needed when θ is treated as an ordinary real variable in physics.\n\nThe derivation is presented as a direct contour integration from the moment integral, using the branch cut of arctan. That approach is concrete and avoids reducing to fitted parameters or self-reference. If the steps are correct, the result is reproducible in principle and could save numerical work in applications.\n\nThe soft spot is the contour itself. The ACG density comes from a bivariate Gaussian after radial integration, so the integrand has a quadratic denominator whose poles depend on the covariance matrix. Nothing in the abstract shows that the chosen contour around the arctan cut stays clear of those poles for the full open set of positive-definite matrices, or that no extra residues appear when a pole crosses the cut. The paper qualifies the result to “relevant parameter values,” but without an explicit domain or independent numerical checks against special cases (isotropic Gaussian, for example), it is not yet clear how broad the formulas actually are.\n\nThis is a narrow but self-contained calculation. Specialists who need linear moments of the ACG for phase-type problems would find the expressions useful if they hold. A referee could verify the contour steps, request the domain statement, and ask for a short numerical table. I would send it to peer review rather than desk-reject it.","headline":"The paper supplies explicit closed forms for the linear moments E[θ] and E[θ²] of the 2D ACG via contour integration, but the contour's validity across all positive-definite parameters needs checking.","tokens_in":2324,"tokens_out":416,"would_cite":false,"duration_ms":16104,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The mean of the two-dimensional angular central Gaussian angle is an arctangent of the parameters and its second moment is the real part of a dilogarithm.","keywords":["two-dimensional angular central Gaussian","linear moments","contour integration","dilogarithm","arctangent","phase distribution"],"falsifier":"Computing the integral for E[θ] numerically with specific parameter values and finding it does not match the proposed arctangent formula would falsify the result.","tokens_in":2581,"feed_emoji":"","tokens_out":565,"duration_ms":18968,"temperature":0.7,"pith_summary":"The paper provides closed-form expressions for the linear moments E[θ] and E[θ²] of the two-dimensional angular central Gaussian distribution where θ lies in (-π/2, π/2). It shows that the mean is a simple arctangent function of the distribution parameters while the second moment is expressed using the real part of a dilogarithm. The derivation relies on contour integration around the branch cut of the arctangent function and these results apply directly in physical contexts treating the angle as a real-valued phase.","feed_headline":"Arctan and dilogarithm give linear moments of angular Gaussian","feed_subtitle":"Mean is arctangent of parameters while second moment is real part of dilogarithm from contour integration.","key_machinery":"Contour integration around the branch cut of arctan z to evaluate the moment integrals.","core_discovery":"By performing a contour integration around the branch cut of arctan z, the integrals defining the linear moments of the ACG density yield E[θ] as an arctangent expression in the parameters and E[θ²] as the real part of a dilogarithm at points determined by those parameters.","pith_inferences":["The contour method could be adapted to compute higher-order linear moments or moments of related distributions.","Connections may exist to integrals arising in other areas of multivariate statistics or phase modeling in physics.","Numerical verification for edge cases of the parameter space would strengthen confidence in the expressions."],"forward_implications":["The moments can be evaluated analytically for any valid parameters without numerical methods.","These closed forms allow exact calculations in applications where linear rather than circular moments of the angle are required.","The approach provides a template for finding moments in similar angular distributions derived from Gaussians."],"fun_headline_variants":["Arctan and dilog for linear moments of ACG","Linear moments of angular central Gaussian in closed form","ACG mean linear moment is simple arctan","Second ACG linear moment from dilog real part","Contour integration closes ACG linear moments"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The contour integration around the branch cut of arctan z produces the desired moments for the relevant parameter values without encountering additional singularities or requiring further restrictions.","fun_headline_variants_meta":{"raw":{"variants":["Arctan and dilog for linear moments of ACG","Linear moments of angular central Gaussian in closed form","ACG mean linear moment is simple arctan","Second ACG linear moment from dilog real part","Contour integration closes ACG linear moments"]},"model":"grok-4.3","cost_usd":0.004124,"raw_usage":{"total_tokens":2079,"prompt_tokens":644,"num_sources_used":0,"completion_tokens":71,"cost_in_usd_ticks":41237000,"prompt_tokens_details":{"text_tokens":644,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1364,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":644,"tokens_out":71,"duration_ms":8960,"temperature":1.0,"reasoning_tokens":1364,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T20:14:16.385959+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Computing the integral for E[θ] numerically with specific parameter values and finding it does not match the proposed arctangent formula would falsify the result.","supporting_citations":[],"review_version":1}