{"id":"afbe8f17-f195-4f44-9c7a-647063b31312","arxiv_id":"2607.29280","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"Generalized Möbius transport—a cosine rearrangement fixed by any rotationally symmetric target—builds tractable anisotropic spherical distributions, including a new Möbius–von Mises–Fisher family.","lead":"This paper identifies the spherical Möbius transformation as a one-dimensional optimal transport and generalizes it to arbitrary rotationally symmetric targets, yielding new anisotropic spherical distributions. The resulting models have closed-form densities and are shown to fit paleomagnetic and comet-orbit data better than standard alternatives.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Paleomagnetic polarity merging could manufacture the apparent skewness that drives the Möbius–vMF outperformance claim.","rationale":"I checked the core mathematical chain carefully: Proposition 2.2's identification M_ρ=F_0^{-1}∘F_ρ is correct; the lift Φ_ρ is indeed the Möbius map (verified via stereographic dilation); Theorem 3.1's density transformation is sound (using T#P0=G gives Jacobian h_G); the conformal uniqueness ODE in §3 has the stated Möbius solutions; and the closed-form projected cdfs in §3.1 and their inverses in Corollary 3.1 check out. I also verified that the spherical Cauchy density is normalized once read as ((1-ρ^2)/(1+ρ^2-2ρ x·μ))^d/ω_d. No internal inconsistency or circular step surfaced. The only vulnerability is the application-level preprocessing of the paleomagnetic data, exactly as the reader flagged: pooling two polarity groups after reflection can create an artificial offset that a skewed model will capture. This is a genuine, concrete risk to the paper's empirical 'outperform' wording, but it does not touch the theoretical construction. A separate within-polarity fit and a two-component mixture comparison would settle it. Since the paper's contribution is primarily theoretical and the empirical sections are illustrative, the ACCEPT verdict stands.","tokens_in":9294,"tokens_out":50807,"duration_ms":393833,"concrete_test":"Refit Table 1 separately for the 82 normal and 157 reversed (after antipodal reflection) directions, and estimate the MvMF skewness parameter ρ̂ and the log-likelihood gain over vMF within each group. Then fit a two-component von Mises–Fisher mixture (common κ, axes separated by the observed 3.2°) to the pooled reflected data and compare AIC/BIC to MvMF. If the single-group fits show ρ̂≈0 with negligible AIC gain, or the two-component mixture is preferred, the pooled skewness/ranking is an artifact of polarity merging.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract's 'outperform' claim for the paleomagnetic dataset rests on fitting MvMF to n=239 directions after reflecting 157 reversed-polarity directions into a common polarity via PmagDiR::common_DI(). The two modal means remain 3.2° apart after reflection. If these are two distinct populations rather than one Fisherian population (e.g., paleosecular variation or incomplete overprint removal), the reflected pooled scatter is slightly bimodal, and a unimodal skewed model like MvMF—which has a dedicated skewness axis μ2—will absorb the between-mode offset as skewness, inflating its likelihood and AIC relative to rotationally symmetric competitors. The paper offers no within-polarity check (is the skewness present in the 82 normal and 157 reversed groups separately?) and no comparison against a two-component mixture that would be the natural model under a two-population interpretation. This does not affect Proposition 2.2 or Theorem 3.1, but it undermines the specific 'outperform' claim on paleomagnetic data.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper shows that the Möbius map generating the spherical Cauchy distribution from the uniform law is the tangent–normal lift of a one-dimensional optimal transport: it acts on the cosine coordinate by the monotone rearrangement F_0^{-1}∘F_ρ and fixes the tangent direction. Proposition 2.2 and Corollary 2.1 formalize this. The authors then generalize the construction by replacing the Cauchy target with any rotationally symmetric law with continuous strictly increasing projected cdf, obtaining the \"generalized Möbius transformation\" M_G. Theorem 3.1 gives a density-transform formula, and Section 3.1 provides closed-form cosine quantile functions for the Poisson kernel and spherical cardioid targets. Section 4 constructs two tractable anisotropic families, the Möbius–von Mises–Fisher and isotropic scaled von Mises–Fisher distributions, with closed-form densities, normalizing constants inherited from vMF, and exact simulation. The empirical sections fit these models to paleomagnetic directions and short-period comet orbit normals, reporting improvements in AIC/BIC over classical and recent alternatives.","tokens_in":9560,"tokens_out":16001,"duration_ms":152334,"significance":"The optimal-transport interpretation of the spherical Cauchy Möbius map is conceptually valuable and, to my knowledge, new. It explains why the Möbius map moves mass along meridians and isolates the role of the projected cdf. The generalization to arbitrary rotationally symmetric targets is elegant and productive: Theorem 3.1's density formula is simple and correct, and the resulting MvMF distribution has an explicit normalizing constant, in contrast to Kent's distribution. The closed-form quantile maps in Corollary 3.1 and the exact simulation schemes are concrete strengths. If the mathematical core is accepted, the paper gives a useful toolkit for anisotropic spherical distributions. The empirical demonstrations are suggestive, but their strength is limited by the data-preprocessing issue discussed in the major comments.","major_comments":[{"comment":"The manuscript cites 'García-Portugués (2026)' and the sphunif package version 1.4.4 as available works. If these are not yet published, please mark them as in press/under review, since the spherical cardioid distribution is used as a building block.","section":"References"}],"minor_comments":[{"comment":"The phrasing 'Möbius transformations are a recurring device' is a bit general; consider naming specifically the spherical Cauchy and wrapped Cauchy examples at the start to set the context.","section":"Section 1"}],"recommendation":"major_revision","confidential_remarks":"The mathematical core is sound: Propositions 2.1–3.1 and Theorem 3.1 are correct, and the closed-form examples check out. The main concern is the paleomagnetic application claim, which depends on a polarity-merging step that is not scrutinized. The missing within-polarity analysis is easy to add, and the ODE uniqueness proof is a one-line addition. I would be comfortable with acceptance after these revisions. Please verify that the cited companion papers and the sphunif data package are accessible, since the spherical cardioid example and the comet data depend on them."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core idea is genuinely new and clean: writing the spherical Cauchy Möbius map as the lift of a one-dimensional optimal transport (Prop 2.2, Eq. (2)) is a real reframing, not a cosmetic one. It turns an analytic transformation into a measure-transport statement, and the generalization to any rotationally symmetric target (Section 3) follows naturally. The density formulas (7)-(10) are explicit, closed-form, and the simulation scheme is immediate. This is a solid contribution to directional statistics. \n\nWhat the paper does well: the proofs are careful and readable. The tangent–normal decomposition is used honestly, the density transform in Theorem 3.1 follows from a plain pushforward argument, and the closed-form quantiles check out. The identification of the isotropic scaled von Mises–Fisher as a Möbius transport (Prop 4.1) is a nice unification. I believe the math. The paper is self-contained and does not rely on circular reasoning.\n\nThe soft spots are mostly at the application level. The paleomagnetic comparison depends on reflecting reversed-polarity directions into a common polarity. The stress-test note worries that two nearly antipodal modes 3.2° apart could be two populations, and the MvMF's skewness axis could absorb the between-mode offset. That concern is legitimate and not manufactured. The paper does not check the two polarity groups separately, nor does it compare against a two-component mixture. So the 'outperform' claim on paleomagnetic data rests on the merging assumption. But that is an application-level assumption, not a flaw in the theory. The comet application is cleaner and the scaled vMF fit there is honestly evaluated with confidence bands. \n\nThe only theoretical gap I notice is the asserted conformal-uniqueness ODE solution in Section 3 (the claim that only Möbius maps solve ψ'(t) = (1−ψ(t)^2)/(1−t^2)). They state it without proof; it looks like a standard Riccati argument, and it is minor, but worth a line in the paper.\n\nWho the paper is for: anyone working on directional distributions, particularly anisotropic models with explicit normalizing constants. The Möbius–von Mises–Fisher is a practical alternative to Kent with a closed-form normalizer, and the OT framing gives it conceptual depth. I would bring it to a reading group and I would cite it. It deserves peer review. The paleomagnetic section needs a robustness check or a softened claim, but the main result is solid.\n\nRecommendation: send it to a serious referee. Expect minor revision on the application section and the uniqueness proof; the theory stands.","headline":"Möbius transport as optimal transport on spheres is a clean, real result; the paleomagnetic application has a genuine preprocessing soft spot, but the core theory holds and deserves peer review.","tokens_in":10062,"tokens_out":669,"would_cite":true,"duration_ms":9271,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62H11","49Q22"],"pacs":[],"model":"deepseek-v4-flash","headline":"The Möbius map that turns the uniform law on a sphere into the spherical Cauchy law is an optimal transport of the cosine coordinate.","keywords":["Möbius transformation","optimal transport","spherical Cauchy distribution","von Mises–Fisher distribution","rotational symmetry","monotone rearrangement","directional statistics","spherical distributions"],"falsifier":"Re-fit the paleomagnetic data on the normal and reversed polarities separately, or with an alternative polarity-reversal criterion; if the Möbius–von Mises–Fisher model no longer achieves the lowest AIC/BIC, the empirical conclusion is an artifact of the merging step. A second check: compute the Jacobian of the generalized map for a Poisson-kernel target on S^2 at a point with cosine not ±1; if it is conformal, the uniqueness claim for the spherical Cauchy target is false.","tokens_in":9183,"feed_emoji":"🌐","tokens_out":7409,"duration_ms":69044,"temperature":0.7,"pith_summary":"This paper shows that a classical device in directional statistics—the Möbius transformation that turns the uniform law on the sphere into the spherical Cauchy law—is an optimal transport in disguise: it moves probability mass by monotonically rearranging the cosine of each point about the location axis and leaving the tangent direction alone. Because this rearrangement is determined entirely by the target's projected cumulative distribution function, any rotationally symmetric law can replace the Cauchy target, producing a generalized Möbius transport. Transporting a von Mises–Fisher base by such a map gives anisotropic spherical distributions with explicit densities, inherited normalizing constants, and exact simulation. The resulting models fit paleomagnetic directions and short-period comet orbit normals better than classical alternatives, capturing skewness and heavy tails that standard models miss.","feed_headline":"Spherical Cauchy's Möbius map is optimal transport","feed_subtitle":"Generalizing it yields anisotropic spherical models, in closed form, that fit paleomagnetic and comet data better.","key_machinery":"The central object is the tangent–normal lift of the monotone rearrangement of the cosine. For a rotationally symmetric target G about a location axis μ, its projected cumulative distribution function F_G defines a map t ↦ F_G^{-1}(F_0(t)) on the cosine coordinate t = x^T μ; applying this map while preserving the tangent vector gives M_G. This single construction is the optimal transport from the uniform to G, specializes to the classical Möbius transformation when G is the spherical Cauchy law, and converts a von Mises–Fisher base into an anisotropic transported distribution with closed-form density.","core_discovery":"The paper establishes that the Möbius transformation generating the spherical Cauchy distribution from the uniform on the sphere is exactly the tangent–normal lift of a one-dimensional optimal transport: the monotone rearrangement of the cosine about the location axis, expressed as F_ρ = F_0 ∘ M_ρ. Since this lift depends on the target only through its projected cumulative distribution function, replacing the Cauchy target by any rotationally symmetric law with a continuous, strictly increasing projected cdf produces a generalized Möbius transformation M_G that carries the uniform to G. Transporting a von Mises–Fisher base by such a map yields anisotropic spherical distributions with closed-","pith_inferences":["Because the transport depends only on the target's projected cdf, one could plug in an empirical cdf estimate to define nonparametric spherical transport maps, potentially yielding distribution-free ranks or tests for directional data—an extension the paper does not explore.","The tangent–normal split suggests a general recipe: any monotone transform of the cosine coordinate lifts to a spherical map, so the construction might serve as a copula-style device that separates the cosine marginal from the uniform tangent structure.","The paper notes but does not pursue compositions M_{G_p} ∘ ... ∘ M_{G_1} with distinct axes; fitting such compositions to bimodal directional data is a direct next step."],"forward_implications":["If the paper is right, the spherical Cauchy map is an explicit optimal transport on the sphere, and the spherical Cauchy law is the unique conformal member of a broader family of cosine-rearranging maps.","Any rotationally symmetric target with a continuous, strictly increasing projected cdf—such as the Poisson kernel or spherical cardioid—yields a tractable generalized Möbius transformation with closed-form map and exact simulation.","Möbius-transported von Mises–Fisher distributions give anisotropic spherical models whose normalizing constants are fully explicit, avoiding the infinite-series constants of the classical Kent distribution.","The isotropic scaled von Mises–Fisher distribution is identified as a Möbius transport, explaining its good fits to paleomagnetic and comet data within the same framework.","Composing generalized Möbius transformations with different axes produces skewed or bimodal spherical densities in closed form, opening a route to multi-axis models."],"fun_headline_variants":["Möbius transport: from Cauchy to any spherical law","Optimal transport view of spherical Möbius maps","Generalizing Möbius maps for anisotropic spheres","Möbius transport outperforms classical spherical models"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The empirical outperformance claim rests on the preprocessing step that reflects reversed-polarity paleomagnetic directions to a common polarity; if that merge is not a faithful representation of a single underlying population, the reported skewness and the AIC/BIC ranking of Möbius–von Mises–Fisher over competitors could be artifacts.","fun_headline_variants_meta":{"raw":{"variants":["Möbius transport: from Cauchy to any spherical law","Optimal transport view of spherical Möbius maps","Generalizing Möbius maps for anisotropic spheres","Möbius transport outperforms classical spherical models"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000606,"raw_usage":{"total_tokens":2620,"prompt_tokens":663,"completion_tokens":1957,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":407,"completion_tokens_details":{"reasoning_tokens":1891}},"tokens_in":407,"tokens_out":1957,"duration_ms":14019,"temperature":1.0,"reasoning_tokens":1891,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T10:07:04.068929+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Re-fit the paleomagnetic data on the normal and reversed polarities separately, or with an alternative polarity-reversal criterion; if the Möbius–von Mises–Fisher model no longer achieves the lowest AIC/BIC, the empirical conclusion is an artifact of the merging step. A second check: compute the Jacobian of the generalized map for a Poisson-kernel target on S^2 at a point with cosine not ±1; if it is conformal, the uniqueness claim for the spherical Cauchy target is false.","supporting_citations":[],"review_version":1}