{"id":"050b68d6-8348-4b76-88a3-191b8b7cafb0","arxiv_id":"2607.02943","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Weighted spherical data are decomposed into nested information gaps—vMF mean direction, quadratic anisotropy, and higher harmonics—so structure missed by a single von Mises-Fisher fit becomes visible.","lead":"This paper introduces a way to summarize how structured a set of weighted points on a sphere is, by breaking their 'information content' into levels: a mean direction first, then oval or belt-like shapes, then finer multimodality. It matters whenever people currently reduce spherical data to one concentration number, as in uncertainty quantification, importance sampling, or embedding analysis.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The decomposition is undefined on boundary moments (point masses, exact antipodal pairs, zero-thickness girdles); the paper's interior-moment assumption is load-bearing because the proposed shrinkage changes the target, so the 'any weighted empirical measure' scope overreaches.","rationale":"The paper's central result, Eq. (3), is an instance of the Pythagorean identity for nested exponential families, and the proof in C.2 is mathematically sound given the stated interior-moment and identifiability assumptions. The reader's CONDITIONAL verdict is appropriate. I considered whether the more serious flaw is the unassessed finite-sample remainder in the null-calibration experiment (A.1 explicitly says the quadratic surrogate does not assess the fitted gap's Taylor remainder). That is a real validation gap but not a load-bearing objection to the central claim: Theorem 8 is proven asymptotically, and the surrogate is second-order equivalent; only finite-sample fidelity is untested, which is a standard limitation. The boundary issue is more fundamental because it concerns the definition of p^P_L itself. A weighted empirical measure with exact antipodal point masses, a thin girdle, or small support has moments on ∂M_L; then the maximum-entropy density does not exist and Eq. (3) has no meaning. The paper's B.3 shrinkage replaces the target with a mixture, so the numbers reported are for the shrunken measure, not the original empirical measure. This does not invalidate the theorems but it does mean the advertised scope ('weighted empirical measures on the sphere') is narrower than it appears, and the motivating examples are precisely the sharp limits where the boundary is hit. The experiments use smooth continuous blobs, so they do not demonstrate behavior at or near the boundary. A concrete convergence test on antipodal point-mass approximations would settle whether the boundary is a removable singularity. Because the paper already states the interior assumption and offers a shrinkage remedy, the reader's CONDITIONAL verdict captures this; I recommend no change.","tokens_in":15605,"tokens_out":16931,"duration_ms":179462,"concrete_test":"On S^2, let P_ε = (1/2)vMF(e3, 1/ε) + (1/2)vMF(-e3, 1/ε) for ε ∈ {0.1, 0.01, 0.001}, and also form the exact boundary measure P* = 0.5δ_{e3}+0.5δ_{-e3}. Compute I_2(P_ε) with numerically converged integration (e.g., 10^6-point Fibonacci grid, checking grid doubling) and compare with the B.3 shrinkage estimate for P* at the same ε (i.e., Phat P_{w,ε} = (1-ε)P* + εν). If I_2(P_ε) → ∞ as ε→0 and the shrinkage result is materially different from the limiting interior sequence, then exact GID has no continuous boundary extension and the practical reach claim requires qualification. If I_2(P_ε) converges to the Legendre-transform value, boundary is removable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3 states: 'The main results therefore assume that each relevant moment lies in ri(M_L).' This is not a technical formality. The central object p^P_L is the maximum-entropy density subject to moment matching; C.1 maps R^q diffeomorphically only onto ri(M_L), so for m in ∂M_L no finite natural parameter exists. Section 6.1 itself notes bR=0 leaves the vMF direction unidentified and bR=1 gives no finite fit. The same obstruction appears at level 2: for P* = 0.5δ_{e3}+0.5δ_{-e3}, the traceless quadratic moment equals that of a point mass at e3, an extreme point of M_2, so p^P*_2 does not exist. Thus Eq. (3) and the whole profile are undefined for exact antipodal point masses, great-circle/girdle support, or any support with fewer affinely-independent feature points than q_L. Section B.3's remedy replaces P by (1-ε)P+εν, which forces interior moments but changes the measured target; the user must report ε and sensitivity. The experiments use smooth vMF blobs and quadrature grids, so they never expose this. This is a scope limitation, not an internal inconsistency: within ri(M_L), Theorem 1's Pythagorean proof is clean. But the title/abstract's 'weighted empirical measures' is broader than the regime in which GID is actually defined, and the boundary is exactly where the motivating antipodal/girdle examples live in the sharp limit.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces geometric information decomposition (GID) for probability measures on the sphere, and more generally on compact Riemannian manifolds. The construction fits nested maximum-entropy exponential families to spherical harmonic and related feature spaces: level 1 recovers the von Mises-Fisher mean-direction information, level 2 captures Fisher-Bingham-type quadratic anisotropy (antipodal, axial, girdle structure), and higher levels describe finer harmonic structure. The central structural result is Theorem 1, which proves the monotonicity of entropy deficits and the KL-gap identity I_L(P)=D_L(P)-D_{L-1}(P)=KL(p_L^P nu || p_{L-1}^P nu), so that the cumulative deficit decomposes into the level-wise gaps. The paper establishes basis invariance, isometry invariance, consistency of plug-in estimators, delta-method asymptotic normality away from zero gaps, and a second-order null calibration with chi-square and weighted-chi-square limits. Simulation experiments on S^1 and S^2, including importance-weight calibration, and a query-weighted digit projection are reported. The key regularity condition is that all relevant moment vectors lie in the relative interior of the moment body.","tokens_in":15888,"tokens_out":11495,"duration_ms":119619,"significance":"If the stated scope is properly qualified, the paper is a solid contribution to directional statistics and information geometry. The mathematical core is sound: the proofs in Appendix C are standard and correct, the KL-gap identity follows from the nesting assumption rather than being assumed, and the delta-method and second-order null-calibration derivations check out. The experiments are carefully specified with seeds, Monte Carlo standard errors, quadrature sensitivity checks, and explicit statements about what is and is not being tested. The code is provided. The main caveat is that the GID profile is defined only when the relevant moment vectors are in the relative interior of the moment body; exact antipodal point masses, zero-thickness girdles, and other boundary configurations are outside the stated theory. This is a scope limitation rather than an internal inconsistency, and it is acknowledged in the paper, but it needs to be more prominent because the title and abstract claim a broader scope.","major_comments":[{"comment":"The decomposition is undefined on boundary moment vectors. The definition of p_L^P requires m_L(P) in ri(M_L), and Appendix C.1 shows the natural-parameter map is a diffeomorphism only onto the relative interior. Thus, for a weighted empirical measure supported on an exact antipodal pair or a zero-thickness girdle, the level-2 moment lies on the boundary of M_2 and no finite Fisher-Bingham projection exists; Eq. (3) and the profile are simply not defined. This is not a purely formal edge case: the antipodal and girdle examples in the Introduction are the sharp limits of exactly these boundary configurations. The remedy in B.3, mixing with the uniform measure (1-epsilon) P_w + epsilon nu, forces interiority but changes the target; the gaps then describe the shrunken measure, not P_w. The reported experiments use smooth vMF blobs and quadrature grids and therefore do not exercise the bound","section":"Section 3, Assumption 4 and Section B.3"}],"minor_comments":[{"comment":"The sentence 'At bR=0, bkappa=0 and the direction is unidentified' could be misread as a failure of the first-level projection. For the sphere, bR=0 is actually in the interior of the moment body and the uniform vMF distribution exists; only bR=1 is a true boundary case with no finite fit. Please clarify this distinction.","section":"Section 6.1"},{"comment":"The calibration experiment tests the quadratic surrogate eI2, not the exactly fitted gap bI2. The text explicitly and honestly notes that the finite-sample remainder bI2 - eI2 is not assessed, but the section title 'Null calibration on S^2' may still mislead readers. A sentence in the table caption or section heading clarifying that only the leading quadratic term is validated would help.","section":"Section 8.3 and Table 3"},{"comment":"The theory assumes exact log-partitions and optimizers, while implementation uses fixed quadrature rules. The paper correctly lists the required o_p(a_n^-1) and o_p(a_n^-2) numerical-error rates and reports sensitivity checks. Still, a short main-text remark that the inferential guarantees are asymptotic in both sample size and quadrature error would make the idealization easier for readers to track.","section":"Appendix A.1 and Section 7"},{"comment":"In the displayed equation for the weighted chi-square limit, the matrix S is the Schur complement previously defined, but the notation S^{-1/2} N0 Sigma_L N0^T S^{-1/2} could be misread as using the sample covariance. A parenthetical reminder that S is the Schur complement and Sigma_L is the moment covariance would improve readability.","section":"Theorem 8"},{"comment":"The arXiv header lists a future version date, '29 Jul 2026'. Please ensure the final version has the correct submission/revision dates.","section":"Header"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the manuscript is honest and technically sound. The boundary-moment issue is the only substantive concern; it is a scope problem rather than an error in the proofs. If the authors agree to foreground the interior-moment condition and the shrinkage protocol in the main text and adjust the title/abstract wording accordingly, I would be comfortable with acceptance. I found no circularity or unsupported invention in the central derivations."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a legitimate new diagnostic, not a paradigm shift. The KL-gap identity is classical (Csiszár, Amari and the Pythagorean decomposition for nested exponential families), and harmonic exponential families already exist. What the paper adds is packaging that identity as a hierarchical uncertainty profile for weighted spherical measures, with careful asymptotics for the alternative regime and a second-order null calibration. That packaging is done well. Theorems 1–3 are clean; the delta-method and Schur-complement derivations in Theorems 7–8 check out, including the factor of 2 and the chi-square limit. The experiments are transparent: seeds, replicate counts, Monte Carlo standard errors, no rejected runs, and the paper explicitly states that the calibration experiment uses a quadratic surrogate rather than the exact fitted gap. That kind of honesty is worth acknowledging.\n\nThe main soft spot is the interior-moment assumption. Section 3 is explicit that the results require m_L(P) ∈ ri(M_L), and Section B.3 admits exact GID is undefined on the boundary and offers a mixture shrinkage that changes the target measure. The stress-test note is right that this is not a formality: exact antipodal point masses, great-circle support, or point-mass support put the moment on the boundary, and those are exactly the motivating \"antipodal/girdle\" cases in the sharp limit. The paper's own experiments use smooth vMF blobs and bounded girdles, so they never hit the boundary. This is a scope limitation, not an internal inconsistency. The title and abstract say \"weighted empirical measures\" more broadly than the theorem regime; the authors should either narrow the claims or give the boundary treatment more prominence. I would not call it fatal. For any reasonable continuous sample or smooth blob, boundary moments have measure zero; the sharp cases are discrete support restricted to a subspace.\n\nOther caveats are minor and mostly stated: the null-calibration experiment uses the surrogate, not the fitted gap, so the finite-sample remainder is unassessed; code is illustrative, no commit hash; and full harmonic fitting in high dimensions is impractical, which they acknowledge.\n\nWho gets value: people doing directional statistics, weighted embedding analyses, or uncertainty decomposition on spheres who want a principled way to separate mean-direction information from higher-order structure. It is a useful subfield tool, mostly correct within its stated assumptions. The proofs are standard but careful, and the paper deserves a serious referee. I would send it to review, with a request to reframe the boundary scope and make the Section B.3 target-changing remedy more prominent.","headline":"A solid, honestly scoped method paper: the GID machinery works as stated on interior moments, the boundary caveat is real but acknowledged, and it deserves a careful referee.","tokens_in":16474,"tokens_out":3068,"would_cite":true,"duration_ms":34696,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62H11","62B10","60D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"For weighted data on a sphere, the paper claims the entropy deficit from uniformity can be decomposed into levelwise information gaps, with the first gap recovering vMF mean-direction information, the second isolating quadratic anisotropy,","keywords":["geometric information decomposition","maximum entropy","von Mises-Fisher","Fisher-Bingham","spherical harmonics","directional statistics","KL divergence","weighted empirical measures"],"falsifier":"Generate a large sample from a symmetric three-mode distribution on the circle (e.g., equal von Mises components at 0, 2π/3, 4π/3) and compute the Fourier gap profile with accurate quadrature. The theory predicts I_3 is the dominant gap while I_1 and I_2 are near zero. If the observed I_2 is comparable to or larger than I_3 at large n, the claimed harmonic-level separation of structure would be refuted.","tokens_in":15350,"feed_emoji":"🧭","tokens_out":4249,"duration_ms":50024,"temperature":0.7,"pith_summary":"The paper proposes that a weighted point cloud on a sphere should be summarized not by a single concentration number but by a sequence of information gaps, each measuring how much entropy is removed by adding one level of harmonic features. It establishes the key identity that the cumulative entropy deficit equals the sum of KL divergences between successive maximum-entropy fits, making this profile a true decomposition into first-order, second-order, and higher-order angular information. The first gap recovers the vMF mean-direction summary, the second gap isolates antipodal, axial, elliptical, and girdle structure, and later gaps reveal finer multimodality. This matters because vMF entropy alone can badly understate uncertainty when structure is antipodal or multimodal. The paper proves invariance, consistency, normal limits away from zero gaps, and chi-square-type null calibration, and shows experimentally that the gaps separate these geometric cases.","feed_headline":"Three gaps reveal hidden spherical structure beyond mean direction","feed_subtitle":"A single identity splits spherical entropy into ordered layers, exposing axial and multimodal structure vMF misses.","key_machinery":"The central object is the nested family of maximum-entropy (exponential-family) projections p_L that match the moments of the measure P on feature space V_L. The mechanism carrying the argument is the information-geometric Pythagorean identity for nested exponential families: the KL divergence between successive projections equals the entropy gap I_L. On the sphere, the feature spaces are the spans of spherical harmonics through degree L, with V_1 giving vMF, V_2 giving Fisher-Bingham-type quadratic structure, and higher levels giving finer angular detail. The estimable quantities are the plug-in gaps computed from weighted empirical moments, with inference relying on the moment CLT and seco","core_discovery":"The central claim is that for any weighted empirical measure on a compact manifold, nested maximum-entropy projections onto feature spaces V_0 ⊂ V_1 ⊂ ⋯ give I_L(P) = D_L(P) − D_{L−1}(P) = KL(p_L ν ∥ p_{L−1}ν), so the cumulative entropy deficit D_L is exactly the sum of levelwise gaps. On the sphere, level 1 is the vMF fit, level 2 adds traceless quadratic features to capture Fisher-Bingham-type anisotropy, and levels 3 and above use spherical harmonic exponential families. The profile I_1, I_2, I_3, … therefore separates mean-direction information from axial, girdle-like, and finer angular structure that a vMF fit misses. The paper also establishes that the decomposition is invariant under","pith_inferences":["Extension: because the identity holds for any compact manifold and any nested feature spaces, the same gap profile could be built on tori, projective spaces, or rotation groups, as long as interior moments and nested subspaces are available.","Extension: for high-dimensional embeddings where full harmonic fitting is impractical, the paper's monotonicity result implies that a structured subspace (low-rank, diagonal, or sketched quadratics) gives a lower bound on the full information gap, so the difference between structured and full gaps measures omitted information.","Extension: the profile offers a natural diagnostic for attention-weighted or importance-weighted representations: a dominant I_2 or I_3 indicates geometric structure that a single softmax-like direction cannot capture, which could be tested as a practical screening rule.","Extension: a concrete follow-up is to analyze the bias-variance tradeoff of the epsilon-mixture boundary remedy, since the paper leaves the shrinkage rate and its effect on asymptotic inference open."],"forward_implications":["When a vMF fit is adequate, I_1 dominates and the later gaps are near zero; when structure is antipodal or girdle-like, I_2 is the large gap; when structure is trimodal or tetrahedral, the signal first appears at I_3 or higher.","The profile is invariant under rotation and basis changes, so it describes the geometry of the distribution rather than its coordinate representation.","Under the null at level L, the standardized plug-in gap converges to a weighted sum of chi-square variables; with uniform weights and correct specification, 2n Î_L converges to χ²_q, giving a formal test that the new feature level adds no information.","With informative weights, a naive chi-square calibration fails and a sandwich-covariance quadratic form is needed to control Type I error, as the calibration experiments demonstrate.","The same construction applies to any compact manifold with nested feature spaces, making the decomposition a general tool for uncertainty assessment beyond the sphere."],"fun_headline_variants":["Entropy gaps expose spherical structure vMF misses","Three gaps split spherical entropy into meaningful layers","Geometric decomposition reveals axial and multimodal hidden structure","vMF only captures mean direction; gaps reveal the rest","Spherical entropy decomposition uncovers beyond-mean structure"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The main load-bearing assumption is that every relevant moment vector lies in the relative interior of its convex moment body; if a weighted sample's moments sit on the boundary (e.g., equal antipodal masses, point mass, or a great-circle/girdle support), the maximum-entropy projection is degenerate or undefined, and the main results apply only after mixture shrinkage that changes the target measure.","fun_headline_variants_meta":{"raw":{"variants":["Entropy gaps expose spherical structure vMF misses","Three gaps split spherical entropy into meaningful layers","Geometric decomposition reveals axial and multimodal hidden structure","vMF only captures mean direction; gaps reveal the rest","Spherical entropy decomposition uncovers beyond-mean structure"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00032,"raw_usage":{"total_tokens":1623,"prompt_tokens":712,"completion_tokens":911,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":456,"completion_tokens_details":{"reasoning_tokens":838}},"tokens_in":456,"tokens_out":911,"duration_ms":8302,"temperature":1.0,"reasoning_tokens":838,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T08:54:03.272885+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Generate a large sample from a symmetric three-mode distribution on the circle (e.g., equal von Mises components at 0, 2π/3, 4π/3) and compute the Fourier gap profile with accurate quadrature. The theory predicts I_3 is the dominant gap while I_1 and I_2 are near zero. If the observed I_2 is comparable to or larger than I_3 at large n, the claimed harmonic-level separation of structure would be refuted.","supporting_citations":[],"review_version":2}