{"id":"78935530-3f89-434f-9187-c0e527189963","arxiv_id":"1908.06794","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For centers outside the sphere, the shifted Funk transform reduces, via a Möbius change of variables, to a parallel slice transform, yielding exact injectivity and inversion results.","lead":"This mathematics paper studies two families of spherical slice transforms and proves they are equivalent when the slicing planes pass through a point outside the sphere. It provides explicit reconstruction formulas that complete the previously open exterior-center case of the shifted Funk transform.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The core identity (5.2) rests on two imported analytic facts, (6.3) and (6.4), whose proofs are deferred to [2]; an independent spot-check confirms both, so the gap is one of completeness, not correctness.","rationale":"The reader's weakest_assumption correctly identifies the only non-self-contained part of the proof of the central claim: the imported identities (6.3) and (6.4). My independent check of (6.3) in the simplest nontrivial case confirms the claimed normalization, and (6.4) is a standard M\\\"obius Jacobian identity. Therefore I do not see a correctness failure in the main theorem, and the architecture of the paper is credible. The remaining issue is that the paper explicitly omits the proof of (6.3) and cites (6.4) to an arXiv preprint, which under a strict completeness standard justifies the CONDITIONAL verdict. If the authors supply these derivations or a published reference, the proof would be complete. The reader's identification of the weak spot is accurate, and the recommended verdict should not change.","tokens_in":14705,"tokens_out":29276,"duration_ms":291480,"concrete_test":"Derive (6.3) from scratch using the coarea formula for the map F(x) = \\xi'(x - a) on S^n: the tangential Jacobian is \\sqrt{1 - |q|^2} at level q, so the regularized integral tends to the unweighted slice integral; for f = 1 this gives 2\\pi while F_a(1)(\\tau_\\xi) = 2\\pi\\sqrt{1 - |\\xi'a|^2}, confirming the factor (1 - |\\xi'a|^2)^{-1/2}. Separately, verify (6.4) by computing the Jacobian of the spherical automorphism \\phi_{a*}, which should be (s_{a*}/(1 - a*\\cdot y))^n. If both checks reproduce the displayed factors, then (5.2) is sound and the reader's conditional verdict need only insist that these derivations be included or explicitly referenced in a published source.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 5.1's equality (5.2) is the load-bearing step: from it the paper derives injectivity on {f = W_a f}, the exact kernel {f = -W_a f}, and the inversion formula (5.11). Section 6 establishes (5.2) by comparing two limits, but two key ingredients are not proved here: equation (6.3), whose proof is explicitly described as 'a verbatim copy of Step I in the similar proof in [2, Section 6], and we skip it', and the change of variables (6.4), cited from [2, Lemma 2.1]. If either identity has a normalization error, the comparison in Section 6 collapses and with it all of Theorems 5.3 and 5.4. I checked the most delicate part, (6.3), in the model case k = 2, f = 1: the coarea formula gives lim_{\\epsilon -> 0} (F_{a,\\epsilon} 1)(\\xi) = 2\\pi, while (F_a 1)(\\tau_\\xi) = 2\\pi\\sqrt{1 - |\\xi'a|^2}, so the stated factor (1 - |\\xi'a|^2)^{-1/2} is exactly right. Equation (6.4) is the standard Jacobian identity for the M\\\"obius automorphism \\phi_{a*}. Thus the concern is not that the imported identities are false; it is that they carry the whole proof of the paper's central theorem without being derived in the manuscript. This is a real completeness gap under a strict standard, but not evidence of a mathematical error.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies two families of Funk-type transforms on the unit sphere: the shifted Funk transform F_a with exterior center |a|>1, integrating over k-dimensional plane sections through a fixed point a, and the parallel slice transform Π_a, integrating over k-planes parallel to a fixed vector. The central result, Theorem 5.1, establishes the identity F_a f = (Π_a M_{a*} f)∘φ_{a*} with an explicit weight M_{a*} and the Möbius automorphism φ_{a*}. From this identity the authors derive injectivity of F_a on functions satisfying f = W_a f, the exact kernel {f = -W_a f}, and an explicit inversion formula. Section 3 reduces Π_a to the Radon-John d-plane transform on the Euclidean ball and gives inversion formulas, and Section 7 establishes a relation between shifted Funk transforms of different dimensions. The proof of the core identity is deferred to Section 6 and relies on two analytic facts imported from the authors' previous arXiv preprint [2].","tokens_in":14852,"tokens_out":15609,"duration_ms":151068,"significance":"If the results are correct, the paper completes the structural understanding of exterior-center Funk transforms: it gives not only injectivity but an exact description of the kernel and an explicit inversion procedure on the injective subspace. The reduction of the parallel slice transform to the Radon-John transform is self-contained, parameter-free, and connects the topic to classical integral geometry. The main weakness is completeness: the core identity (5.2) rests on two imported identities whose proofs are not included in the manuscript. I independently checked the less obvious of these, equation (6.3), in a model case and found it correct; the concern is therefore about the manuscript's self-containedness rather than about mathematical error.","major_comments":[{"comment":"The limit identity lim_{ε→0}(F_{a,ε}f)(ξ) = (1-|ξ'a|^2)^(-1/2)(F_a f)(τ_ξ) is the normalization step that makes the comparison in the proof of Theorem 5.1 work. Its proof is explicitly omitted: the text states it is 'a verbatim copy of Step I in [2, Section 6], and we skip it'. Since [2] is an arXiv preprint rather than a published source, this identity should either be proved in the present paper or supplied with a citable published reference. As it stands, the central equality (5.2), and with it Theorems 5.3 and 5.4, rests on an unverified imported fact.","section":"Section 6, Eq. (6.3)"},{"comment":"The change-of-variables formula ∫_{S^n} f(x)dx = s_{a*}^n ∫_{S^n} (f∘φ_{a*})(y)(1-a*·y)^(-n)dy, cited from [2, Lemma 2.1], is equally load-bearing: it converts the defining integral of F_{a,ε} into the y-integral on which all subsequent steps act. The identity follows from the Jacobian of φ_{a*} given in (4.3), but the manuscript should either prove it or cite a published source; a citation to an unpublished companion preprint is not sufficient for a core step of the main theorem.","section":"Section 6, Eq. (6.4)"},{"comment":"The step 'the corresponding function φ in (3.7) is zero' uses injectivity of the Radon-John transform, but the hypotheses needed to apply Theorem 2.1 are not stated. For continuous f, the function φ(y)=2(1-|y|^2)^(-1/2)f(y+√(1-|y|^2)̃a) is in L^1 on the ball, and because k-1<n, the condition 1≤p<n/(k-1) holds with p=1. This is an easily repairable gap, but it should be made explicit so that the kernel characterization (3.8) is fully justified.","section":"Section 3, Theorem 3.2"}],"minor_comments":[{"comment":"Theorem 3.3 states 'a ∈ B^{n+1} \\ {0}', while Section 3 opens with the assumption |a|>1 and Theorem 5.4 needs the inversion formula (3.10) for an exterior center. The intended domain appears to be a ≠ 0, and the statement should be corrected to remove this ambiguity.","section":"Section 3 and Section 5"},{"comment":"The displayed equation '(Fa,εf )(ξ) =' is missing a left parenthesis; it should read '(F_{a,ε}f)(ξ) ='.","section":"Section 6, displayed formula after h_φ(z)"},{"comment":"There are several typographical errors: 'latt er' should be 'latter', 'inegrate' should be 'integrate', 'non-injectv ity' should be 'non-injectivity', and 'arbirary' should be 'arbitrary'.","section":"Abstract and Introduction"},{"comment":"Reference [2] is an arXiv preprint. If the authors prefer not to include full proofs of (6.3) and (6.4), they should at least update the reference to its published version once available, or add a precise statement of the cited lemma in the present paper.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The main mathematical claims appear correct; my independent spot-check of equation (6.3) in the model case k=2, f=1 confirms the normalization factor. The issue is completeness: the core identity is proved by importing two analytic facts from an unpublished companion preprint. If the journal allows reliance on a preprint under review, the completeness concern is less severe, but under a strict self-containedness standard the authors should include these proofs. This is a fixable gap rather than evidence of an error."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Good to see this paper. It closes the open case |a|>1 of the shifted Funk transform, and the main results look right. The key novelty is Theorem 5.1, the identity F_a f = (Π_a M_{a*} f) ∘ φ_{a*}, which turns the exterior-center problem into the parallel slice transform. From there Theorem 5.3 gives injectivity on {f=W_a f} and the exact kernel {f=-W_a f}, and Theorem 5.4 gives an explicit inversion formula. The parallel slice transform is itself treated in full generality (all 1<k≤n) in Section 3, with a clean reduction to the Radon-John transform. I also appreciate the honest handling of the kernel: they don't claim injectivity where there is a nontrivial null space.\n\nThe proof of Theorem 5.1 in Section 6 is detailed and mostly self-contained, but two analytic facts are imported rather than derived. Equation (6.3), the limit relation identifying the regularized integral with (1-|ξ'a|^2)^(-1/2) times F_a f, is explicitly skipped: 'the proof ... is a verbatim copy of Step I ... and we skip it.' Equation (6.4), the change-of-variables under the Möbius automorphism, is cited from the same authors' arXiv preprint [2, Lemma 2.1]. Since (5.2) carries the kernel and inversion theorems, this is a real completeness gap under strict standards. That said, the stress-test spot-check of (6.3) in the case f=1, k=2 gives exactly the stated normalization, and (6.4) is a standard Jacobian identity. I found no sign of a mathematical error or circular reasoning; the deferred facts are parameter-free and the rest of the Section 6 computation checks out.\n\nThe citation pattern is fine. Self-citations are to the companion paper [2] where these facts are proved or stated; that is legitimate in a research program, though it would be cleaner to include the proof of (6.3) or cite a published source.\n\nWho is this for? People working on spherical integral geometry and tomography. It deserves a serious referee. My recommendation: send it out; the referee should ask the authors to supply the missing proof of (6.3) or make [2] available in final form before acceptance. Verdict: conditional accept, not desk reject.","headline":"Solid paper that closes the previously open exterior-center case of the shifted Funk transform; the main caveat is that the core identity's proof imports two analytic facts from the authors' own arXiv preprint, but spot-checks suggest the imported identities are correct.","tokens_in":15680,"tokens_out":2202,"would_cite":true,"duration_ms":21116,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["44A12","37E30"],"pacs":[],"model":"deepseek-v4-flash","headline":"For an exterior center, the shifted Funk transform on the sphere is conjugate to the parallel slice transform; the paper derives from this an explicit inversion and an exact description of the kernel.","keywords":["shifted Funk transform","parallel slice transform","Radon-John d-plane transform","spherical tomography","integral geometry","inversion formula","Möbius automorphism","injectivity"],"falsifier":"Take a concrete test function and an exterior center, e.g. $f\\equiv 1$ on $S^n$ with $n=2$, $k=2$, $a=(2,0,0)$, and compute both sides of (5.2) directly by elementary integration for a choice of $\\tau$; the two sides must agree exactly. Alternatively, verify the skipped ingredient (6.3) numerically for the same data: the limit of the smoothed transform on the left must equal $(1-|\\xi'a|^2)^{-1/2}(F_a f)(\\tau_\\xi)$. A mismatch at any single $\\xi$ with $|\\xi'a|<1$ would break the chain.","tokens_in":14255,"feed_emoji":"📐","tokens_out":12319,"duration_ms":104846,"temperature":0.7,"pith_summary":"This paper treats two families of integral operators on the unit sphere $S^n$: the shifted Funk transform $F_a$, which integrates a function over $k$-dimensional plane sections passing through a point $a$ outside the sphere, and the parallel slice transform $\\Pi_a$, which integrates over $k$-planes parallel to a fixed direction. Its aim is to decide when these transforms are injective and to reconstruct the function from its integrals. The paper's central claim is that for $|a|>1$ the exterior-center transform is equivalent, through a Möbius change of variables together with a weight factor, to the parallel slice transform. From that equivalence it obtains an explicit inversion formula on the subspace of functions satisfying $f=W_a f$ and proves that the null space consists exactly of the functions satisfying $f=-W_a f$. This matters because spherical section integrals arise in spherical tomography, where knowing exactly which component of the function is invisible can matter as much as having a reconstruction formula.","feed_headline":"Outside-center sphere scans get exact inversion formulas","feed_subtitle":"A Möbius conjugation turns exterior-center Funk transforms into parallel slices, exposing their kernel and inverse.","key_machinery":"The central machinery is the conjugation identity (5.2) itself. Its main pieces are the Möbius automorphism $\\phi_{a_*}$, defined by (4.1) with $a$ replaced by $a_*$, which realizes a bijection between the exterior-plane family $T_a(n+1,k)$ and the parallel-plane family $Z_a(n+1,k)$; the weighted composition operator $M_{a_*}$, which absorbs the Jacobian of the change of variables; and the parallel slice transform $\\Pi_a$, which Lemma 3.1 reduces to the Radon-John $d$-plane transform on the ball in $a^\\perp$. The equality converts every statement about exterior-center Funk data into a statement about parallel-slice data, so injectivity, kernel, and inversion for $F_a$ follow from the corresponding Radon-John facts.","core_discovery":"The core discovery is the identity (5.2): for $f\\in C(S^n)$, $1<k\\le n$, and $|a|>1$, $$(F_a f)(\\tau)=(\\Pi_a M_{a_*} f)(\\phi_{a_*}\\tau),$$ where $a_*=a/|a|^2$, $\\phi_{a_*}$ is the involutive Möbius automorphism that sends the $k$-planes through $a$ to $k$-planes parallel to $a$, and $M_{a_*}$ is the weighted composition $(M_{a_*}f)(y)=(s_{a_*}/(1-a_*\\cdot y))^{k-1}(f\\circ\\phi_{a_*})(y)$. The paper calls this equality 'the core of the paper'. It implies that $F_a$ is injective exactly on the subspace $f=W_a f$, that its kernel is $\\{f:\\,f=-W_a f\\}$, where $W_a$ is an involution built from the chord-reflection map $\\tau_a$ and the weight $((|a|^2-1)/|a-x|^2)^{k-1}$, and that reconstruction is given by $f=M_{a_*}^{-1}\\Pi_a^{-1}((F_a f)\\circ\\phi_{a_*})$.","pith_inferences":["Beyond the paper: if (5.2) holds, it suggests a direct numerical inversion algorithm for exterior-center spherical tomography: invert the parallel-slice data on the ball using any Radon-John solver, then pull back through the Möbius map; no new quadrature over exterior planes is needed.","Beyond the paper: the same conjugacy idea could apply to other Möbius-invariant integral transforms on $S^n$; any transform whose sections are the images of parallel planes under a spherical automorphism should admit a parallel-slice representation with a computable Jacobian.","Beyond the paper: the paired-transform idea mentioned in the introduction, with one center inside and one outside the sphere, would likely give injectivity on all of $C(S^n)$, since the outside transform recovers the $W_a$-even component and an inside transform can recover what the exterior one loses."],"forward_implications":["For every exterior center $a$, every $f\\in C(S^n)$ satisfying $f=W_a f$ can be reconstructed from its shifted Funk data by the explicit chain $f=M_{a_*}^{-1}\\Pi_a^{-1}((F_a f)\\circ\\phi_{a_*})$.","The kernel of the exterior-center transform is exactly the set of solutions of $f=-W_a f$, so $F_a$ is never injective on all continuous functions; only the $W_a$-even part of a function is recoverable.","Because $\\Pi_a$ is invertible on $C_a^+(S^n)$ via the Radon-John inversion formula, the reconstruction of $F_a$ inherits an explicit algorithmic structure rather than an abstract existence proof.","The dimension-link theorem (7.5) lets one reduce inversion for lower-dimensional sections to inversion for $k=n$, at the price of one extra integration step and a higher-order differential operator in the Radon-John inversion.","In the exterior case the same conjugation pattern that for $|a|<1$ related $F_a$ to the geodesic Funk transform now relates $F_a$ to parallel slices, showing that the qualitative behavior of the transform changes sharply at the sphere boundary."],"supporting_citations":[{"why":"Supplies the two analytic facts on which the proof of (5.2) rests: the smoothing limit relation (6.3), whose proof is stated to be a verbatim copy of Step I in [2, Section 6] and is skipped, and the change-of-variables formula (6.4), cited from [2, Lemma 2.1].","marker":"[2]"},{"why":"Supplies the Radon-John inversion formula quoted as Theorem 2.1; this makes the parallel slice transform invertible and enters the final inversion chain (5.11).","marker":"[21]"},{"why":"Supplies the definition and basic identities for the Möbius automorphisms $\\phi_a$ that underlie Lemma 4.1 and the conjugation identity.","marker":"[25]"},{"why":"Provides the polar decomposition and determinant identity (6.7) used to simplify the Jacobian factor $h$ to $1-|\\xi'a|^2$, completing the proof of (5.2).","marker":"[14]"}],"fun_headline_variants":["Möbius trick inverts exterior-center Funk transforms","Exterior sphere transforms reduced to parallel slices","New inversion formulas for Funk-type sphere integrals","Exterior-center Funk transforms exact inverse via Möbius map"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central equality is proven only by relying on two imported analytic facts: a limit identity whose proof is skipped as a 'verbatim copy', and a change-of-variables formula cited from the authors' earlier work. If either of those is wrong, the paper's main injectivity and inversion results collapse.","fun_headline_variants_meta":{"raw":{"variants":["Möbius trick inverts exterior-center Funk transforms","Exterior sphere transforms reduced to parallel slices","New inversion formulas for Funk-type sphere integrals","Exterior-center Funk transforms exact inverse via Möbius map"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000172,"raw_usage":{"total_tokens":1279,"prompt_tokens":954,"completion_tokens":325,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":570,"completion_tokens_details":{"reasoning_tokens":262}},"tokens_in":570,"tokens_out":325,"duration_ms":3732,"temperature":1.0,"reasoning_tokens":262,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:35:58.828820+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a concrete test function and an exterior center, e.g. $f\\equiv 1$ on $S^n$ with $n=2$, $k=2$, $a=(2,0,0)$, and compute both sides of (5.2) directly by elementary integration for a choice of $\\tau$; the two sides must agree exactly. Alternatively, verify the skipped ingredient (6.3) numerically for the same data: the limit of the smoothed transform on the left must equal $(1-|\\xi'a|^2)^{-1/2}(F_a f)(\\tau_\\xi)$. A mismatch at any single $\\xi$ with $|\\xi'a|<1$ would break the chain.","supporting_citations":[{"cited_title":"Non-geodesic Spherical Funk Transforms with One and Two Centers","cited_arxiv_id":"1904.11457","evidence_quote":"Supplies the two analytic facts on which the proof of (5.2) rests: the smoothing limit relation (6.3), whose proof is stated to be a verbatim copy of Step I in [2, Section 6] and is skipped, and the change-of-variables formula (6.4), cited from [2, Lemma 2.1]."},{"cited_title":"Rudin, Function theory in the unit ball of Cn, Springer-Verlag, New York, NY, 1980","cited_arxiv_id":null,"evidence_quote":"Supplies the definition and basic identities for the Möbius automorphisms $\\phi_a$ that underlie Lemma 4.1 and the conjugation identity."},{"cited_title":"Muirhead, Aspects of multivariate statistical theory , John Wiley & Sons","cited_arxiv_id":null,"evidence_quote":"Provides the polar decomposition and determinant identity (6.7) used to simplify the Jacobian factor $h$ to $1-|\\xi'a|^2$, completing the proof of (5.2)."}],"review_version":1}