{"id":"2bf370d7-e586-42b6-a511-1a5b73a822aa","arxiv_id":"2506.18911","paper_version":3,"verdict":"REJECT","confidence":"HIGH","novelty_score":2.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper argues that inverse Radon reconstruction requires complex direct Radon images and proposes a partial Fourier transform trick plus a claimed holonomy effect to generate that complexity.","lead":"This paper studies when Radon transforms, used in CT scans, should give complex rather than purely real image data, and proposes methods to generate that complexity. It matters because the author argues these extra complex parts improve reconstruction from limited-angle measurements.","discovery_kind":"incremental","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 3's 'non-trivial holonomy' is an artifact: (3.6)-(3.9) continue the rotation using R[g2] instead of R[G], while R[G](τ,φ+2π)=R[G](τ,φ) by 2π-periodicity, so the claimed complexity mechanism collapses.","rationale":"The paper's central claim is that direct Radon transforms must be complex for the universal inversion to work. The only argument that this complexity is intrinsic rather than imposed is the 'non-trivial holonomy' of Section 3, so that is the load-bearing point. The Reader's weakest-assumption assessment is exactly right: the mismatch in (3.9) comes from switching functions at the intermediate angle. A fixed G obeys R[G](τ,φ+2π)=R[G](τ,φ); a valid loop can never see holonomy. The paper's own equation (3.7) computes R[g2](τ,φ+2π), not R[G](τ,φ+2π), so no contradiction with 2π-periodicity is established. I also note (3.5) conflates R[g1] with R[g1ΘI], which reinforces the same function-switching pattern. The hybrid Wigner-like construction in Section 5 is mathematically harmless: a Fourier transform in one coordinate turns a real f into complex F, and (5.3) then has complex Radon data. But this is a change of representation, not evidence that complexity is 'needed.' In standard CT the measured projections are real line integrals of f; introducing F is optional and does not follow from the universal inverse representation given here, which relies on the author's earlier papers [3,4]. Thus the central claim is not supported. The paper does contain a correct elementary derivation of the Fourier slice theorem in Section 2 and a correct observation that a partial Fourier transform yields complex intermediate quantities; those parts deserve credit. But the advertised mechanism for intrinsic complexity fails at the holonomy step, and no numerical or independent verification is supplied. The REJECT verdict stands; no change to the Reader's verdict is needed.","tokens_in":7132,"tokens_out":9654,"duration_ms":76298,"concrete_test":"Take g1=g2=1, ΘI={x1≥0,x2≥0}, ΘIII={x1<0,x2<0}, and G=g1ΘI+g2ΘIII. For τ=1 and φ=π/4, evaluate R[G](1,π/4), R[G](1,π/4+π), and R[G](1,π/4+2π) directly from definition (2.3). The last must equal the first by 2π-periodicity. Then compute R[g2](1,π/4+2π)=0, the value used for the second leg in (3.7). If the two 'paths' differ only because the second path replaced R[G] by R[g2], the claimed holonomy is refuted. These integrals reduce to line-segment lengths, so the check is exact and analytic.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing defect is in Section 3. For a fixed function G, the Radon transform satisfies R[G](τ,φ+2π)=R[G](τ,φ) because n_{φ+2π}=n_φ, so no holonomy is possible. Equations (3.5)-(3.9) nevertheless claim one. Granting the support assumptions, the only valid identity at the intermediate angle is R[G](τ,φ+π)=R[g2](τ,φ+π) (with g2 understood with its ΘIII support). Continuing the path must use the same function G: R[G](τ,φ+2π)=R[G](τ,φ)=R[g1](τ,φ). Instead, the paper continues with R[g2](τ,φ+2π) and obtains 0, producing the inequality (3.9). This compares two different functions at the endpoint, not two rotations of the same Radon transform, so it is not holonomy and cannot generate the needed complexity. The same substitution error appears in (3.12). Since the abstract and conclusions advertise the holonomy as a source of complexity, the central claim loses its main independent support; the hybrid Wigner-like construction in Section 5 is a valid but trivial partial Fourier transform and does not show that direct Radon images of the original real object must be complex. In addition, (3.5) silently writes R[g1](τ,φ) for R[g1ΘI](τ,φ), reinforcing the conflation of restricted and unrestricted Radon transforms.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript claims that universal inverse Radon transforms require direct Radon transforms to be complex-valued, and it proposes two mechanisms to generate this complexity: a 'non-trivial holonomy' of Radon transforms for functions with support defects, and the use of 'hybrid (Wigner-like) functions' obtained by Fourier transforming one spatial coordinate. Section 2 re-derives the Fourier slice theorem with a particular ε-regularization of the radial Fourier integral, Section 3 attempts to exhibit a holonomy by rotating a Radon transform by 2π along two paths, Section 4 briefly discusses two-dimensional slices of three-dimensional objects, and Section 5 defines hybrid functions and states a universal inverse formula with real and imaginary parts. The paper concludes that these methods justify treating direct Radon images as complex in reconstruction and optimization.","tokens_in":7477,"tokens_out":4153,"duration_ms":44228,"significance":"If the central claim were established, it would imply that standard tomographic inversion, for real objects, necessarily involves complex-valued Radon data, a nonstandard and potentially impactful statement. However, the paper does not establish this claim. Section 2 correctly re-derives the standard Fourier slice theorem, but this is a textbook result. Section 3's holonomy argument contains a function-substitution error that invalidates the claimed mechanism. Section 5 does show that a partial Fourier transform of a real function is complex, but this is a trivial observation and it changes the object under consideration rather than showing that the Radon transform of the original real function is complex. The manuscript's reliance on the author's prior universal inverse representation [3,4] is also not made self-contained. No machine-checked proofs, reproducible code, or parameter-free derivations are provided; the main novelty claims are therefore not supported by the presented evidence.","major_comments":[{"comment":"The holonomy argument is invalid because it compares Radon transforms of different functions. For a fixed function G, the Radon transform satisfies R[G](τ, φ + 2π) = R[G](τ, φ), since n_{φ+2π} = n_φ. Equation (3.7) computes R[g2](τ, φ + π + π), not R[G](τ, φ + 2π); the latter is R[g1](τ, φ) by (3.8). Thus inequality (3.9) merely states that R[g2](τ, φ + 2π) ≠ R[g1](τ, φ), which is a comparison of two different functions, not a holonomy of a single Radon transform. The same substitution error appears in (3.12). Since the Abstract and Conclusions advertise this holonomy as a source of complexity, the paper's central claim loses its main independent support.","section":"Section 3, Eqs. (3.5)-(3.9)"},{"comment":"The claimed necessity of complex direct Radon transforms is not established for the original real outset function. Section 1 states that the complex measure in the universal inverse representation 'should be compensated' by the complexity of the direct Radon image, but this is an assertion, not a proof. Section 5 then defines a hybrid function F(x1,x2;k) by a one-dimensional Fourier transform of f, notes that F is complex, and computes R[F]. This only shows that the Radon transform of a complex-valued function can be complex; it does not show that the Radon transform R[f] of the original real f must be complex. The reconstruction scheme (5.7) inverts R[F] and then recovers f, but this is a different inverse problem. The argument is therefore circular with respect to the premise from [3,4] and does not provide independent support for the abstract's claim.","section":"Section 1 and Section 5, Eqs. (5.1)-(5.3)"},{"comment":"The assertion that the imaginary part of δ+(η) 'disappears provided the full region of angular integration is considered' is not correct as stated. With δ+(η) = 1/(η - iε), the imaginary part is π δ(η). Substituting this into (2.9) gives an imaginary contribution proportional to ∫ dφ R[f](⟨n_φ, x⟩, φ), which is not generally zero; for example, if f is a point mass at x0, this integral equals 2/|x - x0|. Thus the imaginary part does not vanish under full angular integration, and the claimed cancellation needs an explicit proof or a corrected statement.","section":"Section 2, Eq. (2.10) and following paragraph"}],"minor_comments":[{"comment":"Equation (3.5) writes R[g1](τ, φ) where the integral actually involves g1(⃗x)ΘI(⃗x); the same shorthand appears in (3.6). This notation obscures the distinction between the restricted and unrestricted Radon transforms and contributes to the error in (3.7).","section":"Section 3, Eq. (3.5)"},{"comment":"There are numerous typographical and grammatical errors, including 'r.h.sof', 'semultaneously', 'opimization', 'Ratdon' in the Conclusions, and 'The Introducing' in Section 2. These should be corrected.","section":"Throughout"},{"comment":"The notation 'f.r.' is used without definition; it is later stated to mean 'full regions of variations,' but this should be defined at first use. Similarly, the 'AC-regularization' mentioned in the Introduction is never defined.","section":"Section 1 and Eq. (2.8)"}],"recommendation":"reject","confidential_remarks":"The manuscript's central claim is not supported by its own arguments. The Section 3 holonomy argument is mathematically wrong for the reason stated in the major comments, and Section 5 only replaces the real object by a complex partial Fourier transform. The paper also depends heavily on the author's previous results [3,4] without making the needed universal inverse representation self-contained. This is not a matter of style or presentation; the advertised mechanisms for generating complexity do not survive scrutiny. I recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nRead this one so you don't have to. The punchline: the advertised 'non-trivial holonomy' in Section 3 is an artifact. The paper compares Radon transforms of two different functions and calls the mismatch a holonomy. Once you see that, the central claim loses its main support.\n\nWhat's actually there: Section 2 re-derives the Fourier slice theorem with a particular epsilon-regularization. That's correct but standard. Section 5 introduces a hybrid Wigner-like function: take a partial Fourier transform of f(x1,x2,x3) with respect to x3, get a complex F(x1,x2;k), then take its Radon transform. That's a valid and occasionally useful trick for converting a real reconstruction problem into a complex intermediate one. But it's an elementary operation and doesn't show the original direct Radon image must be complex.\n\nThe soft spot is load-bearing. In (3.5)-(3.9), for G = g1 Theta_I + g2 Theta_III, the authors correctly compute R[G](tau, phi) for tau>0 and phi in [0, pi/2] as R[g1](tau, phi). They then shift to phi+pi and get R[G](tau, phi+pi)=R[g2](tau, phi+pi). Up to here it's fine. The next step shifts R[g2] again, not R[G], and obtains zero, then compares that to R[G](tau, phi+2pi)=R[g1](tau, phi). But R[G] is periodic with period 2pi, and R[G](tau, phi+2pi) is not R[g2](tau, phi+2pi). So (3.9) is just the statement that the Radon transform of g2 at one angle differs from the Radon transform of g1 at another. That's not holonomy, and it can't generate a complex phase. The same substitution error appears in (3.12). The paper's conclusions lean on this result.\n\nThe rest of the complexity argument is circular in a milder way: the claim that direct Radon transforms must be complex is deduced from the author's own universal inverse representation [3,4], and no numerical example is given. The writing is rough, and the JHEP styling doesn't help.\n\nWho is this for? Possibly someone working with the author's prior universal inversion framework who wants a concrete complex representation. The hybrid function trick might be worth citing in passing, but the paper as a whole doesn't support its advertised conclusions. I would not send it to a serious referee without major revision, and the central 'holonomy' claim needs to be withdrawn or completely reworked.\n\nRecommendation: desk reject, or at most a short referee check to confirm the function switch.","headline":"The advertised non-trivial holonomy in Section 3 is an artifact—it compares Radon transforms of two different functions—and the rest of the complexity argument rests on the author's own prior representation, so the central claim is not established.","tokens_in":7983,"tokens_out":5145,"would_cite":false,"duration_ms":46426,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["44A12","42B10","46F10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The universal inverse Radon formula forces direct Radon images to be complex-valued, and hybrid Wigner-like functions supply the missing imaginary part.","keywords":["Radon transform","inverse Radon transform","Fourier slice theorem","hybrid Wigner-like function","complex-valued Radon transforms","computed tomography reconstruction","generalized functions","singular points"],"falsifier":"For a real compactly supported function $f$, compute the hybrid function $F$ from (5.1), its complex Radon image $R[F]$, and invert using only the real principal-value part $F_S$; if the reconstruction reproduces $f$ to the same accuracy as the full inverse $F_S+F_A$, then the imaginary contribution is not needed and the central claim fails.","tokens_in":6905,"feed_emoji":"🩻","tokens_out":8139,"duration_ms":64933,"temperature":0.7,"pith_summary":"The paper argues that the universal, dimension-independent inverse Radon transform contains an imaginary contribution that must be balanced by an imaginary part in the direct Radon image. For a real object, the ordinary direct Radon transform is real, so the needed complexity has to be generated somewhere in the calculation. The paper offers two mechanisms: a non-trivial holonomy of Radon transforms for objects with support defects, and a hybrid Wigner-like function obtained by Fourier transforming one coordinate, which makes the Radon data complex by construction. The main claim is that complex direct Radon transforms are not optional in the universal inversion scheme, and the hybrid-function route gives a natural way to satisfy that requirement.","feed_headline":"Radon inversion forces complex direct images","feed_subtitle":"A Fourier transform over one coordinate turns real CT slice data into complex Radon data, feeding the missing imaginary inverse term.","key_machinery":"The central object is the hybrid (Wigner-like) function $F(x_1,x_2;k)=\\int dx_3\\, e^{-ikx_3} f(x_1,x_2,x_3)$, a phase-space object that is complex even when $f$ is real, together with its discrete-slice analogue $\\tilde F(x_1,x_2;k)=\\sum_n e^{-ikx_{3n}} f(x_1,x_2,x_{3n})$. These functions carry the direct Radon transform into the complex plane, and the inverse formula splits into a principal-value part $F_S$ and an imaginary boundary term $F_A$; the imaginary part of the complex Radon transform supplies $F_A$ naturally. A second mechanism is the non-trivial holonomy of the Radon transform developed in Section 3, where shifting the angle by $\\pi$ twice and comparing two rotation paths produces a mismatch for outset functions supported in different quadrants, which the paper interprets as another source of complexity.","core_discovery":"The paper claims to establish that the universal representation of the inverse Radon transform implies that the direct Radon transform must be complex-valued. The derivation starts from the Fourier slice theorem with an $ε$-regularized radial $λ$-integration, which produces an imaginary contribution beyond the standard principal-value term; in the unified inversion, the direct Radon image enters under both real and imaginary measures. Since the outset function $f$ is real, the paper introduces hybrid functions $F(x_1,x_2;k)$ (the Fourier transform of $f$ in $x_3$) and its slice-sum version $\\tilde F(x_1,x_2;k)$, whose Radon transforms $R[F;\\tilde F](\\tau,\\phi;k)$ are complex and carry the needed imaginary part. The reconstruction scheme $R[F;\\tilde F] \\to \\{F,\\tilde F\\} \\to f$ is the central mechanism: the momentum $k$ acts as an external parameter that aggregates all two-dimensional slices, and the imaginary part of the complex Radon transform feeds the additional term $F_A$ in the inverse formula.","pith_inferences":["A direct numerical test of the central claim is to reconstruct a phantom from complex sinograms with and without $F_A$; if the imaginary term changes the result measurably, the paper's mechanism is confirmed as practically relevant.","The hybrid-function construction effectively performs a continuous interpolation of discrete CT slices, so it may also reduce slice-mismatch artifacts; the paper does not explore this.","The holonomy argument compares $R[g_2](\\tau,\\phi+2\\pi)$ with $R[g_1](\\tau,\\phi)$ for two different outset functions, so its force depends on whether the mismatch persists when a single function is carried around $2\\pi$; a direct check would clarify how much weight this route alone can carry."],"forward_implications":["In the universal inverse scheme, complex-valued direct Radon data are required, so reconstruction pipelines that assume real sinograms are missing a contribution.","The discrete slice method can be replaced by a single complex object $\\tilde F(x_1,x_2;k)$ whose Fourier parameter $k$ serves as a continuous external parameter for optimization.","The additional imaginary contribution $F_A$ changes the inverse formula and therefore the regularization procedure for ill-posed reconstruction problems.","Objects containing localized defects can be separated from a common background by exploiting the holonomy condition on the two rotation paths."],"supporting_citations":[{"why":"Supplies the standard Radon transform definition, the Fourier slice theorem context, and the discrete slice approximation used throughout.","marker":"[1]"},{"why":"Provides the earlier example in QFT where direct Radon transforms possess an imaginary part, motivating the complexity problem.","marker":"[2]"},{"why":"Introduces the universal inverse Radon representation with the extra imaginary measure and the optimization and reconstruction framework this paper builds on.","marker":"[3]"},{"why":"Establishes the universal inverse Radon transforms in arbitrary dimension and the singular-point regularization that the present paper analyzes.","marker":"[4]"},{"why":"The Courant-Hilbert identities whose use the universal representation avoids, setting the baseline against which the new imaginary contribution is identified.","marker":"[5]"},{"why":"Provides the generalized-function regularization tools used for the $ε$-regularized integrations and principal-value handling.","marker":"[6]"},{"why":"Supplies the integral-geometry generalized-function background for Radon transforms and inversion.","marker":"[7]"}],"fun_headline_variants":["Radon inversion demands complex-valued direct transforms","Why Radon inversion requires complex data","Complex Radon transforms: the missing imaginary term","Direct Radon must be complex for inversion to work"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole need for complex direct Radon transforms rests on the assumption that the $ε$-regularized $λ$-integration in the Fourier slice theorem produces a genuine imaginary term in the universal inverse formula that cannot be absorbed by another choice of regularization.","fun_headline_variants_meta":{"raw":{"variants":["Radon inversion demands complex-valued direct transforms","Why Radon inversion requires complex data","Complex Radon transforms: the missing imaginary term","Direct Radon must be complex for inversion to work"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000408,"raw_usage":{"total_tokens":2103,"prompt_tokens":917,"completion_tokens":1186,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":533,"completion_tokens_details":{"reasoning_tokens":1129}},"tokens_in":533,"tokens_out":1186,"duration_ms":7568,"temperature":1.0,"reasoning_tokens":1129,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:40:37.156018+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a real compactly supported function $f$, compute the hybrid function $F$ from (5.1), its complex Radon image $R[F]$, and invert using only the real principal-value part $F_S$; if the reconstruction reproduces $f$ to the same accuracy as the full inverse $F_S+F_A$, then the imaginary contribution is not needed and the central claim fails.","supporting_citations":[{"cited_title":"The Radon Transform and Some of Its Applications,","cited_arxiv_id":null,"evidence_quote":"Supplies the standard Radon transform definition, the Fourier slice theorem context, and the discrete slice approximation used throughout."},{"cited_title":"Inverse Radon transform and the transverse-momentum dependent functions","cited_arxiv_id":"1909.00017","evidence_quote":"Provides the earlier example in QFT where direct Radon transforms possess an imaginary part, motivating the complexity problem."},{"cited_title":"Inverse Radon transforms: analytical and Tikhonov-like regularizations of inversion","cited_arxiv_id":"2405.14897","evidence_quote":"Introduces the universal inverse Radon representation with the extra imaginary measure and the optimization and reconstruction framework this paper builds on."},{"cited_title":"Universal inverse Radon transforms: Inhomogeneity, angular restrictions and boundary","cited_arxiv_id":"2504.01744","evidence_quote":"Establishes the universal inverse Radon transforms in arbitrary dimension and the singular-point regularization that the present paper analyzes."},{"cited_title":"Methods of Mathematical Physics,","cited_arxiv_id":null,"evidence_quote":"The Courant-Hilbert identities whose use the universal representation avoids, setting the baseline against which the new imaginary contribution is identified."},{"cited_title":"Generalized Functions V ol 1 Properties And Operations,","cited_arxiv_id":null,"evidence_quote":"Provides the generalized-function regularization tools used for the $ε$-regularized integrations and principal-value handling."},{"cited_title":"Generalized Functions, V olume 5: Integral Geometry and Representation Theory,","cited_arxiv_id":null,"evidence_quote":"Supplies the integral-geometry generalized-function background for Radon transforms and inversion."}],"review_version":1}