{"id":"3ac818ca-4fd7-473d-9ade-e05f870a13c8","arxiv_id":"2608.13163","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The time derivative of a transit light curve during ingress or egress is a Radon transform of the planet's attenuation map, so a single transit gives at most two Fourier-space slices and a large null space.","lead":"Transit light curves are almost always interpreted as if the planet's shadow is a perfect circle, but real planets can be oblate, ringed, or cloudy. This paper shows that the information one transit can extract about a non-circular shadow is equivalent to just two one-dimensional scans, proving in a clean limit that most of the shadow's shape is unobservable.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the Radon-limit identity is exactly the conditional statement it claims to be, and the paper's own extensions show the two-slice null-space picture is robust to the main realistic corrections.","rationale":"The reader identified Eq. (12) as the weakest assumption, and I agree that it is the key idealization. However, the paper is explicit that Eq. (20) is a limit statement, and it then analyzes departures from that limit in some detail. The two-slice Fourier picture survives the limb-darkening treatment intact, and the curvature treatment introduces only weak transverse sensitivity around the ideal slices rather than genuinely new projection angles. The mathematical derivation is straightforward and internally consistent: the distributional derivative of the Heaviside function is applied correctly, the Radon offset sign convention is consistent through Eq. (20), and the temporal Fourier transform of Eq. (30) reproduces Eq. (31) with the stated sigma_j factor when the integration limits are treated carefully. The numerical comparisons in Figure 3 and the Kepler-51d degeneracy comparison in Figure 7 support the practical relevance of the leading-order picture. The only substantive caveats are reproducibility-related: no code or data artifacts are provided, and the Figure 7 comparison relies on private posterior samples. These affect verification of the examples, not the central analytic claim. A high-confidence acceptance therefore remains appropriate.","tokens_in":809,"tokens_out":1612,"duration_ms":160880,"concrete_test":"Implement the exact half-plane forward model of Eq. (16) on a dense grid for a non-circular A_p, and for a second map f whose 2D Fourier transform vanishes on the two ingress/egress radial slices. Verify that the light-curve derivatives dF_i/dt and dF_e/dt for A_p and for A_p + f agree to numerical precision, or equivalently that the derivatives generated by f alone vanish. This directly tests the load-bearing null-space statement.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection found. The central identity (Eq. 20) is a distributional exactness statement conditional on Eq. (12), and the paper does not overclaim it beyond that limit. The premise that looks weakest -- straight, uniform-brightness stellar limb -- is handled head-on: limb darkening becomes a convolution along the same projection angle (Eqs. 39 and B16), leaving the Fourier slices unchanged (Eq. 42); curvature adds only transverse derivatives about the ideal slices (Eqs. C20 and C24), which the paper explicitly labels as weak. The Fourier-slice and null-space argument (Eqs. 26, 31, and 33) follows from standard Radon theory once the identification is made. I also checked Eq. (31) directly from Eq. (30); the -sigma_j prefactor and the -nu/alpha_j argument are consistent under the stated Fourier conventions. The main limitations are reproducibility items, such as the absence of shipped code and the use of privately shared posterior samples in Figure 7, but these do not bear on the mathematical claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"Transit light curves are usually modeled with a circular sky-projected silhouette. This paper asks what a single transit can reveal when the planetary attenuation map A_p(x) is non-circular. Under a local knife-edge approximation of the stellar limb, with uniform stellar brightness (Eq. 12), the paper derives that the time derivative of the flux during ingress or egress equals the Radon transform of A_p at the projection angle set by the limb normal, multiplied by the limb-normal velocity component (Eq. 20). Section 3 uses the Fourier-slice theorem, together with a constant-velocity approximation, to show that the temporal Fourier transform of the ingress/egress derivative samples the 2D Fourier transform of A_p along at most two radial lines (Eqs. 30-31); hence the null space of a single transit consists of maps whose Fourier transforms vanish on those two slices (Eq. 33). Section 4 examines departures: limb darkening acts as a convolution along the same projection direction without introducing new projection angles; finite exposure time and sampling smooth and limit the accessible spatial frequencies along the slices; stellar-limb curvature adds weak sensitivity to transverse Fourier derivatives about the ideal slices; time-dependent planetary maps make the problem even more underconstrained. Section 5 applies the framework to oblateness inference (with a comparison to the Kepler-51d posterior of Liu et al.","tokens_in":21230,"tokens_out":39204,"duration_ms":327308,"significance":"If the result holds, this paper provides a compact, parameter-free explanation of several known degeneracies in transit shape inference and sharpens them into a single statement: a single transit constrains only two radial slices through the Fourier transform of the attenuation map. The central identity (Eq. 20) is derived from stated assumptions with no fitted constants, and I verified the key steps, including the Fourier-transform prefactor in Eq. (31), the limb-darkening convolution and slice invariance (Eqs. B16-B17), and the curvature expansion (Eqs. C15-C20); the algebra is clean throughout. The paper is also commendably honest about its frame: the two-slice null-space statement is explicitly conditional on the idealized half-plane model, and the extensions in Sections 4.1-4.4 quantify how realistic effects lift the degeneracy only weakly. The numerical checks against three independent codes (Figure 3) and the alignment of the analytic degeneracy curves with the published Kepler-51d posterior (Figure 7) strengthen confidence in the leading-order picture.","major_comments":[{"comment":"The text quotes the agreement between the Radon approximation and the model codes at the level of the light-curve residuals (about 100 ppm and 20 ppm for the two adopted area ratios), but the quantity that carries the Radon information in Eq. (20) is the time derivative of the light curve. The derivative residuals are displayed in the lower rows of Figure 3 but are neither quantified nor interpreted. From the curvature expansion in Eq. (C20), the leading curvature correction to the derivative is of order s/(2R_s) relative to the leading Radon term, i.e., up to about R_p/(2R_s) of the local derivative and R_p/(4R_s) of the peak derivative, which for the R_p/R_s = 0.1 case is a few percent, much larger in relative terms than the quoted ppm-level flux residuals. Please report the derivative residuals for the three comparisons and state explicitly that Eq. (20) holds only at the percent level for the largest area ratio considered, so that readers can gauge the accuracy of the derivative-level identification.","section":"§2.3, Figure 3"}],"minor_comments":[{"comment":"The paper uses posterior samples from Q. Liu et al. (2024) with their permission, but it does not state whether the samples are publicly available or describe how the 'selected reference samples' in Figure 7 were chosen; please add a data-availability statement (or deposit the samples) and specify the selection procedure so that the comparison is reproducible.","section":"§5.1, Figure 7"},{"comment":"The paper lists the software versions used (jaxoplanet, squishyplanet, catwoman, JAX, etc.) but provides no code for the Radon-approximation light curves, the non-uniqueness examples of Figure 5, or the analytic degeneracy curves of Figures 6 and 7; a short repository or even a pseudocode listing would make the numerical claims independently checkable.","section":"Sections 2.3, 3.2, 5.1"},{"comment":"The abstract motivates non-circular silhouettes by listing 'planetary rotation, tides, rings, or atmospheric inhomogeneities,' but the body does not discuss ringed silhouettes except for a passing reference in Section 5.1; either add a brief treatment of rings or remove them from the motivating list in the abstract.","section":"Abstract and Section 1"},{"comment":"The Heaviside function is defined only for x>0 and x<0, leaving H(0) unspecified; stating H(0) (or noting that values on a measure-zero set are irrelevant for the distributional argument) would close a small analytic gap.","section":"Equation (13)"},{"comment":"The null-space condition is written with 'θ ∈ {θ_i, θ_e}' inside the universal quantifier; writing the two slice conditions separately (θ = θ_i and θ = θ_e) would be easier to read.","section":"Equation (33)"},{"comment":"There are small typesetting artifacts (e.g., 'Radon-T ransform' in the title line and the spacing in 'Keywords:Exoplanets') that should be cleaned up in the final version.","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"The central derivation is sound and the paper is a good fit for the journal. The paper's own stated limitations are handled honestly, and the main requests I would press on are the derivative-level validation (major comment) and the data-availability statement for Figure 7. I have no concerns about the reference list; the citations are appropriate and the self-citation to Tada et al. (2025) is directly relevant. The disclosure of ChatGPT-assisted language editing is adequate. One point for the editor's awareness: if the Liu et al. (2024) posterior samples remain private, Figure 7's comparison cannot be reproduced by third parties, which matters for a paper whose selling point is a clean analytic insight; asking the authors for a public repository would resolve this."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper earns its place. The central identity (Eq. 20) is a real result, not a repackaging. In the straight-limb, uniform-brightness limit, the ingress/egress flux derivative is exactly a Radon projection of the planetary attenuation map, which makes the information limit visual and quantitative: two projection angles, two Fourier slices, a large null space. The paper is also honest about what breaks the limit. Limb darkening becomes a convolution along the same slice, curvature adds weak transverse-derivative sensitivity, and time-dependent maps make the inverse problem even more underconstrained. Each modification is derived, not hand-waved.\n\nWhat is genuinely new: the explicit Fourier-slice null space (Eqs. 31-33) and the curvature correction that interprets the leading error as sensitivity to transverse Fourier derivatives. The payoff is a clean explanation of the known b ~ 0.7 preference in oblateness constraints: maximal complementarity between ingress and egress projections at b ~ 1/sqrt(2). I also appreciate that the Kepler-51d comparison in Figure 7 is anchored to selected posterior samples, not fitted to them, and it matches the posterior contours well enough to show the dominant degeneracy.\n\nThe numerical comparisons in Figure 3 are a real strength. The Radon approximation reproduces jaxoplanet, squishyplanet, and catwoman light curves at the quoted few-tens-of-ppm level, and the curvature-error scaling is checked against that residual. The derivation is parameter-free, so the circularity burden is low. The paper states its approximations explicitly and repeatedly, which is exactly what a framework paper should do.\n\nSoft spots are mostly reproducibility, not correctness. No code or data artifacts are shipped, so Figures 3 and 7 are hard to rebuild. Figure 7 uses privately shared posterior samples from Liu et al., which means a reader cannot fully reproduce the comparison without contacting the authors. That is a minor blemish, not a fatal one, and it does not affect the analytic claims. The citation pattern is solid: it engages the non-parametric degeneracy literature, eclipse mapping, and recent oblateness work, and the one self-citation (Tada et al. 2025) is directly on point for the separate-ingress/egress discussion.\n\nWho is this for? Transit modelers working on oblateness, ringed planets, or morning-evening asymmetries. It is a framework paper: it will likely be cited as the standard way to state the information limit. I agree with the reader's ACCEPT. It deserves a serious referee and publication after minor revision; the main requests should be shipping the code and either releasing the posterior samples or making the comparison fully specified.","headline":"A clean and honest Radon-transform reformulation of transit shape inference; the core identity is correct and the paper deserves refereeing and publication.","tokens_in":21782,"tokens_out":1871,"would_cite":true,"duration_ms":18922,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"In a clean limit, an exoplanet's ingress and egress are two Radon projections of its shadow, so a single transit fixes the attenuation map on only two radial Fourier slices.","keywords":["exoplanets","exoplanet structure","transit photometry","Radon transform","Fourier-slice theorem","inverse problems","non-circular silhouettes","planetary oblateness"],"falsifier":"Take two physically allowed silhouettes that the Radon limit declares identical, such as an ellipse and a more complicated shape from the paper's Figure 5, compute noiseless light curves with a full limb-darkened, curved-limb transit model, and check whether the ingress and egress derivatives differ by more than the paper's curvature-level residuals (roughly tens of ppm for $R_p/R_s \\sim 0.1$); if they differ measurably, the practical null space is strictly smaller than the two-slice claim.","tokens_in":20817,"feed_emoji":"🪐","tokens_out":9645,"duration_ms":81402,"temperature":0.7,"pith_summary":"In a clean limiting case, this paper establishes exactly what a single transit light curve can tell us about a non-circular planet. When the stellar limb is treated as a locally straight, uniformly bright knife edge, the time derivative of the light curve during ingress or egress becomes literally one Radon projection, a line integral, of the planet's two-dimensional attenuation map, with the projection direction set by the local normal to the stellar limb. Since ingress and egress supply at most two projection angles, the Fourier-slice theorem says the data constrain the map's two-dimensional Fourier transform only along two radial slices, leaving a large null space of invisible map features. The paper then shows that limb darkening and finite exposure only blur or reweight those slices rather than adding new angles, while limb curvature adds only weak sensitivity to transverse Fourier derivatives, so the non-uniqueness is structural. If correct, this reframes silhouette and atmosphere-asymmetry claims from transits: the inferred shape is always a prior-dependent choice among a continuum of maps that fit the same light curve.","feed_headline":"Transit light curves see only two slices of a planet's shadow","feed_subtitle":"Ingress and egress reveal at most two projections; many non-circular silhouettes fit the same data.","key_machinery":"The load-bearing machinery is the two-dimensional Radon transform, $R[f](s,\\,\\theta)=\\int_{\\mathbb{R}^2} f(x)\\,\\delta(n(\\theta)\\cdot x - s)\\,dx\\,dy$, combined with the Fourier-slice theorem, which identifies the one-dimensional Fourier transform of a projection with a slice of the map's two-dimensional Fourier transform. In the transit context, the orbital velocity maps time into the Radon offset $s_j(t)$ and supplies the prefactor $n_j\\cdot v(t)$, so each limb crossing mechanically scans out one projection of $A_p$. The inverse-problem consequence is the two-slice null-space condition of Eq. (33): any map whose transform vanishes on the two radial slices is silent. Departures from the ideal limit are handled by the same machinery: limb darkening convolves along the offset direction, finite exposure multiplies the slice by a sinc, and limb curvature adds terms built from tangential moments $M_1$ and $M_2$, which correspond to transverse derivatives of $\\widehat{A}_p$ about the ideal slices.","core_discovery":"On the paper's own terms, the discovery is that a transit ingress or egress is an exact Radon measurement in the knife-edge, uniform-brightness limit. The time derivative of the flux satisfies $\\dot{F}_j(t) = -I_0\\,(n_j\\cdot v(t))\\,R[A_p](s_j(t),\\,\\theta_j)$, so each event gives one projection of the attenuation map $A_p$; the temporal Fourier transform of that segment then samples $\\widehat{A}_p(-\\nu\\cos\\theta_j/\\alpha_j,\\,-\\nu\\sin\\theta_j/\\alpha_j)$, a single radial line through the origin of the map's Fourier plane. Ingress and egress therefore constrain at most two radial slices, and the null space of the forward map contains every attenuation map whose Fourier transform vanishes on those two slices. Consequently, no unique non-circular silhouette can be recovered from the light curve alone; physically allowed maps that differ only off those slices are exactly indistinguishable in this limit. The same structure carries over to secondary eclipses with the planetary brightness map in place of the attenuation map.","pith_inferences":["Beyond the paper, the same two-slice picture gives a quantitative explanation for why eclipse-mapping analyses need phase curves or rotation information: the eclipse analog constrains the brightness map on only two radial slices, so any claimed longitudinal map is largely prior-driven.","If the curvature correction scales as the paper estimates, then near-grazing transits of large planets around small stars, where curvature residuals reach tens to hundreds of ppm, may be the practical place to recover some off-slice Fourier structure from high-precision photometry alone.","The framework suggests a testable prediction: a flexible non-parametric inversion of a high-signal-to-noise transit should return a continuous family of maps aligned with the Radon-invariant directions, and any point estimate that looks unique is a property of the prior, not the data.","One could extend the analysis by treating the two slices as weak measurements of the map's second moments; this would make quantitative the intuition that ingress and egress constrain a planet's projected area and its projections onto two directions, and could be turned into a model-independent moment estimator."],"forward_implications":["A single transit can never pin down a unique non-circular silhouette: many physically allowed attenuation maps produce exactly the same ingress and egress light-curve derivatives in the Radon limit.","Oblateness inference is inseparable from the orbit: changing projected oblateness can be compensated by changing impact parameter and scaled semi-major axis, so external stellar and orbital constraints are required to break the degeneracy.","Limb darkening adds no new information; it only reweights the same two Fourier slices, so an unknown limb-darkening profile can bias the recovered attenuation map.","Finite exposure time and discrete sampling set a resolution floor along each slice; structures narrower than roughly twice the corresponding offset spacing cannot be recovered.","Stellar-limb curvature slightly softens the strict null space by making the data weakly sensitive to transverse Fourier derivatives around the ideal slices, with the most complementary coverage at impact parameter $b\\sim 1/\\sqrt{2}$, which explains why oblateness constraints tend to be strongest at high impact parameters."],"supporting_citations":[{"why":"Defines the Radon transform that Eq. (20) identifies in the light-curve derivative.","marker":"J. Radon (1917, 1986)"},{"why":"Supplies the Fourier-slice theorem that turns one projection angle into one radial slice of the map's two-dimensional Fourier transform.","marker":"R. N. Bracewell (1956)"},{"why":"Establishes the standard circular-silhouette transit model that the paper's non-circular framework generalizes.","marker":"K. Mandel & E. Agol (2002)"},{"why":"Supplies the catwoman semicircle-silhouette model used as a benchmark for the Radon approximation.","marker":"N. Espinoza & K. Jones (2021)"},{"why":"Supplies the squishyplanet non-spherical model used to validate the Radon approximation on elliptical silhouettes.","marker":"B. Cassese et al. (2024)"},{"why":"Provides the earlier oblateness degeneracy and the near-b=0.7 detectability peak that the Radon picture explains.","marker":"J. W. Barnes & J. J. Fortney (2003)"},{"why":"Provides the Kepler-51d posterior samples whose shape-orbit degeneracy matches the analytic Radon-predicted degeneracy curve.","marker":"Q. Liu et al. (2024)"},{"why":"Supplies the forward-modeling oblateness analysis whose best-constrained impact parameters align with the two-slice complementarity argument.","marker":"S. Dholakia et al. (2025)"}],"fun_headline_variants":["Transit light curves: just two slices of a planet's shadow","Two projections max: what ingress and egress reveal","Exoplanet shape ambiguity from transit light curves","Radon view: transits capture at most two shadow slices","Non-unique silhouettes from transit light curves"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The result depends on treating the stellar limb near ingress and egress as a straight, uniformly bright knife edge with a static planet shadow; if the limb is curved or limb-darkened beyond a tiny correction, the light-curve derivative is no longer a pure Radon projection and the two-slice null-space statement is only approximate.","fun_headline_variants_meta":{"raw":{"variants":["Transit light curves: just two slices of a planet's shadow","Two projections max: what ingress and egress reveal","Exoplanet shape ambiguity from transit light curves","Radon view: transits capture at most two shadow slices","Non-unique silhouettes from transit light curves"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000239,"raw_usage":{"total_tokens":1542,"prompt_tokens":998,"completion_tokens":544,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":614,"completion_tokens_details":{"reasoning_tokens":479}},"tokens_in":614,"tokens_out":544,"duration_ms":5068,"temperature":1.0,"reasoning_tokens":479,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:22:42.912054+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take two physically allowed silhouettes that the Radon limit declares identical, such as an ellipse and a more complicated shape from the paper's Figure 5, compute noiseless light curves with a full limb-darkened, curved-limb transit model, and check whether the ingress and egress derivatives differ by more than the paper's curvature-level residuals (roughly tens of ppm for $R_p/R_s \\sim 0.1$); if they differ measurably, the practical null space is strictly smaller than the two-slice claim.","supporting_citations":[{"cited_title":"1917, Berichte ¨ uber die Verhandlungen der K¨ oniglich-S¨ achsischen Gesellschaft der Wissenschaften zu","cited_arxiv_id":null,"evidence_quote":"Defines the Radon transform that Eq. (20) identifies in the light-curve derivative."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Fourier-slice theorem that turns one projection angle into one radial slice of the map's two-dimensional Fourier transform."}],"review_version":1}