{"id":"7224e0c7-9118-4f30-a950-b5449824e8ef","arxiv_id":"2608.13105","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Finite-speed particle scattering can reconstruct a force field once the particle kinetic energy exceeds the largest potential difference between the domain interior and its boundary, supported by analytic examples and ML experiments.","lead":"The paper asks how slowly particles can be fired through an unknown force field while still permitting reconstruction of that field from scattering data. It argues that the required kinetic energy equals the largest potential-energy gap between the interior and the boundary, and tests this on exactly solvable examples and neural network reconstructions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (1.1) generalizes two solvable examples into a universal threshold without proving necessity or sufficiency; the paper's own Section 7.7.1 concedes that the energy indicator ignores reachable-set connectivity and topological barriers.","rationale":"The reader's weakest_assumption identifies the same sufficiency gap: the energy criterion assumes local energetic access implies dynamical reachability, while Section 7.7.1 explicitly disclaims connectivity and topological restrictions. This is the most load-bearing concern because the headline Eq. (1.1) is phrased as a general minimal-energy condition for restoring the potential, yet the theoretical derivations cover only harmonic and constant-force potentials and the numerical section is explicitly diagnostic. The exact harmonic and constant-force calculations are internally consistent and the perturbative consistency check in Section 4.3 is strong evidence for those examples, so the paper deserves credit where it is careful. The overreach is the leap from 'these expansions break down around this energy' and 'this ML model's error rises around this energy' to 'this is the minimal kinetic energy required to restore the potential.' The self-flagged limitation in Section 7.7.1 marks precisely where a barrier, a closed invariant region, or a missing connectivity argument could break the claim. Therefore the conditional verdict stands, but the paper should either restrict Eq. (1.1) to the studied potential classes or provide a general reachability and invertibility argument.","tokens_in":23690,"tokens_out":9077,"duration_ms":100374,"concrete_test":"Run an exact noiseless numerical experiment on the unit disk for the harmonic potential U(r)=r^2 and for an annular-barrier central potential, e.g. U(r)=r^2 + 4 exp(-((r-0.5)/0.1)^2). Generate dense boundary data (traversal time, exit angle, exit velocity) at speeds below and above the Eq. (1.1) threshold, and reconstruct with a non-perturbative least-squares/optimization method rather than the 1/v series or FFPM. If the harmonic potential is recovered below v_c = sqrt(2), Eq. (1.1) is not necessary; if the annular-barrier potential is not recoverable above the local-energy threshold, Eq. (1.1) is not sufficient.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim Eq. (1.1) is stated as a universal condition, but the evidence establishes only that the 1/v perturbative expansion (Section 3.2) and the FFPM reconstruction error degrade when E_kin approaches max|U_in - U_out|. Convergence of a particular expansion and performance of one trained architecture do not set an information-theoretic boundary for the inverse scattering problem. Sufficiency is also assumed rather than proven: Section 7.7.1 explicitly says the local energy indicator 'does not take into account the connectivity of the set of reachable points' and that topological restrictions are 'intentionally ignored.' For non-monotone or non-central fields, effective-potential barriers such as U_eff(r) = U(r) + L^2/(2r^2), or invariant tori more generally, can exclude subsets of the interior even when the local energy inequality holds, while deep attractive wells can be probed by low-energy radial orbits even when the inequality fails. Because no general theorem, exhaustive search, or falsification attempt is supplied, Eq. (1.1) is a plausible heuristic from the harmonic and constant-force examples, not an established limit of the inverse problem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the inverse scattering problem for Newtonian particles at finite speed on the unit disk, asking how low the initial kinetic energy can be while the conservative force field remains reconstructible from boundary-to-boundary flight data. After recalling the infinite-speed Radon limit, the authors develop a perturbative expansion of the finite-speed vector ray integral in powers of δU/(mv²), analyze two exactly solvable examples (harmonic potential and constant force), conjecture a universal non-linear relation between finite- and infinite-speed ray integrals (Eq. (6.1)), and train a GAN-based Force-Field Prediction Model on synthetic random potentials. The central claim is Eq. (1.1): the minimal kinetic energy required to restore the potential equals max(|U_in − U_out|). The harmonic example is treated in detail and reproduces the critical speed v_c = √2; the constant-force example, the ML experiments, and the out-of-distribution paraboloid test are used as additional evidence for the same energy-scale transition.","tokens_in":24023,"tokens_out":15954,"duration_ms":139423,"significance":"If Eq. (1.1) were established, it would give a simple and practically attractive criterion for planning finite-speed particle-tomography experiments: the required kinetic energy is set by the maximal potential difference between boundary and interior. The paper has real strengths: the harmonic-oscillator analysis in Section 4 is explicit and internally consistent; the perturbative framework suggests useful single-speed iteration and multi-speed extrapolation procedures; and the authors provide code and datasets, with a substantial ML engineering effort. However, as written, Eq. (1.1) is a conjecture rather than a proven limit. One of the two exact examples contains a concrete algebraic error, and the conjectured universal formula of Section 6 is contradicted by the corrected constant-force kinematics. The ML validation is also partly circular. The paper is best viewed as a well-motivated conjecture with supporting numerics, not an established theorem.","major_comments":[{"comment":"The exact constant-force ray integral is dimensionally inconsistent and does not reduce to the infinite-speed limit. With m=1, J_v has units of velocity squared, but the right-hand side of Eq. (5.8), √2 v²/g times a quantity of units velocity, has units length×velocity and diverges as Lv as v→∞, instead of approaching J∞=Lg of Eq. (5.9). The correct expression is J_v(L)=√2 v √(v²+Lg∥−√((v²+Lg∥)²−g²L²)) (g/g). Expanding the corrected expression gives J_v = Lg − L² g∥ g/(2v²)+O(v^{-4}), with the opposite sign to Eq. (5.10). Consequently the claimed agreement of Q[F] with the general formula in §5.2, and the support this example is said to lend to Eq. (1.1), are not valid as written.","section":"§5.1, Eq. (5.8)"},{"comment":"The conjectured universal relation (6.1) is not exact for the constant-force family, contrary to the claim that algebraic inversion of (5.8) leads to it. For a constant force parallel to the chord, the exact kinematics give J∞ = J_v + J_v²/(2v²) (as vector relation along τ), whereas Eq. (6.1) with J_v·n=0 reduces to J_v√(1−J_v²/(4v⁴)). The two differ already at O(v^{-2}). Thus Eq. (6.1) is not a general exact relation even for the two solvable families; it is exact only for the harmonic case, where it collapses to Eq. (6.2). This undermines the paper's presentation of (6.1) as a universal structure.","section":"§6, Eq. (6.1)"},{"comment":"The central claim Eq. (1.1) is not established for general potentials. The perturbative expansion of §3.2 has no convergence theorem for general fields; the only critical-speed derivation is the harmonic example of Section 4. Section 7.7.1 explicitly states that the local energy indicator f_acc 'does not take into account the connectivity of the set of reachable points' and that topological restrictions are 'intentionally ignored.' Angular-momentum barriers can therefore exclude interior regions even when the pointwise energy inequality holds, so sufficiency of Eq. (1.1) is assumed rather than proven. The numerical support is partly circular: the constants C_1..C_5 in Eq. (7.12) are computed from the same synthetic-field ensemble used for validation, metric 4 is selected post hoc, and Figs. 5–6 min-max scale the 'inaccessible area' curve to fit the error curve, so no independent falsifiable test of Eq. (1.1) is provided. The paper should either prove the criterion for a well-defined class (for example monotone central potentials), or state Eq. (1.1) as a conjecture and validate it with an independently calibrated, pre-registered energy scale.","section":"Eq. (1.1) and §7.7.1"},{"comment":"The perturbative derivation, while suggestive, is not self-contained enough to justify the transition claim. The half-integer expansion of T is justified by a described iterative procedure but not written out, no general convergence radius ϵ_c is given for arbitrary F, and the value ϵ_c=1/2 is obtained only for the harmonic oscillator. The sentence 'Note that T^2 = ...' appears in the middle of the derivation without a clear role and reads as an unfinished note. Since Eq. (3.29) is the theoretical bridge to Eq. (1.1), this gap should be addressed directly rather than left as an implicit assumption.","section":"§3.2, Eqs. (3.12)–(3.29)"}],"minor_comments":[{"comment":"The author names and affiliations contain corrupted characters ('Fediniπe', 'iInstitute') that should be cleaned before publication.","section":"§1, author line"},{"comment":"The Img2Vec amplitude prediction is shown only as a scatter plot; please report quantitative metrics such as R² or mean absolute error in log10A.","section":"§7.6, Fig. 4"},{"comment":"The 'inaccessible area, min–max scaled to fit' curves are rescaled to match the error curves; this makes the visual agreement uninformative. The inaccessible fraction should be shown on its own axis or with a stated, physically motivated normalization.","section":"§7.7, Figs. 5 and 6"},{"comment":"The statement that future work will address 'independent test sets and repeated training runs' confirms that the current ML results come from single training runs; this limitation should be stated in Section 7.7.1 rather than only in the conclusions.","section":"§8, Conclusions"},{"comment":"The notation uses v both as the scalar launch speed and implicitly inside the vector v_in; please distinguish the scalar |v| from the vector v_in to avoid confusion.","section":"§3.2, Eq. (3.19)"}],"recommendation":"major_revision","confidential_remarks":"The paper has a useful perturbative framework, a correct and reproducible harmonic-oscillator verification, and a substantial ML engineering effort. My recommendation of major revision, rather than rejection, reflects that the central claim Eq. (1.1) is presented too strongly and that Section 5 contains a concrete algebraic error which can be fixed. However, if the authors are unwilling either to correct the constant-force calculation and the universal formula of Section 6 or to reframe Eq. (1.1) as a clearly labeled conjecture, the paper would not be publishable in its current form. I would also encourage the editor to require an independent calibration of the energy-scale constants before the numerical claim is accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper's genuine content is the exact finite-speed ray integrals for the harmonic and constant-force potentials, the systematic 1/v^2 expansion, and the clean observation that the harmonic scattering map switches from two-to-one to one-to-one coverage at v_c = sqrt(2). Those checks are internally consistent and the expansion matches direct computation. The machine-learning study is also described carefully, with code and data released, and the out-of-distribution paraboloid test is a nice control. If the paper only claimed the threshold for the studied potential classes, I would have few complaints.\n\nThe soft spot is Eq. (1.1), which generalizes two examples to every conservative field without proof. The paper's own Section 7.7.1 concedes that the energy indicator \"does not take into account the connectivity of the set of reachable points\" and that topological restrictions are \"intentionally ignored.\" That is not a minor caveat: angular-momentum barriers in central potentials can exclude the interior even when the local energy inequality holds, and deep wells can be probed by low-energy radial orbits when it fails. So the criterion is not an information-theoretic boundary; it is a heuristic supported by two analytic examples and by ML experiments that use energy-scale constants computed from the same fields being validated, with metric 4 selected post hoc and no repeated runs. The ML evidence is real but weaker than the claim.\n\nI also think there is a mismatch between the abstract's question — how low can speed be — and the answer, which concerns a particular perturbative expansion and one trained architecture. That does not invalidate the work, but it means the title overpromises.\n\nWho is this for? People working on Newtonian inverse scattering and particle-beam tomography will find the exact formulas useful and the energy-scale phenomenology worth considering. The paper deserves a serious referee, but I would expect major revision: either restrict the universal claim to the studied classes or supply a general argument, and at minimum add repeated training runs and honest treatment of reachability. The analytic sections can mostly stand as they are.","headline":"Solid examples, shaky universal threshold: the harmonic and constant-force calculations are real contributions, but Eq. (1.1) is a plausible heuristic, not an established limit.","tokens_in":24458,"tokens_out":1777,"would_cite":false,"duration_ms":18815,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","44A12","81U40","65R32"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that finite-speed Newtonian scattering can recover a force field exactly when the particle's kinetic energy exceeds the potential gap between the interior and the boundary of the region.","keywords":["inverse scattering","Newtonian tomography","vector ray integral","Radon transform","finite-speed particles","potential reconstruction","critical energy threshold","machine learning tomography"],"falsifier":"Construct a disk potential containing a deep, wide well surrounded by a high but narrow barrier, and launch particles with energy above the local potential difference inside the well; if angular momentum prevents them from crossing the barrier, the reconstruction should fail despite satisfying Eq. (1.1). A numerical experiment of this kind would settle whether the energy criterion is sufficient or only necessary.","tokens_in":23513,"feed_emoji":"⚛️","tokens_out":10665,"duration_ms":88382,"temperature":0.7,"pith_summary":"The paper studies a Newtonian version of tomography: particles are fired into a disk and the unknown conservative force field inside is reconstructed from boundary data such as entry point, initial speed, traversal time, exit point, and exit velocity. The authors claim that this reconstruction is possible only when the particle's kinetic energy exceeds a characteristic potential gap between the interior and the boundary, $E=\\max(|U_{\\rm in}-U_{\\rm out}|)$. Two exactly solvable examples, a harmonic well and a constant force, together with a machine-learning reconstruction pipeline, support the claim: above the threshold the inverse problem behaves like a corrected Radon transform, while below it the scattering map loses coverage and the reconstruction error rises steeply. If the claim is right, it gives a practical lower bound on beam energy for particle-based tomography rather than an absolute speed scale.","feed_headline":"Particle tomography works only above a potential-energy gap","feed_subtitle":"The threshold equals the potential gap between the disk's interior and border, confirmed by exact examples and neural reconstruction.","key_machinery":"The load-bearing object is the vector ray integral $J_v(L)=v\\,\\Delta v(L)$, the speed-weighted momentum change of a particle crossing the disk along a chord $L$; in the infinite-speed limit it reduces to the Radon transform of the force field, which is inverted componentwise. The argument proceeds by a fixed-boundary perturbative expansion in $\\varepsilon=\\delta U/(mv^2)$: the trajectory and flight time are expanded with pinned entry and exit points, generating explicit functionals $Q[F]$ and $P[F]$ that correct the ray integral at finite speed. The harmonic and constant-force problems are solved exactly and confirm the expansion; the machine-learning pipeline, the Force-Field Prediction Model, reconstructs the force field from boundary measurements plus a Radon prior, with the kinetic-to-potential energy ratio $\\kappa=E_{\\rm kin}/E_{\\rm pot}$ as the controlling parameter.","core_discovery":"The central claim is that finite-speed Newtonian boundary scattering can recover a conservative force field on the unit disk exactly when the initial kinetic energy is at least the absolute potential variation between the interior and the boundary, as stated in Eq. (1.1). The paper develops a systematic expansion of the measured vector ray integral, $J_v = v\\,\\Delta v = J_\\infty + (\\delta U/mv^2)Q[F] + (\\delta U/mv^2)^2P[F] + \\cdots$, about the infinite-speed Radon limit, with the correction functionals $Q[F]$ and $P[F]$ integrated along the unperturbed chord. For the harmonic potential $U=r^2$, the first correction vanishes, the next term is $J_\\infty^3/(8v^4)$, and the series converges only for $v\\ge\\sqrt{2}$ when $\\delta U=m=1$; this same speed separates two-to-one scattering coverage from one-to-one coverage. Numerical experiments on synthetic random fields and on a held-out paraboloid show reconstruction error growing sharply in the window $E_{\\rm kin}/E_{\\rm pot}\\in[1,3]$, which brackets the predicted threshold.","pith_inferences":["A testable extension: in non-central potentials the local energy indicator ignores angular-momentum barriers, so the true recoverability threshold may exceed Eq. (1.1); a simulation with a deep well shielded by a narrow barrier would separate the energy condition from the reachability condition.","The residence-time channel is the boundary measurement most correlated with field amplitude, so time-of-flight data alone may serve as a practical detector of the threshold in experimental setups.","If the conjectured nonlinear relation between $J_v$ and $J_\\infty$ holds approximately for general fields, it would provide a preprocessing step that removes most finite-speed distortion before any Radon inversion.","A practical design rule follows: choose the beam energy to exceed the maximum interior-to-boundary potential drop of the target potential class, rather than tuning to an absolute speed."],"forward_implications":["Above the threshold, the inverse Radon formula used for X-rays can be corrected order by order in $1/v^2$, giving an iterative reconstruction scheme that converges for speeds above a critical value.","Below the threshold, some exit points are unreachable or multiply covered, so the boundary data no longer determine the force field; for the harmonic well the dividing speed is exactly $v_c=\\sqrt{2\\,\\delta U/m}$.","Measurements at two different speeds can be combined algebraically to eliminate the first correction and recover the field directly, without knowing the correction functional in advance.","Reconstruction quality is governed by the ratio $E_{\\rm kin}/E_{\\rm pot}$, so the same transition is predicted for slow particles in shallow potentials and fast particles in deep ones.","The paper conjectures the universal nonlinear relation $J_\\infty = \\frac{2(1-|J_v|^2/(4v^4))}{\\sqrt{4-(J_v\\cdot n)^2/v^4}-J_v\\cdot\\tau/v^2}\\,J_v$, and shows it is exact for the harmonic and constant-force cases, suggesting a data-only correction step before Radon inversion."],"supporting_citations":[{"why":"Supplies the classical result that single-energy scattering determines a spherically symmetric potential outside the closest-approach radius, motivating the energy threshold.","marker":"[4]"},{"why":"Establishes the high-energy uniqueness theorem that the X-ray transform of the force field is recoverable, the baseline the finite-speed corrections are built on.","marker":"[6]"},{"why":"Extends high-energy uniqueness to the relativistic Newton equation and clarifies which components of scattering data are recoverable.","marker":"[3]"},{"why":"Provides fixed-energy uniqueness for Newton equations in electromagnetic fields, the fixed-energy counterpart of the paper's expansion.","marker":"[7]"},{"why":"Supplies the generator architecture used by the Force-Field Prediction Model in the numerical experiments.","marker":"[20]"},{"why":"Makes the code and datasets for the numerical experiments available so the threshold can be reproduced.","marker":"[17]"}],"fun_headline_variants":["Particle tomography hits a wall below potential gap","Speed threshold for particle tomography equals potential gap","Newtonian scattering recovers forces only above a speed floor","Particle tomography requires kinetic energy above potential swing","Threshold speed for particle tomography set by potential difference"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The threshold assumes that a particle with kinetic energy above a local potential difference can actually reach every interior point, ignoring angular-momentum barriers and topological obstructions that can keep particles out of some regions; the paper states these restrictions are intentionally ignored.","fun_headline_variants_meta":{"raw":{"variants":["Particle tomography hits a wall below potential gap","Speed threshold for particle tomography equals potential gap","Newtonian scattering recovers forces only above a speed floor","Particle tomography requires kinetic energy above potential swing","Threshold speed for particle tomography set by potential difference"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000244,"raw_usage":{"total_tokens":1496,"prompt_tokens":876,"completion_tokens":620,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":492,"completion_tokens_details":{"reasoning_tokens":548}},"tokens_in":492,"tokens_out":620,"duration_ms":6015,"temperature":1.0,"reasoning_tokens":548,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:41:14.368621+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a disk potential containing a deep, wide well surrounded by a high but narrow barrier, and launch particles with energy above the local potential difference inside the well; if angular momentum prevents them from crossing the barrier, the reconstruction should fail despite satisfying Eq. (1.1). A numerical experiment of this kind would settle whether the energy criterion is sufficient or only necessary.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the classical result that single-energy scattering determines a spherically symmetric potential outside the closest-approach radius, motivating the energy threshold."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the high-energy uniqueness theorem that the X-ray transform of the force field is recoverable, the baseline the finite-speed corrections are built on."},{"cited_title":"Jollivet","cited_arxiv_id":null,"evidence_quote":"Extends high-energy uniqueness to the relativistic Newton equation and clarifies which components of scattering data are recoverable."},{"cited_title":"On inverse problems in electromagnetic field in classical mechanics at fixed energy","cited_arxiv_id":"math-ph/0701008","evidence_quote":"Provides fixed-energy uniqueness for Newton equations in electromagnetic fields, the fixed-energy counterpart of the paper's expansion."},{"cited_title":"Fedin, K","cited_arxiv_id":null,"evidence_quote":"Makes the code and datasets for the numerical experiments available so the threshold can be reproduced."}],"review_version":1}