{"id":"4138a906-3e71-4118-a140-582c720dc044","arxiv_id":"2606.00799","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Analogous Weyl-type uniqueness theorems hold for conformal structures and torsion-free connections in Galilei and Carroll geometries.","lead":"The paper extends Weyl's 1921 theorem on metrics determined by conformal and projective structures to suitably defined conformal structures in Galilei and Carroll geometries. These are the non-relativistic and ultra-relativistic limits of Lorentzian geometry, so the work may interest researchers modeling spacetime approximations or limits in physics.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Suitability of chosen conformal structures for Galilei/Carroll cases is the load-bearing assumption","rationale":"The reader's weakest_assumption matches the load-bearing point exactly. The abstract-only review correctly flags that the result stands or falls on whether the 'suitably defined' structures are the right ones; the full text would need to supply the justification and the explicit proof to lower the correctness risk. No other internal inconsistency is visible from the given material.","tokens_in":1612,"tokens_out":369,"duration_ms":14880,"concrete_test":"Extract the precise definitions of conformal structure and projective structure given for the Galilei and Carroll cases (probably §§2–3); check whether they are obtained by taking the appropriate limit of the Lorentzian conformal class or by a canonical construction from the degenerate metric and its automorphisms; then verify whether the uniqueness proof in the main theorem invokes only those properties or requires additional non-degenerate assumptions.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that the paper's definitions of conformal structure on Galilei and Carroll geometries (degenerate metrics with respective Newton-Cartan or Carroll connections) are the natural analogues that allow a Weyl-type uniqueness result: a torsion-free connection compatible with the conformal class is determined by its projective structure. This holds in the Lorentzian case because the conformal class plus unparametrised geodesics fix the Weyl connection uniquely. For the degenerate cases the paper must exhibit definitions (likely via weighted bundles or tractor-like constructions adapted to the degenerate metric) that preserve this uniqueness without extra structure, and must argue they arise canonically from the non-/ultra-relativistic limits. If the definitions are chosen primarily to make the algebraic identity work rather than from group-invariant or limiting considerations, the analogy is weaker than claimed.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript extends Weyl's 1921 theorem—that a Weyl metric is uniquely determined by its conformal and projective structures, or equivalently that a torsion-free connection compatible with a pseudo-Riemannian conformal class is fixed by its projective structure—to Galilei and Carroll geometries. It introduces suitably defined notions of conformal structure on the degenerate metrics equipped with Newton-Cartan or Carroll connections (arising as non-relativistic and ultra-relativistic limits of Lorentzian geometry) and claims that the same uniqueness holds: a torsion-free connection compatible with the conformal class is determined by its projective structure.","tokens_in":1759,"tokens_out":517,"duration_ms":20051,"significance":"If the chosen conformal structures are the canonical ones obtained from the limiting procedure and the uniqueness proofs are complete, the result supplies a precise mathematical statement of how conformal and projective data interact in degenerate geometries. This could serve as a foundation for studying conformal invariants or tractor calculus in non-relativistic and Carrollian settings, with possible relevance to condensed-matter or gravitational models in those limits. The explicit use of limiting constructions from the Lorentzian case is a methodological strength when carried through rigorously.","major_comments":[{"comment":"§2 (Definitions of conformal structure): The paper must demonstrate that the chosen weighted degenerate metric (or tractor-like object) for the Galilei case arises canonically from the non-relativistic limit of the Lorentzian conformal class without auxiliary choices; if the definition is selected primarily to make the algebraic uniqueness identity hold, the claimed analogy to Weyl's theorem is weaker than asserted.","section":"§2"},{"comment":"§4 (Uniqueness proof for Carroll geometry): The argument that the torsion-free connection compatible with the Carroll conformal class is fixed by the projective structure must explicitly handle the degeneracy of the metric; the standard Lorentzian counting of degrees of freedom does not apply directly, and any additional assumptions needed to close the proof should be stated and justified as natural.","section":"§4"}],"minor_comments":[{"comment":"Notation for the weighted bundles or densities used in the Galilei and Carroll conformal classes should be introduced with a short comparison table to the Lorentzian case to improve readability.","section":null},{"comment":"A brief remark on whether the results reduce exactly to the classical Weyl theorem in the appropriate limit would strengthen the narrative.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major comment below and have revised the manuscript to strengthen the presentation of the limiting constructions and the uniqueness arguments.","responses":[{"response":"We agree that demonstrating the canonical origin of the definition is essential. In the revised version we add an explicit computation in §2 (new subsection 2.3) showing that the weighted degenerate metric is obtained directly by taking the non-relativistic limit of the Lorentzian conformal class, with no auxiliary choices introduced. This makes the analogy to Weyl’s theorem fully rigorous.","revision_made":"yes","referee_comment":"[§2] §2 (Definitions of conformal structure): The paper must demonstrate that the chosen weighted degenerate metric (or tractor-like object) for the Galilei case arises canonically from the non-relativistic limit of the Lorentzian conformal class without auxiliary choices; if the definition is selected primarily to make the algebraic uniqueness identity hold, the claimed analogy to Weyl's theorem is weaker than asserted."},{"response":"We accept that the original argument in §4 relied on an implicit non-degenerate counting. The revised proof now works directly with the degenerate Carroll metric, provides an adapted degree-of-freedom count, and states the two natural assumptions (torsion-freeness and compatibility with the Carroll conformal class) explicitly, justifying them as the direct analogues of the Lorentzian conditions.","revision_made":"yes","referee_comment":"[§4] §4 (Uniqueness proof for Carroll geometry): The argument that the torsion-free connection compatible with the Carroll conformal class is fixed by the projective structure must explicitly handle the degeneracy of the metric; the standard Lorentzian counting of degrees of freedom does not apply directly, and any additional assumptions needed to close the proof should be stated and justified as natural."}],"tokens_in":1290,"tokens_out":407,"duration_ms":15246,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This paper takes the classic Weyl theorem—that a torsion-free connection compatible with a pseudo-Riemannian conformal class is fixed by its projective structure—and looks for parallel statements once the metric degenerates in the non-relativistic and ultra-relativistic limits. The central move is to introduce suitable notions of conformal structure on Galilei and Carroll geometries and then argue that the same uniqueness holds.\n\nWhat is new is the explicit treatment of these degenerate cases. The original result does not apply directly, so the authors have to supply fresh definitions, likely via weighted bundles or adapted tractor constructions, and show that the algebraic identity still goes through. That step is the actual contribution.\n\nThe paper is clear about the motivation and states the analogous claims without overreaching. It stays inside mathematical physics and does not pretend the result reshapes broader gravity research.\n\nThe main soft spot is exactly the one the stress-test flags: whether the chosen conformal structures are the mathematically natural or physically relevant ones, or whether they were selected mainly because they let the uniqueness proof go through. If the definitions follow from the limiting procedure or from the underlying group structure, the analogy is solid. If they are more ad hoc, the result is narrower. The abstract gives no proof details, so the full text needs to make this transparent.\n\nThis is for people already working on geometric formulations of non-Lorentzian field theories or gravity. A reader who cares about projective geometry in degenerate metrics will get something out of it. It is narrow enough that it does not demand broad attention, but the technical extension is worth checking, so it deserves a serious referee.","headline":"The paper extends Weyl's 1921 uniqueness result to Galilei and Carroll geometries by defining analogous conformal structures, but the value rests on how canonically those definitions arise from the limits.","tokens_in":2252,"tokens_out":409,"would_cite":false,"duration_ms":14923,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"A torsion-free connection compatible with the conformal structure on Galilei or Carroll geometry is uniquely determined by its projective structure.","keywords":["Weyl theorem","Galilei geometry","Carroll geometry","conformal structure","projective structure","torsion-free connection","non-relativistic limit","ultra-relativistic limit"],"falsifier":"A concrete counterexample would be a Galilei (or Carroll) manifold equipped with two distinct torsion-free connections that are both compatible with the same conformal structure yet share the same set of unparametrised geodesics.","tokens_in":2500,"feed_emoji":"","tokens_out":693,"duration_ms":18996,"temperature":0.7,"pith_summary":"The paper extends Weyl's 1921 theorem, which states that a Weyl metric is fixed by its conformal and projective structures, to Galilei and Carroll geometries. These geometries are the non-relativistic and ultra-relativistic limits of Lorentzian spacetime. The authors introduce suitable notions of conformal structure for each case and prove that a torsion-free linear connection compatible with the conformal structure is fixed by the set of unparametrised geodesics. A reader would care because the result supplies uniqueness statements for connections in physical regimes that arise as limits of ordinary relativity. The work therefore supplies a direct analogue of the relativistic case without invoking a Lorentzian metric.","feed_headline":"Weyl uniqueness extends to Galilei and Carroll geometries","feed_subtitle":"Torsion-free connections compatible with the defined conformal structure are fixed by their projective structure alone.","key_machinery":"The suitably defined conformal structure for Galilei and Carroll geometry, which makes a torsion-free connection compatible with it and thereby lets the projective structure fix the connection uniquely.","core_discovery":"A classic theorem of Weyl states that a Weyl metric is uniquely determined by its conformal and projective structures. An equivalent formulation is that a torsion-free linear connection compatible with a pseudo-Riemannian conformal structure is uniquely determined by its projective structure. The paper establishes the same uniqueness for suitably defined notions of conformal structure on Galilei and Carroll geometries, which arise as the non-relativistic and ultra-relativistic limits of Lorentzian geometry.","pith_inferences":["The uniqueness statements could be applied to construct invariants or conserved quantities in Galilean or Carrollian field theories that rely on projective data.","One could check whether the same pattern persists for other limits of Lorentzian geometry or for modified connections that retain torsion.","The results suggest that projective geometry may serve as a unifying ingredient when comparing conformal properties across different relativistic regimes."],"forward_implications":["The projective structure alone fixes the compatible torsion-free connection once the Galilei conformal structure is given.","The same uniqueness holds once the Carroll conformal structure is given.","Unparametrised geodesics together with the defined conformal data determine the connection in both the non-relativistic and ultra-relativistic limits.","The result supplies a direct parallel to the Lorentzian Weyl theorem without requiring a full Lorentzian metric."],"fun_headline_variants":["Weyl uniqueness holds in Galilei and Carroll geometries","Weyl theorem generalizes to Galilei Carroll geometries","Uniqueness holds for Galilei and Carroll conformal structures","Galilei Carroll geometries feature Weyl uniqueness"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The suitably defined notions of conformal structure for Galilei and Carroll geometry are the mathematically natural and physically relevant analogues of the pseudo-Riemannian conformal structure used in the original Weyl theorem.","fun_headline_variants_meta":{"raw":{"variants":["Weyl uniqueness holds in Galilei and Carroll geometries","Weyl theorem generalizes to Galilei Carroll geometries","Uniqueness holds for Galilei and Carroll conformal structures","Galilei Carroll geometries feature Weyl uniqueness"]},"model":"grok-4.3","cost_usd":0.006683,"raw_usage":{"total_tokens":2978,"prompt_tokens":556,"num_sources_used":0,"completion_tokens":61,"cost_in_usd_ticks":66828000,"prompt_tokens_details":{"text_tokens":556,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2361,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":556,"tokens_out":61,"duration_ms":16283,"temperature":1.0,"reasoning_tokens":2361,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T18:03:21.454595+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A concrete counterexample would be a Galilei (or Carroll) manifold equipped with two distinct torsion-free connections that are both compatible with the same conformal structure yet share the same set of unparametrised geodesics.","supporting_citations":[],"review_version":1}