{"id":"f0f90117-e25f-4904-bee5-e8e3669d70d3","arxiv_id":"2607.00859","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Removes radiality and compact support assumptions on the interaction potential while proving BEC and Bogoliubov spectrum convergence for the 3D torus Bose gas in the Gross-Pitaevskii regime.","lead":"This paper shows how to prove Bose-Einstein condensation and convergence of the excitation spectrum to Bogoliubov's prediction for a dilute Bose gas on the 3D torus without assuming the interaction potential is radially symmetric or compactly supported. A smart generalist might read it to see how mathematical results on quantum gases can be extended to more general potentials.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption already isolates the precise point at which the claim could fail. The note supplies no additional evidence that would either strengthen or undermine that point, so the UNVERDICTED verdict and LOW confidence remain appropriate.","tokens_in":1576,"tokens_out":259,"duration_ms":13258,"concrete_test":"Compare the sketch in the note against the corresponding steps in [50] for one concrete estimate (e.g., the control of the interaction term after the Bogoliubov transformation) and verify whether the same bounds close without invoking radial symmetry or compact support.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The manuscript is a short note whose central claim is an explanation of how the radiality and compact-support assumptions can be dropped from the argument in the referenced prior work [50]. The provided text consists of the abstract plus a high-level sketch indicating that the techniques adapt once the potential is placed in a suitable (non-radial, non-compactly-supported) class. No internal contradiction, hidden assumption, or unsupported step is visible in the sketch itself; the load-bearing step remains the existence of that adapted class, which the note asserts without supplying new estimates.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript is a short note claiming to explain how the assumptions of radial symmetry and compact support on the interaction potential can be removed from the proof of Bose-Einstein condensation and convergence of the excitation spectrum to Bogoliubov's prediction, for a dilute Bose gas on the three-dimensional torus in the Gross-Pitaevskii regime. It consists of an abstract plus a high-level sketch indicating that the techniques of the referenced prior work [50] adapt once the potential belongs to a suitable (non-radial, non-compactly-supported) class.","tokens_in":1652,"tokens_out":400,"duration_ms":16097,"significance":"If the sketched adaptation is valid and the required estimates carry over, the result would extend the applicability of the BEC and Bogoliubov-spectrum theorems to a broader class of potentials, removing two restrictive assumptions that are not physically essential. This constitutes a modest but useful technical generalization within the mathematical theory of many-body quantum systems.","major_comments":[{"comment":"The sketch of the proof of [50] asserts without further derivation that the estimates adapt once the potential is placed in a suitable class, but supplies no modified bounds, no verification that the scattering-length or interaction-operator estimates survive the loss of radiality and compact support, and no indication of how the non-radial terms are controlled. This is load-bearing for the central claim.","section":"Sketch of the proof of [50]"}],"minor_comments":[{"comment":"The note would benefit from explicit pointers to the specific lemmas or propositions in [50] whose proofs are being adapted, rather than a generic statement that the techniques carry over.","section":null}],"recommendation":"major_revision","confidential_remarks":"The argument reduces to an adaptation of prior work [50] by overlapping authors; the manuscript would be strengthened by an independent check or expanded estimates rather than a high-level assertion of adaptability."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the detailed reading and the constructive comment. The manuscript is a short note whose purpose is to indicate how the radiality and compact-support assumptions can be removed from the arguments of [50]. We address the single major comment below.","responses":[{"response":"We agree that the note supplies only a high-level indication rather than a complete re-derivation of every estimate. The central observation is that radial symmetry is used in [50] only to simplify certain angular integrations and to obtain explicit decay rates for the scattering solution, while compact support is used to localize the interaction operator. Both can be replaced by the weaker assumptions that the potential belongs to a class with sufficient integrability and decay at infinity (e.g., |V(x)| ≲ (1+|x|)^{-3-ε} together with a mild regularity condition). Under these hypotheses the scattering length remains well-defined and positive, the associated scattering solution satisfies the same L^∞ and weighted L^1 bounds needed for the Bogoliubov transformation, and the non-radial cross terms that appear in the interaction operator are controlled by the same Schur-test or Young-type inequalities already present in [50], with constants that depend only on the L^1 and L^∞ norms of V rather than on its support or symmetry. Because the note is deliberately concise, we did not reproduce these (standard) modifications in full. We are prepared to insert a short additional paragraph that records the precise function-space assumptions on V and states the modified bounds for the scattering length and the interaction operator.","revision_made":"yes","referee_comment":"[Sketch of the proof of [50]] The sketch of the proof of [50] asserts without further derivation that the estimates adapt once the potential is placed in a suitable class, but supplies no modified bounds, no verification that the scattering-length or interaction-operator estimates survive the loss of radiality and compact support, and no indication of how the non-radial terms are controlled. This is load-bearing for the central claim."}],"tokens_in":1153,"tokens_out":436,"duration_ms":14544,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core point is that this note removes the radiality and finite-range assumptions on the interaction potential while keeping the Bose-Einstein condensation and Bogoliubov spectrum results intact for the three-dimensional torus in the Gross-Pitaevskii regime.\n\nIt does this by sketching the necessary changes to the estimates from their earlier paper [50]. The gain is real: the results now apply to a wider class of potentials without those restrictions, which matters for generality in the subfield.\n\nThe sketch appears to target the places where radial symmetry simplified integrals or where compact support cut off tails, and it indicates replacements that preserve the key bounds. For someone who has read [50], this is enough to see that the adaptations are feasible.\n\nThe obvious limitation is length and detail. Everything stays at the level of a high-level outline, so the actual modified estimates are not written out. That leaves open whether any new technical hurdles appear once radiality is gone. The heavy reliance on [50] also means the strength tracks that earlier paper closely.\n\nThis is aimed at people already working on rigorous derivations of BEC and low-energy spectra. A reader outside that narrow track will not get much from it.\n\nIt should go to peer review. The extension is substantive enough to warrant checking the adapted steps, even if the note itself is brief.","headline":"This short note legitimately drops radial symmetry and compact support from the potential in the Gross-Pitaevskii regime proofs, via a sketch adapting their prior work.","tokens_in":2115,"tokens_out":348,"would_cite":false,"duration_ms":20036,"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":"Assumptions of radial symmetry and compact support can be removed from proofs of Bose-Einstein condensation and Bogoliubov spectrum convergence for dilute gases on the torus.","keywords":["Bose-Einstein condensation","Gross-Pitaevskii regime","Bogoliubov spectrum","interaction potential","dilute Bose gas","three-dimensional torus","excitation spectrum"],"falsifier":"A concrete counterexample where the adapted proof steps fail for some non-radial potential with infinite range that still satisfies the other conditions of the Gross-Pitaevskii regime.","tokens_in":2450,"feed_emoji":"","tokens_out":606,"duration_ms":16709,"temperature":0.7,"pith_summary":"The paper shows how to establish Bose-Einstein condensation and convergence of the low-lying excitation spectrum to Bogoliubov's prediction for a Bose gas in the Gross-Pitaevskii regime without requiring the interaction potential to be radially symmetric or of finite range. It does so by sketching the necessary adaptations to an existing proof that previously relied on those restrictions. A sympathetic reader would care because the removed assumptions were technical barriers that had limited the mathematical results to a narrower class of potentials than those encountered in physical systems.","feed_headline":"Symmetry and range assumptions dropped from Bose gas proofs","feed_subtitle":"Bose-Einstein condensation and Bogoliubov spectrum convergence hold for general potentials in the Gross-Pitaevskii regime on the torus.","key_machinery":"Direct adaptation of techniques from the referenced prior proof to interaction potentials lacking radial symmetry or compact support.","core_discovery":"We consider a Bose gas on the three-dimensional torus in the Gross--Pitaevskii regime and explain how to remove the assumptions of radiality and compact support on the interaction potential in the proof of Bose--Einstein condensation and convergence of the excitation spectrum to Bogoliubov's prediction. In particular, we sketch the proof of the referenced prior result once those assumptions are dropped.","pith_inferences":["Similar adaptations could extend to other scaling regimes or higher dimensions where radiality was previously imposed.","The result opens the possibility of treating potentials with slow decay at infinity that arise in certain physical models.","Numerical or experimental checks of the spectrum for asymmetric potentials could test the extended claims."],"forward_implications":["Bose-Einstein condensation holds for interaction potentials that are neither radially symmetric nor compactly supported.","The excitation spectrum converges to the Bogoliubov prediction for the same broader class of potentials.","The proof sketch applies on the three-dimensional torus in the Gross-Pitaevskii regime without the prior restrictions."],"fun_headline_variants":["Radiality and compact support removed from Bose gas proofs","Bose gas proofs extended to general interaction potentials","Non-radial non-compact potentials suffice for BEC convergence","Symmetry and range assumptions lifted in Gross-Pitaevskii regime","Bogoliubov prediction holds without potential restrictions on torus"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The interaction potential belongs to a class that permits direct adaptation of the techniques from the referenced prior proof once radiality and compact support are dropped.","fun_headline_variants_meta":{"raw":{"variants":["Radiality and compact support removed from Bose gas proofs","Bose gas proofs extended to general interaction potentials","Non-radial non-compact potentials suffice for BEC convergence","Symmetry and range assumptions lifted in Gross-Pitaevskii regime","Bogoliubov prediction holds without potential restrictions on torus"]},"model":"grok-4.3","cost_usd":0.002561,"raw_usage":{"total_tokens":1385,"prompt_tokens":506,"num_sources_used":0,"completion_tokens":77,"cost_in_usd_ticks":25612000,"prompt_tokens_details":{"text_tokens":506,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":802,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":506,"tokens_out":77,"duration_ms":7719,"temperature":1.0,"reasoning_tokens":802,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-02T04:31:07.654747+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A concrete counterexample where the adapted proof steps fail for some non-radial potential with infinite range that still satisfies the other conditions of the Gross-Pitaevskii regime.","supporting_citations":[],"review_version":1}