{"id":"aa6123a3-f58f-422c-b43a-4e09b872e3ac","arxiv_id":"2605.31397","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Classification of boost-symmetric CMC surfaces in the sub-Lorentzian Heisenberg group yields a family of smooth acausal surfaces conjectured to be isoperimetric maximizers.","lead":"The paper derives the first-variation formula for horizontal area under volume-preserving radial variations in the sub-Lorentzian Heisenberg group and classifies smooth boost-symmetric constant horizontal mean curvature surfaces. It identifies a specific family of acausal surfaces conjectured to solve the isoperimetric problem as analogues of Pansu bubbles.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.3","headline":"Conjecture that the boost-symmetric family maximizes isoperimetric ratio lacks any symmetry-forcing argument or comparison to asymmetric competitors","rationale":"The reader's weakest_assumption exactly locates the gap between the classification result and the isoperimetric conjecture. No other internal inconsistency appears from the abstract-level description; the symmetry restriction is the single load-bearing assumption for the strongest claim.","tokens_in":1684,"tokens_out":291,"duration_ms":8281,"concrete_test":"Construct a one-parameter family of C^2 perturbations of one of the proposed surfaces that break boost symmetry while preserving the enclosed volume to first order; numerically evaluate the first variation of horizontal area (using the formula derived in the paper) and check whether any perturbation yields a negative first variation; if so, the surface is not even a local maximizer.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the identified two-sheeted acausal boost-symmetric CMC surfaces are natural candidates for isoperimetric maximizers (and the conjecture is stated explicitly). The first-variation formula and the classification are restricted to boost-symmetric surfaces; nothing in the derivation shows that a volume-preserving variation that breaks boost symmetry cannot increase the horizontal area-to-volume ratio, nor is there a comparison theorem or calibration that would force maximizers to be symmetric. Without such a step the conjecture remains formally unsupported even if the classification itself is correct.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper derives a first-variation formula for horizontal area under volume-preserving radial variations in the sub-Lorentzian Heisenberg group, shows that smooth isoperimetric candidates have constant horizontal mean curvature away from the characteristic set, and gives a complete classification of smooth boost-symmetric constant mean curvature surfaces (their characteristic sets, causal behaviour, and isometry classes). It identifies a specific family of smooth, acausal, boost-symmetric nonzero-CMC surfaces realized as two-sheeted graphs over the exterior of a future hyperbola and conjectures that this family consists of the isoperimetric maximizers.","tokens_in":1795,"tokens_out":310,"duration_ms":22190,"significance":"If the first-variation formula and the classification of boost-symmetric CMC surfaces are correct, the work supplies a direct sub-Lorentzian analogue of the Pansu bubbles together with an explicit first-variation tool that can be used for further isoperimetric analysis in this geometry.","major_comments":[{"comment":"Abstract: the conjecture that the identified two-sheeted boost-symmetric family 'gives the isoperimetric maximisers' is load-bearing for the paper's stated motivation, yet the classification and first-variation formula are derived only under the boost-symmetry assumption; no symmetry-forcing result, calibration argument, or comparison with non-symmetric competitors is supplied to justify that maximizers must lie in this class.","section":"Abstract"}],"minor_comments":[],"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 identifying this point concerning the abstract. We respond to the major comment below.","responses":[{"response":"The first-variation formula is derived without any symmetry assumption, and the classification is explicitly restricted to the boost-symmetric case; both results stand independently of the conjecture. The conjecture itself is framed as arising from the classification together with the known analogy to the rotationally symmetric Pansu bubbles in the sub-Riemannian Heisenberg group. The manuscript does not assert, nor does it attempt to prove, that every isoperimetric maximizer must be boost-symmetric; no symmetry-forcing theorem, calibration, or direct comparison with non-symmetric surfaces is claimed or supplied. We therefore view the conjecture as an open statement motivated by the explicit family we construct, rather than a proven characterization. Because the paper already presents the statement as a conjecture and does not rely on it for any of the proved results, we do not believe a change to the abstract is required.","revision_made":"no","referee_comment":"[Abstract] Abstract: the conjecture that the identified two-sheeted boost-symmetric family 'gives the isoperimetric maximisers' is load-bearing for the paper's stated motivation, yet the classification and first-variation formula are derived only under the boost-symmetry assumption; no symmetry-forcing result, calibration argument, or comparison with non-symmetric competitors is supplied to justify that maximizers must lie in this class."}],"tokens_in":1232,"tokens_out":321,"duration_ms":17777,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main thing to know is that this paper classifies smooth boost-symmetric constant horizontal mean curvature surfaces in the sub-Lorentzian Heisenberg group and singles out one explicit family as a candidate for isoperimetric maximizers.\n\nThey start by deriving the first-variation formula for horizontal area under volume-preserving radial variations, then use it to classify the boost-symmetric CMC surfaces completely, including their characteristic sets, causal behavior, and ambient isometry classes. From the list they extract a family of smooth, acausal surfaces with nonzero constant mean curvature, realized as two-sheeted graphs over the exterior of a future hyperbola. This is presented as the natural sub-Lorentzian version of the Pansu bubbles and leads to the stated conjecture.\n\nThe classification under the symmetry assumption and the concrete graph description appear new relative to the referenced literature. The variation formula and the direct symmetry reduction are handled in a straightforward way, and the paper is clear about working only inside the boost-symmetric class.\n\nThe soft spot is the conjecture itself. Nothing in the work shows that isoperimetric maximizers must be boost-symmetric or rules out better asymmetric competitors, so the claim that this family maximizes the ratio stays unsupported even if the classification holds. That gap is load-bearing for the strongest statement.\n\nThe paper is for specialists in sub-Lorentzian or sub-Riemannian geometric analysis who care about CMC surfaces and isoperimetric questions in stratified groups. Readers looking for explicit constructions will get something concrete from the classification.\n\nIt deserves a serious referee because the classification is a tangible step forward in an underexplored setting. I would send it to peer review, with the expectation that the conjecture section will need either a symmetry argument or a more cautious statement.","headline":"The paper classifies boost-symmetric CMC surfaces in the sub-Lorentzian Heisenberg group and flags an explicit two-sheeted graph family as isoperimetric candidates, but the maximizer conjecture has no symmetry argument.","tokens_in":2273,"tokens_out":440,"would_cite":false,"duration_ms":19471,"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 family of boost-symmetric surfaces with constant horizontal mean curvature in the sub-Lorentzian Heisenberg group is conjectured to solve the isoperimetric problem.","keywords":["constant mean curvature","sub-Lorentzian Heisenberg group","isoperimetric problem","boost-symmetric surfaces","horizontal mean curvature","Pansu bubbles","characteristic set"],"falsifier":"Existence of a smooth isoperimetric surface that is not boost-symmetric and achieves a strictly larger isoperimetric ratio than the identified family.","tokens_in":2590,"feed_emoji":"","tokens_out":644,"duration_ms":17038,"temperature":0.7,"pith_summary":"The paper derives the first-variation formula for horizontal area under volume-preserving radial variations and concludes that smooth isoperimetric candidates must have constant horizontal mean curvature away from the characteristic set. It then classifies all smooth boost-symmetric constant mean curvature surfaces by their characteristic sets, causal behaviour, and ambient isometry classes. From the classification the authors isolate one family of smooth, acausal, nonzero-constant-mean-curvature surfaces that can be expressed as two-sheeted graphs over the exterior of a future hyperbola. These surfaces are presented as the direct sub-Lorentzian analogue of the Pansu bubbles. The authors conjecture that the family realises the isoperimetric maximum in the sub-Lorentzian Heisenberg group.","feed_headline":"Boost-symmetric CMC surfaces conjectured as isoperimetric maximizers","feed_subtitle":"A two-sheeted graph family over the exterior of a future hyperbola is the sub-Lorentzian counterpart to Pansu bubbles.","key_machinery":"The complete classification of smooth boost-symmetric constant mean curvature surfaces by characteristic sets, causal behaviour, and sub-Lorentzian isometry classes.","core_discovery":"From this classification, we single out a family of smooth, acausal, boost-symmetric surfaces with nonzero constant mean curvature. Written as a two-sheeted graph over the exterior of a future hyperbola, this family is a natural sub-Lorentzian analogue of the Pansu bubbles and leads us to conjecture that it gives the isoperimetric maximisers in the sub-Lorentzian Heisenberg group.","pith_inferences":["If isoperimetric maximisers need not be boost-symmetric, asymmetric competitors could exceed the ratio attained by the identified family.","The first-variation formula and classification method may apply directly to other sub-Lorentzian or sub-Riemannian three-dimensional geometries.","Numerical computation of the isoperimetric ratio attained by the two-sheeted graph family would provide a concrete test of the conjecture."],"forward_implications":["Smooth isoperimetric candidates have constant horizontal mean curvature away from the characteristic set.","The identified family consists of acausal surfaces that remain regular for the isoperimetric problem.","These surfaces supply explicit examples against which the isoperimetric conjecture in the sub-Lorentzian Heisenberg group can be tested.","The classification organises all boost-symmetric constant-mean-curvature surfaces into finitely many isometry classes."],"fun_headline_variants":["Boost-symmetric CMC surfaces classified in sub-Lorentzian Heisenberg group","Acausal boost-symmetric surfaces conjectured as isoperimetric maximizers","Two-sheeted CMC graphs over future hyperbola as Pansu analogues","Classification of boost-symmetric CMC surfaces in sub-Lorentzian group"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The classification and conjecture rest on restricting attention to boost-symmetric surfaces.","fun_headline_variants_meta":{"raw":{"variants":["Boost-symmetric CMC surfaces classified in sub-Lorentzian Heisenberg group","Acausal boost-symmetric surfaces conjectured as isoperimetric maximizers","Two-sheeted CMC graphs over future hyperbola as Pansu analogues","Classification of boost-symmetric CMC surfaces in sub-Lorentzian group"]},"model":"grok-4.3","cost_usd":0.004799,"raw_usage":{"total_tokens":2326,"prompt_tokens":598,"num_sources_used":0,"completion_tokens":74,"cost_in_usd_ticks":47987000,"prompt_tokens_details":{"text_tokens":598,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1654,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":598,"tokens_out":74,"duration_ms":10628,"temperature":1.0,"reasoning_tokens":1654,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T20:52:56.054964+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Existence of a smooth isoperimetric surface that is not boost-symmetric and achieves a strictly larger isoperimetric ratio than the identified family.","supporting_citations":[],"review_version":1}