{"id":"cdbea2bf-626c-4bbb-8336-f115e3770256","arxiv_id":"2606.02426","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves that eigenfunctions in Lebesgue-almost-every generic Bunimovich stadium fail to equidistribute in physical space along a subsequence, via a constructed observable Q that converts Hassell's phase-space obstruction into a physical-space statement.","lead":"The paper shows that for almost every stadium in a one-parameter family of Bunimovich stadia, there are eigenfunctions whose mass fails to equidistribute in physical space with respect to some smooth observable. This strengthens an earlier phase-space result and answers Tao's question on physical-space scarring in the generic setting.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Conversion of Hassell's phase-space scarring to physical non-equidistribution may fail if position marginals of the scarring measures remain Lebesgue","rationale":"The reader's weakest assumption is precisely the conversion step from phase-space obstruction to a physical observable Q. The concern is internal to the argument (not external consensus) and is the only place where the strengthening could fail even if Hassell's phase-space result holds. If the explicit constructions in the manuscript confirm that ν deviates from Lebesgue in every case, the result stands; otherwise it requires restriction to a subset of t.","tokens_in":1725,"tokens_out":407,"duration_ms":32349,"concrete_test":"In the full manuscript, for each case of Hassell's classification, extract the explicit scarring measure μ and compute its position marginal ν = π_* μ; check whether there exists smooth mean-zero Q with ∫ Q dν ≠ 0 and with the zero set of Q having smooth relative boundary inside the stadium. If ν = Lebesgue in any case, the headline claim fails for that case.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Hassell's classification produces, for a.e. t, a phase-space semiclassical measure μ ≠ Liouville that is a weak-* limit point of |u_j|^2 dx dθ. The paper constructs, in each case, a smooth mean-zero Q(x) on the stadium such that lim ⟨Q u_j, u_j⟩ = ∫ Q d(π_* μ) ≠ 0, where π projects to position. This requires π_* μ ≠ Lebesgue (otherwise every position observable equidistributes). The abstract asserts the construction succeeds in every case of the classification, but supplies no explicit verification that the position marginal deviates from uniform Lebesgue on a set with smooth relative boundary; if any case has μ supported on a momentum-localized set whose position projection is already Lebesgue, no such Q exists and the physical-space claim collapses.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves that for Lebesgue-almost every t ∈ [1,2] in the family of Dirichlet stadia S_t (rectangular part of height π and half-length π t/2), there exist real eigenfunctions u_j and a smooth mean-zero physical observable Q such that ⟨Q u_j, u_j⟩ admits a non-zero subsequential limit. Consequently, eigenfunction mass fails to equidistribute on a fixed interior region whose relative boundary is smooth. The argument proceeds by invoking Hassell's classification of generic stadia and, in each case, constructing a suitable Q that converts the known phase-space semiclassical measure obstruction into a non-vanishing integral against the position marginal.","tokens_in":1886,"tokens_out":587,"duration_ms":21740,"significance":"If the constructions are correct, the result supplies an affirmative answer to Tao's question on physical-space scarring within Hassell's generic-stadia framework and strengthens the phase-space non-QUE theorem to a concrete physical observable. The work is significant for quantum chaos because it exhibits explicit position-space observables that detect scarring without requiring microlocal analysis at the final step. The reuse of the existing classification is efficient; the novelty resides in the case-specific Q constructions that guarantee the position projection deviates from Lebesgue.","major_comments":[{"comment":"§1 (and the case-analysis section): the central claim requires that, for every case in Hassell's classification, the position marginal π_* μ of the scarring measure satisfies π_* μ ≠ Lebesgue on a set of positive measure with smooth relative boundary. The manuscript asserts that an appropriate Q exists in each case, but does not supply an explicit verification (or a short argument) that the support of μ forces the marginal to be non-uniform; this step is load-bearing for the physical-space conclusion.","section":"§1 and case-analysis section"},{"comment":"The construction of Q in the 'bouncing-ball' or 'whispering-gallery' cases (whichever appear in the classification) must be checked to ensure ∫ Q d(π_* μ) ≠ 0 while Q remains smooth and mean-zero; if the paper only invokes existence without exhibiting the sign or support condition on the marginal, the reduction from phase-space to physical space remains formal.","section":"case-analysis section"}],"minor_comments":[{"comment":"Notation for the stadium family S_t and the projection π should be introduced once with a diagram or explicit formula to aid readers unfamiliar with the geometry.","section":"Introduction"},{"comment":"A short table or enumerated list mapping each Hassell case to the corresponding Q would improve readability of the case analysis.","section":"case-analysis section"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and for identifying the need for greater explicitness in the case-by-case verification. We agree that the reduction from phase-space measures to physical-space observables requires a short, self-contained argument showing that each scarring measure in Hassell's classification has a non-uniform position marginal on a set of positive measure with smooth relative boundary. We will supply this in the revision.","responses":[{"response":"We accept the point. In the revised manuscript we will insert, immediately after the invocation of Hassell's classification in §1 and again at the start of the case-analysis section, a uniform short argument: for each of the (finitely many) possible supports of the scarring measures appearing in Hassell's list, the projection π_* μ is supported on a proper closed subset of the stadium whose complement has positive area and whose boundary is smooth (or can be smoothed by a small perturbation that does not affect the integral). This immediately implies the existence of a smooth mean-zero Q with ∫ Q d(π_* μ) ≠ 0. The argument uses only the explicit geometric description of the supports already present in Hassell’s work.","revision_made":"yes","referee_comment":"[§1 and case-analysis section] §1 (and the case-analysis section): the central claim requires that, for every case in Hassell's classification, the position marginal π_* μ of the scarring measure satisfies π_* μ ≠ Lebesgue on a set of positive measure with smooth relative boundary. The manuscript asserts that an appropriate Q exists in each case, but does not supply an explicit verification (or a short argument) that the support of μ forces the marginal to be non-uniform; this step is load-bearing for the physical-space conclusion."},{"response":"We will make the constructions fully explicit for every case, including any bouncing-ball or whispering-gallery measures that arise. For each such measure we will define Q by taking a smooth cutoff that is strictly positive on the region where the density of π_* μ exceeds its average value and strictly negative on a complementary set of equal measure, then normalize to have mean zero. Because the support of π_* μ is a proper subset with smooth relative boundary, such a Q exists and satisfies ∫ Q d(π_* μ) ≠ 0 by construction. These explicit choices will be written out in the case-analysis section.","revision_made":"yes","referee_comment":"[case-analysis section] The construction of Q in the 'bouncing-ball' or 'whispering-gallery' cases (whichever appear in the classification) must be checked to ensure ∫ Q d(π_* μ) ≠ 0 while Q remains smooth and mean-zero; if the paper only invokes existence without exhibiting the sign or support condition on the marginal, the reduction from phase-space to physical space remains formal."}],"tokens_in":1457,"tokens_out":607,"duration_ms":15469,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that this work turns the known phase-space obstruction from Hassell into a statement about physical observables failing to equidistribute for generic stadia. For almost every t they produce real eigenfunctions and a smooth mean-zero Q such that the matrix element has a nonzero subsequential limit, so mass does not spread uniformly on a fixed interior region with smooth boundary.\n\nWhat the paper does well is the explicit conversion step. It takes each case in Hassell's classification and supplies a Q that makes the position projection of the scarring measure give a nonzero integral. That directly answers Tao's question in this setting without new semiclassical analysis.\n\nThe soft spots are modest. The abstract asserts the constructions succeed in every case, which implies the position marginals deviate from Lebesgue in all of them, so the stress-test worry about uniform projections does not appear to bite. Still, without the explicit Q formulas or the short calculations showing the integral is nonzero, it is hard to judge how delicate the choices are. The full text presumably contains those details.\n\nNo circularity or fitting issues show up. The argument stays within the existing classification and adds only the position observables.\n\nThis is for specialists in quantum ergodicity and billiard eigenfunctions. Anyone who has read Hassell will see the value immediately. It deserves a serious referee because the strengthening is concrete and the construction, if it checks out, is a useful addition to the literature.","headline":"The paper converts Hassell's phase-space scarring into physical non-equidistribution by building case-by-case observables Q that pick up non-uniform position marginals.","tokens_in":2379,"tokens_out":367,"would_cite":false,"duration_ms":20693,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"For Lebesgue-almost every t in the stadium family, certain eigenfunctions fail to spread evenly over some fixed interior region.","keywords":["Bunimovich stadium","eigenfunction scarring","quantum ergodicity","non-equidistribution","Dirichlet eigenfunctions","physical space","billiard domains"],"falsifier":"A concrete value of t together with a proof that every smooth mean-zero observable Q has ⟨Q u_j, u_j⟩ tending to zero along every eigenfunction sequence.","tokens_in":2605,"feed_emoji":"","tokens_out":689,"duration_ms":19985,"temperature":0.7,"pith_summary":"The paper proves that in the one-parameter family of Bunimovich stadia with rectangular half-length scaled by t, for almost every t there exist sequences of real eigenfunctions whose mass does not equidistribute in physical space. This is established by building, for each stadium in Hassell's generic classification, a smooth mean-zero observable Q whose expectation against the eigenfunctions stays away from zero along a subsequence. The construction converts an existing phase-space obstruction into a concrete physical-space integral over a region with smooth boundary inside the stadium. The result strengthens the known failure of quantum unique ergodicity by making the non-equidistribution visible without reference to momentum. It thereby answers Tao's question affirmatively for generic stadia.","feed_headline":"Generic stadia exhibit physical scarring of eigenfunctions","feed_subtitle":"For almost every t, some eigenfunctions fail to spread evenly over fixed interior regions with smooth boundaries","key_machinery":"A smooth mean-zero physical observable Q, constructed case-by-case from Hassell's classification of generic stadia, that converts the known phase-space obstruction to QUE into a non-vanishing physical-space integral.","core_discovery":"For the family of Dirichlet stadia S_t whose rectangular part has height π and half-length π t/2 with t in [1,2], for Lebesgue almost every t there exist real eigenfunctions u_j and a smooth mean-zero physical observable Q such that ⟨Q u_j, u_j⟩ has a non-zero subsequential limit; consequently the eigenfunction mass fails to equidistribute on a fixed region whose relative boundary in the interior of the stadium is smooth.","pith_inferences":["The same case-by-case construction of Q may extend to other billiard families where phase-space scarring has already been established.","Physical-space non-equidistribution could be used to test numerical eigenfunction computations on stadia for concrete values of t.","The result raises the question whether an explicit, non-generic Q can be written down for at least one concrete stadium."],"forward_implications":["Eigenfunction mass fails to equidistribute on a fixed interior region whose relative boundary is smooth.","This gives a physical-space strengthening of Hassell's non-QUE theorem for generic stadia.","The construction supplies an affirmative answer to Tao's question inside Hassell's generic setting."],"fun_headline_variants":["Physical scarring in generic Bunimovich stadia","Generic stadia scar eigenfunctions in physical space","Physical space scarring for generic Bunimovich stadia","Stadia eigenfunctions non-equidistribute in physical space"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"Hassell's classification of generic stadia permits, in each case, the construction of a suitable smooth mean-zero physical observable Q that converts the known phase-space obstruction into a non-vanishing physical-space integral.","fun_headline_variants_meta":{"raw":{"variants":["Physical scarring in generic Bunimovich stadia","Generic stadia scar eigenfunctions in physical space","Physical space scarring for generic Bunimovich stadia","Stadia eigenfunctions non-equidistribute in physical space"]},"model":"grok-4.3","cost_usd":0.007854,"raw_usage":{"total_tokens":3568,"prompt_tokens":638,"num_sources_used":0,"completion_tokens":55,"cost_in_usd_ticks":78537000,"prompt_tokens_details":{"text_tokens":638,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2875,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":638,"tokens_out":55,"duration_ms":26026,"temperature":1.0,"reasoning_tokens":2875,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T13:34:03.018118+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A concrete value of t together with a proof that every smooth mean-zero observable Q has ⟨Q u_j, u_j⟩ tending to zero along every eigenfunction sequence.","supporting_citations":[],"review_version":1}