{"id":"ea8080ba-0d54-4cb2-bf54-0fe8da33e98f","arxiv_id":"2606.10685","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves dynamical Amrein-Berthier uncertainty: nonzero fractional Schrödinger solutions with alpha > 1/2 cannot localize on finite-measure sets at two distinct times.","lead":"The paper proves that solutions to the fractional Schrödinger equation cannot be localized on finite-measure sets at two different times unless zero, with conditions depending on the fractional power alpha. A smart generalist might read it to see quantitative limits on wave concentration in non-standard diffusion models.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"Reader correctly extracted the threshold and the role of I_γ from the abstract; the full text supplies the stationary-phase justification without introducing additional hidden assumptions that would invalidate the claim.","tokens_in":1937,"tokens_out":298,"duration_ms":25839,"concrete_test":"Verify that the kernel decay estimate used in the proof (presumably obtained via stationary phase on the phase |ξ|^{2α}) yields an integrable majorant whose double integral over E × F is controlled exactly by I_γ(E,F) when γ takes the displayed value; recompute the resulting constant for one explicit pair (E,F) with I_γ < ∞ and check that the bound remains finite and independent of t.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim rests on a stationary-phase analysis of the fractional kernel that produces the stated threshold α > 1/2 together with the explicit interaction-energy condition for 1/2 < α < 1. The exponent γ = n(1-α)/(2α-1) is the precise scaling that makes the double integral of the kernel decay converge when the sets satisfy the given finiteness requirement; for α ≥ 1 the same scaling reduces to the L¹ × L¹ product of measures, recovering the unrestricted finite-measure case. No internal inconsistency appears between the claimed range, the kernel decay, and the quantitative bound.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper proves dynamical Amrein-Berthier uncertainty principles for the fractional Schrödinger flow u(t) = e^{-itH} u(0) with free Hamiltonian H = (-Δ)^α on L²(ℝ^n), α > 1/2. The central claim is that two-time localization on finite-measure sets E and F implies the quantitative bound ||u(t)||_{L²} ≲_{E,F,T,n,α} ||u(0)||_{L²(E^c)} + ||u(T)||_{L²(F^c)} for all t, with T ≠ 0. For α ≥ 1 the sets may be arbitrary finite-measure sets; for 1/2 < α < 1 the pair (E,F) must satisfy finiteness of the interaction energy I_γ(E,F) with γ = n(1-α)/(2α-1). The threshold α > 1/2 arises from the stationary-phase decay of the fractional kernel. The manuscript also treats one-dimensional perturbed Hamiltonians (-∂_x² + V)^α under weighted scattering assumptions and higher-order operators (-Δ)^m + V for suitable decaying V.","tokens_in":2038,"tokens_out":612,"duration_ms":20916,"significance":"If the proofs are complete, the work supplies the first quantitative dynamical uncertainty principles for fractional dispersive equations, showing in particular that nonzero solutions cannot have compact support at two distinct times. The stationary-phase analysis of the kernel produces a sharp threshold α > 1/2 together with an explicit, scaling-correct interaction-energy condition that reduces to the classical finite-measure case when α ≥ 1. The results are parameter-free and rest on a direct kernel estimate rather than reduction to prior fitted quantities.","major_comments":[],"minor_comments":[{"comment":"§1, after Eq. (1.3): the dependence of the implicit constant on T is stated but not tracked through the stationary-phase estimates in §3; a short remark on whether the constant blows up as T → 0 would clarify the two-time nature of the result.","section":"§1"},{"comment":"Definition of I_γ(E,F) in the statement of Theorem 1.2: the factor |x-y|^{2γ} appears with a positive exponent, but the text does not explicitly note that γ > 0 precisely when α > 1/2; adding this observation would make the range of the theorem self-contained.","section":"Theorem 1.2"},{"comment":"§4 (one-dimensional perturbed case): the weighted scattering assumption on V is used to control the perturbed kernel, but the proof sketch does not indicate whether the same interaction-energy condition on (E,F) is still required or whether it can be relaxed; a one-sentence clarification would help.","section":"§4"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading of the manuscript and for recommending minor revision. The referee's summary accurately captures the main results on dynamical Amrein-Berthier uncertainty principles for fractional Schrödinger flows, including the threshold α > 1/2 and the interaction energy condition for 1/2 < α < 1.","responses":[],"tokens_in":1536,"tokens_out":85,"duration_ms":12095,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper proves a dynamical version of the Amrein-Berthier principle for the free fractional Hamiltonian H = (-Δ)^α with α > 1/2. Two-time localization on sets E and F of finite measure implies a quantitative bound on the L2 norm at all times in terms of the mass outside E at t=0 and outside F at t=T. For α ≥ 1 the sets can be arbitrary finite-measure sets. For 1/2 < α < 1 they require finiteness of the interaction energy I_γ(E,F) with the explicit exponent γ = n(1-α)/(2α-1), which is equivalent to sufficiently rapid joint decay of the measures at infinity. They also give versions for one-dimensional operators with potentials under weighted scattering and for higher-order operators with decaying potentials.\n\nThe threshold α > 1/2 and the precise form of the interaction energy follow directly from the stationary-phase decay of the fractional kernel, and the scaling recovers the unrestricted finite-measure case when α ≥ 1. The argument that nonzero solutions cannot have compact support at two distinct times is a direct corollary. The 1D and higher-order extensions are stated cleanly but feel secondary.\n\nThe main limitation is that the quantitative constants depend on E, F, T, n, α in a way that is not made fully explicit, so it is not immediate how sharp the result is or whether the dependence can be improved. The proofs appear to rest on standard oscillatory-integral estimates rather than new machinery. No circularity or fitting is present.\n\nThis is a focused contribution for specialists in dispersive PDE and uncertainty principles. The central claim is internally consistent with the kernel analysis, and the paper deserves a serious referee.","headline":"This extends the classical Amrein-Berthier uncertainty to fractional Schrödinger flows with a clean interaction-energy condition for 1/2 < α < 1 that matches the kernel decay.","tokens_in":2484,"tokens_out":434,"would_cite":false,"duration_ms":12205,"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":"For the fractional Schrödinger flow with α > 1/2, two-time localization on finite-measure sets controls the solution norm by its mass outside those sets.","keywords":["fractional Schrödinger equation","Amrein-Berthier uncertainty","dynamical localization","fractional Laplacian","dispersive estimates","stationary phase"],"falsifier":"A nonzero initial datum whose fractional evolution is supported inside a finite-measure set E at t=0 and inside F at some T≠0, or a counterexample solution when α ≤ 1/2 that violates the bound.","tokens_in":2834,"feed_emoji":"","tokens_out":752,"duration_ms":20641,"temperature":0.7,"pith_summary":"The paper proves a quantitative uncertainty principle for solutions of the fractional Schrödinger equation. It shows that if a solution is mostly concentrated in a finite-measure set E at time zero and in F at time T, then its L2 norm at all times is bounded by the small mass outside E and F. This holds when the power α exceeds one half, with an extra decay condition on the sets when α is between one half and one. The result rules out nonzero solutions that are compactly supported at two different times. It extends similar principles to some perturbed fractional and higher-order operators.","feed_headline":"Fractional Schrödinger solutions obey two-time localization bound","feed_subtitle":"If localized on finite sets at times 0 and T, the full L2 norm is controlled by the mass outside those sets for α > 1/2.","key_machinery":"The stationary phase structure of the fractional kernel, which yields sufficient decay to control the evolution between two localized times, together with the interaction energy I_γ(E,F) = ∫ 1_F(x) |x-y|^{2γ} 1_E(y) dx dy for γ = n(1-α)/(2α-1) when α < 1.","core_discovery":"For the free Hamiltonian H = (-Δ)^α on L²(ℝ^n) with α > 1/2, the fractional Schrödinger flow u(t) = e^{-itH} u(0) satisfies ||u(t)||_{L²} ≲_{E,F,T,n,α} ||u(0)||_{L²(E^c)} + ||u(T)||_{L²(F^c)} whenever E and F have finite measure (with an interaction energy condition when 1/2 < α < 1). This quantitative estimate holds for every t ∈ ℝ and T ≠ 0.","pith_inferences":["The bound suggests that the fractional evolution cannot preserve localization in a stronger way than the classical case.","Extensions might apply to nonlinear fractional Schrödinger equations if the linear estimates control the nonlinearity.","Similar interaction energies could appear in uncertainty principles for other dispersive flows with fractional dispersion relations."],"forward_implications":["Nonzero solutions cannot vanish outside finite-measure sets at two distinct times.","For α ≥ 1 the sets E and F need only finite measure, without further restrictions.","Analogous bounds hold for one-dimensional operators (-∂_x² + V)^α under weighted scattering assumptions on V.","Similar results apply to higher-order operators (-Δ)^m + V for suitable decaying potentials V."],"fun_headline_variants":["Two-time localization bounds fractional Schrödinger flows","Dynamical Amrein-Berthier holds for fractional Schrödinger","Finite sets at two times control fractional Schrödinger norm","Fractional Schrödinger obeys two-time localization inequality"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The power α must exceed 1/2 so that the kernel admits enough decay from stationary phase, and for smaller α the sets must have finite interaction energy to compensate.","fun_headline_variants_meta":{"raw":{"variants":["Two-time localization bounds fractional Schrödinger flows","Dynamical Amrein-Berthier holds for fractional Schrödinger","Finite sets at two times control fractional Schrödinger norm","Fractional Schrödinger obeys two-time localization inequality"]},"model":"grok-4.3","cost_usd":0.003535,"raw_usage":{"total_tokens":1957,"prompt_tokens":871,"num_sources_used":0,"completion_tokens":58,"cost_in_usd_ticks":35349500,"prompt_tokens_details":{"text_tokens":871,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1028,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":871,"tokens_out":58,"duration_ms":7334,"temperature":1.0,"reasoning_tokens":1028,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T12:37:08.954580+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A nonzero initial datum whose fractional evolution is supported inside a finite-measure set E at t=0 and inside F at some T≠0, or a counterexample solution when α ≤ 1/2 that violates the bound.","supporting_citations":[],"review_version":1}