{"id":"b98666fd-aa7d-4091-bc54-70b8f7085b9d","arxiv_id":"2412.14698","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Fractional Helmholtz operators admit high-frequency geometrical optics solutions, and for s≥1/2 these give Hölder stable recovery of the potential from multi-frequency boundary Cauchy data.","lead":"The paper constructs geometrical-optics wave solutions for a nonlocal fractional version of the Helmholtz equation and uses them to prove a Hölder stability estimate for recovering an unknown potential from boundary measurements. This is a substantial improvement over the logarithmic stability previously available for fractional wave and Schrödinger inverse problems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.3's Hölder exponent γ independent of k is unsupported: the proof's final step requires ‖Q‖_{H^{t_M}} for arbitrarily large t_M, but q1,q2 are only assumed in H^k.","rationale":"Good-faith read: the paper's main construction is a genuine extension of geometrical optics to fractional Helmholtz operators; Theorems 1.1 and 1.2 and the resolvent estimates in Section 3 are substantial, and I do not see a circularity or a fabricated step there. The concern is confined to the application in Theorem 1.3. The reader's weakest assumption is exactly the right one. The proof's Step 5 uses smallness of the Cauchy-data distance to control the geodesic ray transform, but to extract the potential one applies a stability estimate for I*I in H^{-1}, and the interpolation step requires a high Sobolev norm of Q. The phrase 't_M can be taken arbitrarily large' is what makes γ independent of k; however, t_M is tied to the order M of the amplitude expansion, and the higher-order amplitudes are only bounded in L2 by H^{t_M} norms of the principal amplitude. Since b1 is chosen as I(I*IQ)ν_p·θ, that norm is controlled only by a Sobolev norm of Q of the same order. Hence the proof's conclusion requires q_j to be smooth, or at least to have uniformly bounded norms in all the Sobolev spaces needed as M grows, not merely H^k. This does not undermine the parametrix itself or the stability strategy; it is a formulation gap that can likely be repaired by either assuming q_j ∈ C∞ with bounds or by stating γ = γ(s,k). Therefore I keep the reader's CONDITIONAL verdict.","tokens_in":31718,"tokens_out":4571,"duration_ms":42878,"concrete_test":"Track t_M through the induction in Step 4: for a fixed desired approximation order M, compute the minimal t_M such that ‖a_j‖_{L2} ≲ ‖a0‖_{H^{t_M}} for j ≤ A_M, and check whether t_M → ∞ as M → ∞. If yes, then the line 'using the high order Sobolev estimates for q1,q2 we get ‖Q‖_{H^{t_M}} ≲ 1' is unusable under the stated hypothesis q_j ∈ H^k. A direct way to expose the gap: choose q1 = 0 and q2 ∈ H^k with compact support but q2 ∉ H^{t_M} for the M that would be needed to approach the claimed γ; then the displayed bound in Step 5 fails exactly at the interpolation estimate, locating the missing regularity assumption.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing issue is the regularity hypothesis in Theorem 1.3, not the geometrical optics parametrix. In the last step of the proof (Section 4, Step 5) the authors need the bound ‖Q‖_{H^{t_M}} ≲ 1, where Q=(q1−q2)e^{−2f+iJ(q1−q2)}r^{−n} and t_M is chosen as large as needed so that the final Hölder exponent t_M/(t_M+1) approaches 1. The only hypotheses on the potentials are q1,q2 ∈ H^k with ‖q_j‖_{H^k} ≤ N; for fixed k this gives no control in H^{t_M} once t_M > k. The choice b1 = I(I*IQ)ν_p·θ makes the principal amplitudes inherit exactly this H^{t_M} regularity, so the issue cannot be bypassed by a different choice of b1. The manuscript itself flags the gap: the proof ends with 'However, this implies that we need to assume higher order Sobolev estimates for q1,q2.' Thus the stated independence of γ from k is not established; the proof supports either smooth potentials with all-order bounds or an exponent γ=γ(s,k) depending on k. The introduction assumes q∈C∞ while the theorem states H^k, and this inconsistency should be resolved.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs high-frequency geometrical optics (GO) approximate solutions for the fractional Helmholtz equation ((-Δ)^s - τ^{2s} r^{2s} + q)u = 0, with a detailed stationary-phase expansion that separates the regimes s∈(0,1/2) and s∈[1/2,1). In the latter regime the eikonal equation agrees with the classical one and the nonlocality affects only the zeroth-order term; in the former regime additional phase functions appear. The approximate solutions are upgraded to exact solutions by a semiclassical resolvent estimate obtained through a positive-commutator argument. The parametrix is then used to prove Theorem 1.3, a Hölder stability estimate for recovering q from multi-frequency Cauchy data when r is fixed and simple, which would improve the usual logarithmic stability arising from Runge approximation.","tokens_in":31956,"tokens_out":6755,"duration_ms":58404,"significance":"If Theorem 1.3 is established at the stated regularity, this is a substantial advance: it transfers classical GO and geodesic-ray-transform techniques to a nonlocal operator whose semiclassical symbol is not smooth at ξ=0, and it gives the first Hölder-type multi-frequency stability in this fractional setting. The stationary-phase expansions in Section 2 and the frequency-cutoff positive-commutator resolvent argument in Section 3 are detailed, parameter-free, and credible. A particular strength is that the parametrix remainder is controlled in semiclassical Sobolev norms with explicit dependence on the approximation order. The main load-bearing defect is in the final regularity step of Theorem 1.3, not in the GO construction itself.","major_comments":[{"comment":"The final interpolation step is not justified under the hypotheses of Theorem 1.3. After the estimate involving ‖Q‖_{H^{-1}} and ‖Q‖_{H^{t_M}}, the proof states: 'Observe now that Q is independent of τ, and therefore using the high order Sobolev estimates for q1,q2 we get ‖Q‖_{H^{t_M}} ≲ 1.' But Theorem 1.3 only assumes q1,q2∈H^k, while t_M may be arbitrarily large; for fixed k there is no uniform control in H^{t_M} once t_M>k. The manuscript itself flags this gap in the very next sentence: 'However, this implies that we need to assume higher order Sobolev estimates for q1,q2.' Since t_M also enters the final Hölder exponent through t_M/(t_M+1), the stated conclusion that γ depends only on s is not established by the proof. The theorem should either assume q1,q2 are smooth with uniform C^M-bounds, or state γ=γ(s,k) and set t_M=k, with a quantitative proof of the preceding estimates in terms of k.","section":"§4, Step 5 (end of proof of Theorem 1.3)"},{"comment":"The proof of Theorem 1.3 uses smooth-potential results without a smoothing argument, despite the theorem assuming only q1,q2∈H^k. Proposition 2.2 and Proposition 3.1 are stated for r,q∈C∞(Ω), and Step 2 invokes the Section 2 approximate solutions and Proposition 3.1 directly. In particular, the transport equations and the bounds ‖a0,j‖_{H^{t_M}} are derived for smooth q. Either Theorem 1.3 should be restated for smooth potentials with uniform bounds, or a regularization argument with quantitative control in H^k must be supplied. This is a regularity-consistency issue separate from, but related to, the final H^{t_M} gap.","section":"§4, Step 2 and Propositions 2.2, 3.1"}],"minor_comments":[{"comment":"The introduction states 'q∈C∞(Rn)', while Theorem 1.3 assumes q1,q2∈H^k; these hypotheses should be reconciled, and the regularity assumptions in the introduction should match the theorem.","section":"Introduction, p. 1"},{"comment":"In the statement of Theorem 1.2, the transport equation is written as '2∇ϕ·∇a0 + b_s a0 = 0', but only ϕ0 has been introduced; this should be ϕ0, or ϕ should be defined explicitly.","section":"Theorem 1.2"},{"comment":"Proposition 2.2 uses the norm H^β_scl(Ω) before the semiclassical Sobolev norm is defined in Section 3; the norm should be defined or referenced before its first use.","section":"Proposition 2.2 and Section 3"},{"comment":"The notation b1(θ) := I(I*IQ)ν_p·θ is not explained; the paper should state precisely how the normal ν_p is used and which stability estimate for I*I on simple manifolds is being invoked.","section":"§4, Step 5"}],"recommendation":"major_revision","confidential_remarks":"This is a strong paper with credible GO and resolvent arguments. The only serious obstacle is the regularity statement in Theorem 1.3: the proof requires arbitrarily high Sobolev regularity for q1,q2, while the theorem assumes only H^k, and the gamma independent of k is therefore not established. The reliance on [38] is appropriate, since that is the local-case analogue providing the ray-transform stability input; I see no circularity. If the authors either strengthen the assumption to smooth potentials with uniform bounds or make gamma depend on k, the paper would be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a real advance, with one genuinely broken corner in the statement of Theorem 1.3. The parametrix construction for the fractional Helmholtz equation is new, and the bulk of the analysis looks solid. The eikonal/transport structure, the s=1/2 threshold, and the multi-phase behavior for s<1/2 are all worked out in detail, and the resolvent upgrade in Section 3 is a serious piece of work. If the construction survives a careful referee, it replaces logarithmic stability with Hölder stability in this setting, which is the right kind of payoff.\n\nThe soft spot is exactly where the stress-test put it. At the end of Step 5 in Section 4 the proof needs Q = (q1-q2)e^{-2f+iJ(q1-q2)}r^{-n} to be bounded in H^{t_M} for t_M arbitrarily large in order to make γ independent of k. The theorem only assumes q1,q2 ∈ H^k with a fixed k. The manuscript even admits this: 'However, this implies that we need to assume higher order Sobolev estimates for q1,q2.' That is not a hole in the GO machinery; it is a mismatch between the theorem statement and the proof. The fix is likely one of two: assume q1,q2 smooth (or H^∞ with bounds) and state γ depending only on s, or state γ = γ(s,k,N). Either way the result remains significant.\n\nTwo smaller issues. The remarks about propagation of singularities are informal; the paper only proves the parametrix, not a rigorous propagation theorem. The introduction assumes q ∈ C^∞ while Theorem 1.3 states H^k; that inconsistency is part of the same regularity gap. The reliance on [38] is not circular — the resolvent and ray transform estimates are published independent support, and the self-citations to fractional elasticity are background.\n\nWho should read this: anyone working on fractional Calderón problems or on extending geometric inverse methods to nonlocal operators. It deserves a serious referee; the right verdict is conditional, not reject. I would send it out and ask for the regularity statement to be repaired or made precise.","headline":"A genuine extension of geometrical optics to the fractional Helmholtz equation, with a real but localized regularity gap in the main stability theorem that needs fixing before the stated Hölder exponent can stand.","tokens_in":32483,"tokens_out":1624,"would_cite":true,"duration_ms":14607,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","35R11","35S30"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper constructs geometrical optics solutions for the fractional Helmholtz equation and derives Hölder stability for recovering a potential from multi-frequency Cauchy data when the fractional order is at least 1/2.","keywords":["fractional Helmholtz equation","geometrical optics solutions","eikonal equation","transport equation","Hölder stability","multi-frequency Cauchy data","geodesic ray transform","fractional Calderón problem"],"falsifier":"At the end of Section 4, the estimate ‖q1−q2‖_{L2}^{2+2/t_M} ≲ $δτ^{{2s}}$+$τ^{{−α1}}$‖Q‖_{$H^{{t_M}}$} is combined with ‖Q‖_{$H^{{t_M}}$}≲1. If q1,q2 are only in H^k, the bound ‖Q‖_{$H^{{t_M}}$}≲1 holds only for t_M≤k; setting t_M=k makes the final exponent α1/(4s+2α1)·k/(k+1) explicitly k-dependent. Determining whether this k-dependence is unavoidable—for instance by constructing $H^{1}$ potentials whose stability modulus is not Hölder with any exponent independent of k—would settle the claim as stated.","tokens_in":31458,"feed_emoji":"🌊","tokens_out":12383,"duration_ms":79647,"temperature":0.7,"pith_summary":"The paper constructs high-frequency approximate solutions, called geometrical optics solutions, for the fractional Helmholtz equation, and uses them to prove a stability estimate for an inverse problem. For fractional order s in [1/2,1) the solutions are essentially classical: the phase solves the usual eikonal equation |∇φ|=r and the nonlocal character of the operator appears only as an s-dependent term in the transport equation, so the wave geometry is governed by the geodesics of the metric $r^{2}$δ. For s in (0,1/2) the potential is a strong perturbation and a single-phase ansatz is impossible, forcing a hierarchy of phase functions oscillating at fractional powers of the frequency. As an application, the paper shows that the potential can be recovered from multi-frequency Cauchy data with fixed simple refraction index at Hölder rate, improving on the logarithmic stability given by the usual Runge-approximation argument for fractional wave equations. The result matters because Hölder stability is the modulus that makes numerical recovery of the coefficient feasible.","feed_headline":"Hölder stability proved for fractional Helmholtz potential recovery","feed_subtitle":"With s ≥ 1/2 the nonlocal equation keeps classical wave geometry, so multi-frequency data give polynomial stability instead of logarithmic.","key_machinery":"The engine is a stationary-phase parametrix—an approximate high-frequency solution ansatz—for the nonlocal operator. Because the symbol |ξ|^{2s} of the fractional Laplacian is singular at ξ=0, the expansion splits the symbol into a cut-off piece and a smooth piece, applies nonstationary and stationary phase estimates, and yields the eikonal and transport equations that fix φ and the amplitudes a_l; the approximate solutions are then upgraded to exact ones by a semiclassical resolvent estimate proved with a positive-commutator argument on nontrapping domains. On the inverse side, the decisive objects are the same solutions written in polar normal coordinates of the simple metric g=$r^{2}$δ: the phase φ0=±ρ is geodesic distance, the principal amplitude is a0=b(θ)$e^{{−f+iJ(q)}}$, and substituting these into the Alessandrini identity turns the boundary-data distance δ into the geodesic ray transform I of Q. A stability estimate for the normal ray transform I*I on simple manifolds closes the argument.","core_discovery":"The central discovery is that the fractional Helmholtz operator with s∈[1/2,1) admits geometrical optics solutions u=$e^{{iτφ}}$(a0+$τ^{{-α1}}$a1+...+$τ^{{-αN}}$aN)+R_N with ‖R_N‖_{H^s}=O($τ^{{-C_s N}}$), where the phase satisfies the same eikonal equation as in the classical case and the first transport equation is modified only by a zeroth-order term b_s (with the potential entering at s=1/2). For s∈(0,1/2), the same ansatz provably fails, and the paper constructs instead solutions with a sum of phase functions φ_j, showing that the potential changes the propagation of singularities. Using the former regime, the paper proves Hölder stability: ‖q1−q2‖_{L2(Ω)} ≤ C[sup_{τ≥τ0} δ(C^τ_{r,q1},C^τ_{r,q2})]^γ with γ depending only on s, obtained by converting the difference of Cauchy data, through the Alessandrini identity applied to specially chosen solutions, into a geodesic ray transform of a weighted potential difference and then optimizing over the frequency τ.","pith_inferences":["The regularity gap in the final interpolation step suggests the clean version of Theorem 1.3 may carry an exponent depending on the Sobolev order k of the potentials; a natural test is to let t_M=k and read off the k/(k+1) factor in the exponent.","If the regularity issue is resolved, the ray-transform representation of the data opens the way to explicit reconstruction of q from multi-frequency data, not just stability.","The multi-phase structure below s=1/2 resembles long-range scattering phenomena; the lower-order phase corrections carry extra τ-dependent information that a future inverse strategy might exploit.","The same multi-frequency mechanism could upgrade the fractional elasticity uniqueness result to a stability estimate if the parametrix extends to the vector-valued fractional elasticity operator the authors point to as future work."],"forward_implications":["For s∈[1/2,1), high-frequency solutions of the fractional Helmholtz equation have a single phase satisfying the classical eikonal equation, so the wave geometry follows the geodesics of the metric r^2δ.","Multi-frequency Cauchy data at fixed simple refraction index determine the potential with Hölder stability, a polynomial-in-data improvement over the logarithmic modulus from fixed-frequency Runge approximation.","For s∈(0,1/2), no single-phase geometrical optics solution can exist; the phase must be a sum of functions oscillating at fractional powers of the frequency, reflecting the potential acting as a strong perturbation.","Via the fractional Liouville reduction, the same parametrix applies to the fractional conductivity equation, producing approximate solutions with amplitude prefactor γ^{-1/2}."],"supporting_citations":[{"why":"Supplies the large-frequency stability method and the positive-commutator resolvent estimates that the proof of Theorem 1.3 follows in Section 4.","marker":"[38]"},{"why":"Establishes the fractional Calderón problem framework: well-posedness of the Dirichlet problem, the DN map, and the Alessandrini identity used to relate data to coefficient differences.","marker":"[28]"},{"why":"Provides the stationary-phase and nonstationary-phase theorems used to expand the fractional Laplacian acting on oscillatory test functions.","marker":"[30]"},{"why":"Provides the semiclassical calculus and positive-commutator machinery for the resolvent estimate upgrading approximate to exact solutions.","marker":"[57]"},{"why":"Supplies the definition and properties of simple manifolds and the geodesic ray-transform stability used in the final step.","marker":"[41]"},{"why":"Gives the fixed-frequency logarithmic stability result that Theorem 1.3 improves to Hölder by using many frequencies.","marker":"[46]"}],"fun_headline_variants":["Hölder stability for fractional Helmholtz inverse problem","Fractional wave geometry yields Hölder stability","Nonlocal Helmholtz: log to Hölder stability","Geometric optics for fractional Helmholtz inversion","Classical eikonal powers fractional Helmholtz recovery"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the weighted coefficient Q=(q1−q2)$e^{{−2f+iJ(q1−q2)}}$$r^{{−n}}$ can be treated in arbitrarily high Sobolev norm $H^{{t_M}}$ while the theorem only assumes q1,q2∈H^k; unless the potentials are smooth or the exponent is allowed to depend on k, the advertised s-only Hölder exponent is not what the proof delivers.","fun_headline_variants_meta":{"raw":{"variants":["Hölder stability for fractional Helmholtz inverse problem","Fractional wave geometry yields Hölder stability","Nonlocal Helmholtz: log to Hölder stability","Geometric optics for fractional Helmholtz inversion","Classical eikonal powers fractional Helmholtz recovery"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000554,"raw_usage":{"total_tokens":2698,"prompt_tokens":1061,"completion_tokens":1637,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":677,"completion_tokens_details":{"reasoning_tokens":1575}},"tokens_in":677,"tokens_out":1637,"duration_ms":8412,"temperature":1.0,"reasoning_tokens":1575,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:00:53.136293+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"At the end of Section 4, the estimate ‖q1−q2‖_{L2}^{2+2/t_M} ≲ $δτ^{{2s}}$+$τ^{{−α1}}$‖Q‖_{$H^{{t_M}}$} is combined with ‖Q‖_{$H^{{t_M}}$}≲1. If q1,q2 are only in H^k, the bound ‖Q‖_{$H^{{t_M}}$}≲1 holds only for t_M≤k; setting t_M=k makes the final exponent α1/(4s+2α1)·k/(k+1) explicitly k-dependent. Determining whether this k-dependence is unavoidable—for instance by constructing $H^{1}$ potentials whose stability modulus is not Hölder with any exponent independent of k—would settle the claim as stated.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the large-frequency stability method and the positive-commutator resolvent estimates that the proof of Theorem 1.3 follows in Section 4."},{"cited_title":"Ghosh, M","cited_arxiv_id":null,"evidence_quote":"Establishes the fractional Calderón problem framework: well-posedness of the Dirichlet problem, the DN map, and the Alessandrini identity used to relate data to coefficient differences."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the semiclassical calculus and positive-commutator machinery for the resolvent estimate upgrading approximate to exact solutions."},{"cited_title":"Paternain, M","cited_arxiv_id":null,"evidence_quote":"Supplies the definition and properties of simple manifolds and the geodesic ray-transform stability used in the final step."},{"cited_title":"R¨ uland and M","cited_arxiv_id":null,"evidence_quote":"Gives the fixed-frequency logarithmic stability result that Theorem 1.3 improves to Hölder by using many frequencies."}],"review_version":1}