{"id":"b0f7f332-11cc-47b0-b5fa-ec759840b2a0","arxiv_id":"2607.24076","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Time-trace observations from Hausdorff-thick zero-measure sets determine Airy solutions on the real line for every observation time, and finite point observability on the torus is characterized by a Kalman rank condition.","lead":"This paper proves that Airy-type dispersive wave equations can be fully reconstructed from time-recordings at very thin, fractal-like spatial sets on the real line — a regime where heat-equation theory requires thick measurement sets. On the torus it gives an exact linear-algebra test for when a finite array of point sensors can observe the wave.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2 rests on the imported uniform propagation estimate (3.2) from Zhu [52]; if its exponent depends on more than (s,h), Lemma 3.1 and the low-frequency separation in Lemma 6.1 fail.","rationale":"I read the full manuscript and checked the structural logic of the main proofs. The free-Airy proof is internally consistent: the TT* local smoothing argument gives the trace admissibility, the regular-sequence Hautus estimates close the high-frequency observability, the Gevrey time filters produce the needed e^{-c N^{3/σ}} exterior tails, and the low-frequency Hausdorff–Remez estimate (assuming Lemma 3.1) is integrated correctly with the absorption step in Section 7. The torus part appears coherent: the Riesz-projection spectral decomposition, the Bari–Markus reduction to a finite-dimensional invariant subspace, and the Kalman rank criterion all line up, and the sharp 3-vs-2 point result for linear KdV follows from the stationary obstruction. I found no internal inconsistency or missing proof among the claims presented. The only genuine weak point is the imported local disk estimate (3.2). The reader identified exactly this assumption, and I agree it is the most load-bearing. However, relying on a cited theorem is standard practice, and the cited result is plausible and, in the fixed-scale use made here, likely correct. The author flags the point, and the proof of Lemma 3.1 gives a precise reduction to Zhu's inequality. I do not see a concrete reason to distrust the uniformity as stated, and therefore I do not recommend changing the ACCEPT verdict. A targeted verification of (3.2) would raise confidence from MODERATE to HIGH.","tokens_in":47264,"tokens_out":36369,"duration_ms":292053,"concrete_test":"Inspect [52, Prop. 2.3] and its proof. Verify that for fixed s>0, h>0, and 0<R<1/4, the exponent α_Z in the two-constants inequality (3.2) can be chosen independent of the particular set G (with H^s_C(G)≥h) and of the center z0, and that the constant remains finite for the chain-of-balls localization used in Lemma 3.1. If the proof gives an exponent depending on the number of balls needed to cover G, check whether that number is bounded by a function of h alone; if not, construct a counterexample to (3.2) with a sparse fractal G and F(z)=e^{λ z}, λ large. If (3.2) cannot be verified, Lemma 3.1—and with it Lemma 3.3, Lemma 6.1, and Theorem 1.2—is unsupported at its base.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.2 is structured so that the entire low-frequency estimate flows through Lemma 3.3, whose proof invokes Lemma 3.1. Lemma 3.1 in turn depends on the local disk inequality (3.2), which is stated as a consequence of [52, Prop. 2.3] but not re-proved. The paper asserts that Zhu's proposition yields a uniform exponent α_Z(s,h)∈(0,1) whenever H^s_C(G)≥h, with the inequality holding on B(z0,R) vs B(z0,4R) for F holomorphic in B(z0,5R). This uniformity is load-bearing in two places. First, in Lemma 3.1 the localization argument selects one subinterval A_ν0 with H^s content ≥ h0 = m0/(2^s N); if α_Z actually depended on the covering number of G at scale ρ, that step would not control the exponent. Second, in Lemma 3.3 the same constant C* is used for every interval I_j; any hidden dependence of C* on the particular set E_j (rather than only on s, m, R) would make the estimate (3.25) non-uniform in j, and the sum over all blocks in (3.16) would fail. Because the final absorption in Lemma 6.1 requires e^{C(1+N)} to be beaten by e^{-c N^{3/σ}} (σ<3), a blow-up of C* as a function of the set or of the scale would reopen the low-frequency error. The paper gives no derivation of (3.2) from the cited text; it is an external dependency not re-verified. This is the single most load-bearing imported step, and it is structurally distinct from the claimed new ideas.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies observability inequalities for the Airy equation. On the real line, Theorem 1.2 establishes, for every T>0, an observability inequality from Hausdorff-thick sets E in a time-trace sense, using the block-supremum functional O_{E,T}(u_0)=Σ_j sup_{x∈E_j} ||Tr_x S(·)u_0||^2_{L^2(0,T)}. Proposition 1.3 treats periodic Borel-measure observations, and Proposition 1.7 gives analogous results for the linear KdV equation, including a negative result for fixed lattice observations. On the torus, Theorem 1.10 gives a necessary and sufficient Kalman-rank condition for finite point observability for ∂_t+∂_x^3+p(x) with bounded real potential, Proposition 1.11 proves the sharp 3-vs-2 point result for periodic linear KdV, and Corollary 1.14 yields observability from sets with an accumulation point under finite regularity assumptions on p. The proof strategy combines a Hilbert-valued Hausdorff–Remez estimate, Gevrey time filters with subexponential tails e^{-cN^{3/σ}}, a unitary Hautus high-frequency estimate, and a finite-dimensional spectral reduction.","tokens_in":47588,"tokens_out":35529,"duration_ms":284933,"significance":"If the results are correct, the paper is a substantial contribution. The Airy flow is unitary and lacks the parabolic decay that powers previous Hausdorff-type observability results, so the block-supremum time-trace norm and the Gevrey-filter method are genuinely new tools. The torus results, including the treatment of non-normal operators with bounded potentials and the finite-dimensional Kalman criterion, are clean and sharp, with explicit counterexamples showing optimality of the number of points for linear KdV. The paper is also commendably self-contained in most parts: full proofs are given for the trace admissibility, the Gevrey cutoff construction, the high-frequency separation, and the spectral decomposition. The main caveat is that one load-bearing estimate, the uniform planar propagation-of-smallness bound (3.2), is imported from Zhu [52] and not re-proved in the manuscript.","major_comments":[{"comment":"The proof of the central low-frequency estimate rests on the uniform planar propagation-of-smallness estimate imported from Zhu [52, Prop. 2.3]. The paper states that “[52]’s proposition, together with its proof, implies” the uniform form (3.2), with α_Z depending only on (s,h), but it does not reproduce the proposition or prove this uniformity. The point is load-bearing: Lemma 3.1 uses (3.2) to define C*, Lemma 3.3 uses C* to obtain the e^{C(1+N)} factor in (3.16), and Lemma 6.1 then needs e^{C(1+N)} to be absorbed by the Gevrey tail e^{-c_T N^{3/σ}} with σ<3. If α_Z(s,h) in (3.2) depended on the covering data of G or on the scale in a way not controlled by H^s_C(G)≥h, the low-frequency separation and the absorption in §7 would fail. Please either state the precise theorem from [52] and give a full derivation of (3.2), including the exact dependence on H^s_C(G), or provide a self-contai","section":"§3, Eq. (3.2)"}],"minor_comments":[{"comment":"The adjoint spectral projection P_n^* is associated with the conjugate spectral value \\bar{λ_n} (equivalently, with the reflected contour \\overline{D_n}), not with λ_n. The subsequent formulas are correct, but the wording “associated with λ_n” is confusing.","section":"§10, after (10.9)"},{"comment":"The constant B is said to depend only on E, m, R; since Definition 1.1 also involves s, the dependence on s should be stated.","section":"Remark 1.4(iv)"},{"comment":"The notation H^s_C(E) is introduced inside the proof of Lemma 3.1. It would be clearer to define it near the statement or in the notation section, since it is not the same as the H^s_∞ used in Definition 1.1.","section":"§3, Lemma 3.1"}],"recommendation":"major_revision","confidential_remarks":"This is a strong manuscript with new ideas and careful proofs. My sole substantive concern is the verifiability of the imported uniform estimate (3.2). If the authors can provide a self-contained derivation or a precise statement and proof of the needed uniformity from Zhu [52], I would support acceptance. I do not see circularity, post-hoc selection, or overclaiming in the main theorems."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"In short: this is a strong, unusually complete research preprint that delivers the advertised results. The block-supremum time-trace functional and the Gevrey-filter/sampling-Hautus combination are genuinely new and well engineered. I checked the structural logic: the e^{C(1+N)} cost from the Hausdorff–Remez step is beaten by the e^{-c N^{3/σ}} tail because σ<3, and the absorption in §7 closes. The torus part, with the Bari–Markus decomposition into a finite-dimensional invariant space plus high-frequency eigenfunctions, is coherent, and the Kalman criterion is the right necessary and sufficient condition. The sharp 3-vs-2 points result for linear KdV is a clean and satisfying coda.\n\nThe soft spot is exactly the one the stress-test note identifies: the uniform propagation estimate (3.2) imported from Zhu [52]. It carries the entire low-frequency estimate through Lemma 3.1, and the constant C* must depend only on s, m0, U. The paper states this uniformity as a consequence of [52, Prop. 2.3] but does not reprove it. The author does flag it as known; if Zhu's proof indeed yields the claimed uniformity, everything holds. But this is a load-bearing external peg. A referee should be asked to confirm that (3.2) follows from [52] with the claimed uniformity and that the constant does not secretly depend on the covering number of G or on the scale. I see no reason to believe it fails, but the text does not verify it.\n\nOther concerns are minor. The paper is long and notation-heavy; a few steps, like the bad-good interval decomposition in Lemma 3.3 or the exact constants in the Gevrey tail, are sketched rather than fully expanded, but they point to the correct mechanisms. Corollary 1.14's k0 is said to depend only on ||p||∞ but is not explicitly computed; the author refers to Prop 12.3 for the determination, so that is acceptable.\n\nOverall: the mathematics is honest, the main theorems are new and significant, and the proof is dense but structurally sound. The paper deserves a serious referee; I would send it to review with a request to verify the Zhu import and to add a remark on the explicit k0. For anyone working on observability of dispersive equations, this is worth reading and citing.","headline":"A dense, serious preprint that completes the Hausdorff-observability program for Airy on the line and point observability on the torus; the new machinery is original and the proofs largely hold, with one load-bearing imported estimate (Zhu's uniform propagation) that deserves explicit referee scrutiny.","tokens_in":48228,"tokens_out":2337,"would_cite":true,"duration_ms":23900,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q53","93B07","93C20"],"pacs":[],"model":"deepseek-v4-flash","headline":"Hausdorff-thick sets of zero measure fully determine Airy initial data from L² time traces for every T>0; on the torus, finite point observability is a Kalman rank condition.","keywords":["Airy equation","Hausdorff-thick sets","time-trace observability","propagation of smallness","Gevrey filters","Kalman criterion","linear KdV equation","point observability"],"falsifier":"Test the uniformity of the propagation-of-smallness exponent: search numerically or analytically for a family of sets G_k ⊂ B(0,1) with H^s_C(G_k) ≥ h and holomorphic functions F_k with ‖F_k‖_{L∞(B(0,4))} ≤ e^B ‖F_k‖_{L∞(B(0,1))} yet ‖F_k‖_{L∞(B(0,1))} ≥ e^{c_k B} ‖F_k‖_{L∞(G_k)} with c_k→∞; if such examples exist, the constant C* in Lemma 3.1 cannot be chosen uniformly, and the paper's absorption step (coefficient 1/16 in Lemma 6.1) would not close. A cheaper check: verify the Hadamard three-circle step in Lemma 3.1 for an explicit function like g(z)=exp(Bz) with A a Cantor-type set of prescr","tokens_in":47010,"feed_emoji":"🌊","tokens_out":8686,"duration_ms":80168,"temperature":0.7,"pith_summary":"This paper answers two long-standing observability questions for the Airy equation. On the real line, it proves that if a zero-measure set E has Hausdorff content at least m in every block [3jR,3jR+R], then the block-supremum of L² time traces over E controls the full L² norm of the initial data, for every observation horizon T>0—despite the equation being unitary and having no pointwise smoothing of L² data. On the torus, it proves that finite-point observability for the Airy equation with a bounded real potential is equivalent to a Kalman rank condition on an explicitly constructed finite-dimensional invariant subspace; from this, one point observes the free periodic Airy wave, while the periodic linear KdV wave requires exactly three distinct points. The proofs replace the parabolic decay used in heat-type results with two new devices: Gevrey time-frequency filters whose subexponential tails beat the exponential cost of Hilbert-valued Hausdorff–Remez propagation, and a unitary Hautus/resolvent criterion for high frequencies.","feed_headline":"Observe the Airy wave from any Hausdorff-thick set","feed_subtitle":"Time traces on sparse sets recover all L² data; on the torus, point observability is a Kalman rank check.","key_machinery":"The load-bearing objects are two. (1) On the line, the block-supremum time-trace functional O_{E,T}(u₀)=Σ_j sup_{x∈E_j}‖Tr_x S(·)u₀‖²_{L²(0,T)}: the natural L²-admissible replacement for pointwise-in-space observations, since L² data have no point values but do have time traces at each spatial point. The proof splits frequencies: low frequencies are handled by a Hilbert-valued Hausdorff–Remez propagation-of-smallness estimate with cost e^{C(1+N)}, and high frequencies by a unitary Hautus criterion for regular sampling sequences. (2) The bridge between the two regimes is a Gevrey time filter of order 1<σ<3: because the dispersion relation τ=ξ³ identifies spatial and temporal frequencies, the","core_discovery":"The central discovery is that zero-measure sets can determine the whole solution of the Airy equation. Theorem 1.2 states: if E⊂R is Hausdorff-thick in the sense H^s_∞(E∩[3jR,3jR+R]) ≥ m for every j, then for every T>0 there are C,C′>0 such that ‖u₀‖²_{L²(R)} ≤ C Σ_j sup_{x∈E_j} ‖Tr_x S(·)u₀‖²_{L²(0,T)} ≤ C′‖u₀‖²_{L²(R)}. This is the first observability inequality for the Airy equation from a zero-measure set, and it achieves any positive observation time. The second discovery, Theorem 1.10, is a necessary and sufficient condition on the torus: for p∈L^∞(T;R), finite point observability of ∂³_x+p holds if and only if the finite-dimensional system (C_F, A=L_p|_{X_0}) on the natural invariant","pith_inferences":["The same geometric trick—comparing the exponential cost of a low-frequency propagation estimate with the subexponential tail of a Gevrey filter—should transfer to other dispersive equations whose phase is τ=ξ^a, giving observability from Hausdorff-thick sets whenever a Gevrey order σ<a is available; the cubic case a=3 is the first instance.","The torus result suggests a general principle: for self-adjoint operators with polynomial spectral gaps, adding a bounded perturbation confines all obstructions to point observability to a finite-dimensional invariant subspace, so finite-point observability is always a finite-rank linear algebra condition.","One could test whether the block-supremum trace functional can be replaced by a single supremum over all of E (rather than blockwise) or by weighted variants; the paper's admissibility constants suggest the blockwise structure is essential for L² data, so this would be a genuine open problem.","The sharp 3-vs-2 point threshold for linear KdV is explained by the three-dimensional stationary mode n∈{-1,0,1}; analogous thresholds for other 'transport + dispersion' equations on the torus should equal the dimension of the stationary eigenspace of the leading dispersion polynomial."],"forward_implications":["Every Hausdorff-thick zero-measure set yields an observability inequality for the Airy equation on ℝ with any observation time T>0, with an explicit constant of the form B exp(BT^{−1}) for small T.","Periodic Borel measure observations—including single spatial point sequences—give two-sided trace estimates, and the same holds for the linear KdV equation along moving traces x+t.","Fixed-time periodic point observations fail for the linear KdV on ℝ, so the transport term w_x destroys time-trace observability for stationary trace points.","On the torus, finite-point observability for Airy with potential p is time-independent and decidable by a finite rank test (Kalman on X_0); in particular free Airy is observable from any single point, while linear KdV needs exactly three distinct points.","Any observation set with an accumulation point observes the toroidal Airy equation with potential of sufficiently high (finite) regularity, for every T>0."],"fun_headline_variants":["Airy waves observed from zero-measure sets","Hausdorff-thick sets yield Airy observability","Point observability on torus: Kalman condition","Time traces on sparse sets recover Airy data","Zero-measure observability for Airy equation"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"Everything rests on an imported uniformity statement: a cited planar propagation-of-smallness result (stated as (3.2)) is assumed to hold with an exponent bounded away from zero uniformly for all sets whose s-dimensional Hausdorff content is at least h; the paper invokes it without re-proof, and Lemma 3.1, Lemma 3.3, and therefore the low-frequency estimate (and the absorption that closes Theorem 1.2) would fail if that uniformity were false.","fun_headline_variants_meta":{"raw":{"variants":["Airy waves observed from zero-measure sets","Hausdorff-thick sets yield Airy observability","Point observability on torus: Kalman condition","Time traces on sparse sets recover Airy data","Zero-measure observability for Airy equation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000244,"raw_usage":{"total_tokens":1444,"prompt_tokens":891,"completion_tokens":553,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":635,"completion_tokens_details":{"reasoning_tokens":478}},"tokens_in":635,"tokens_out":553,"duration_ms":5238,"temperature":1.0,"reasoning_tokens":478,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T23:08:21.625489+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test the uniformity of the propagation-of-smallness exponent: search numerically or analytically for a family of sets G_k ⊂ B(0,1) with H^s_C(G_k) ≥ h and holomorphic functions F_k with ‖F_k‖_{L∞(B(0,4))} ≤ e^B ‖F_k‖_{L∞(B(0,1))} yet ‖F_k‖_{L∞(B(0,1))} ≥ e^{c_k B} ‖F_k‖_{L∞(G_k)} with c_k→∞; if such examples exist, the constant C* in Lemma 3.1 cannot be chosen uniformly, and the paper's absorption step (coefficient 1/16 in Lemma 6.1) would not close. A cheaper check: verify the Hadamard three-circle step in Lemma 3.1 for an explicit function like g(z)=exp(Bz) with A a Cantor-type set of prescr","supporting_citations":[],"review_version":1}