{"id":"eb595d35-bc62-47f4-ad96-253b5d0093fe","arxiv_id":"2607.18491","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Block-diagonal operators made of uniformly bounded Jordan blocks generate frames from a single vector exactly when their eigenvalues form a Carleson sequence and the vector's leading component stays bounded away from zero; with positive spectrum, sampling along any sequence of positive natural densi","lead":"This mathematics paper identifies exactly when the family of vectors generated by repeatedly applying a block-diagonal operator to one starting vector forms a frame, a set that lets every signal be reconstructed stably. It shows the eigenvalues must be well separated inside the unit disk, and that for positive spectra, thinning time samples to any sequence with positive density still leaves a frame.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1 and hence the sufficiency direction of Theorem 4.1 rest on an unproved real-power Carleson lemma asserted via [9] with 'proof easily generalizable'.","rationale":"The reader identifies the same load-bearing concern: Theorem 3.1 and the sufficiency direction of Theorem 4.1 depend on an unproved real-power Carleson lemma. This is explicitly acknowledged in the manuscript only by the phrase 'the proof is easily generalizable,' which is not a proof. The other main results are supported by detailed arguments—the classification Theorem 1.2 follows from the commutant description (Proposition 2.3), the canonical-vector frame result (Proposition 3.3), and the cited orbit-frame lemma (Lemma 3.4); the density criterion Theorem 4.1 has a mostly self-contained sufficiency proof once Theorem 3.1 is granted, and its necessity is sketched via [10]. Thus the central gap is exactly the unproved lemma. Because the lemma is likely true—the power map is an approximate hyperbolic isometry near the unit circle—the appropriate verdict remains CONDITIONAL, not REJECT. The reader's verdict already reflects this, so no verdict change is recommended. A complete proof of the real-power Carleson lemma, or a counterexample to it, would settle the matter.","tokens_in":21653,"tokens_out":34690,"duration_ms":300173,"concrete_test":"Independently prove the missing lemma. A natural route: show that for each δ>0, z↦z^r is bi-Lipschitz with respect to the pseudohyperbolic metric on A_δ={δ≤|z|<1}, with constants depending only on δ and r, using ρ(a,b)=|a-b|/|1-\\bar a b| and the asymptotic ρ(z^r,w^r)/ρ(z,w)→1 as |z|,|w|→1 away from the branch cut. Then argue that all but finitely many terms of a Carleson sequence lie in A_δ and that bounded multiplicative distortion of the factors preserves the Carleson product condition. If such an estimate cannot be established—e.g., because points on opposite sides of the branch cut make the ratio unbounded—construct a concrete counterexample family z_{2k}=a_k e^{i(π-ε_k)}, z_{2k+1}=-a_k(1+cε_k)e^{i(π-ε_k)} with r just above 1 and check whether {z_i^r} still satisfies the product condition.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 3.1 (Section 3.2), the authors assert that if {z_i}⊂D is Carleson and {|z_i|} is Carleson, then {z_i^r} is Carleson for every real r>0. They cite [9], while noting that [9] only proves the integer-power case and that 'the proof is easily generalizable,' but they supply no argument. This lemma is load-bearing: Theorem 3.1 is proved by reducing {J^{rk}g} to the frame {⊕_i J_{n_i}(z_i^r)^k f}, and Theorem 4.1's sufficiency uses Theorem 3.1 with r=L^{-1}. The lemma is not automatic: for non-integer r, z↦z^r is not a Möbius map, has a branch cut in D, and is not injective for r>1. The cited result concerns integer powers, so the generalization needs proof. If the lemma fails, the advertised redundancy theorems lose their proof. A smaller delegation occurs in Proposition 4.5, whose necessity proof is 'nearly identical' to [10]; the real-power lemma is the more central gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces block diagonal Carleson frames, i.e. singly generated dynamical frames {J^k g}_{k≥0} where J = ⊕_i J_{n_i}(z_i) is a direct sum of Jordan blocks of uniformly bounded size. The central classification result, Theorem 1.2, states that {J^k g} is a frame for ℓ²(N₀) iff {z_i} is a Carleson sequence and g has the coefficient form c_{i,j}√(1−|z_i|²)δ_{i,j} with inf_i |c_{i,0}|>0 and sup_{i,j}|c_{i,j}|<∞. The paper then proves redundancy results: Theorem 3.1 extends the integer-power redundancy theorem of [9] to real powers r>0 when {|z_i|} is also Carleson; Theorem 4.1 gives a complete density criterion for positive spectrum: with Λ⊂[0,∞) having natural density L, {J^{λ_k}g} is a frame iff 0<L<∞ (with the additional condition λ_j=j for j<n_0 when 0∈σ_p(J)). Section 5 proves completeness results, including that the Müntz condition is sufficient but not necessary for completeness of sampled orbits.","tokens_in":21963,"tokens_out":22141,"duration_ms":311629,"significance":"If all claims hold, the paper is a substantial and valuable extension of Carleson-frame theory from diagonal normal generators to nonnormal block-diagonal generators. Theorem 1.2 is a genuine block-diagonal generalization, not a reduction to the diagonal case, and Theorem 4.1 gives a clean asymptotic-density answer for positive spectrum. The proof strategy is mostly sound: the commutant classification, the use of Karamata's Tauberian theorem, and the Müntz-Szász theorem for C^m are appropriate and, for the most part, carefully executed. The paper also contains several independently useful technical tools, such as Proposition 2.3 and Corollary 2.6. However, one load-bearing analytic lemma in the proof of Theorem 3.1 is only asserted by reference to an integer-power case with the remark that the proof 'is easily generalizable'; this needs to be supplied. There are also a few places where necessary details are delegated to a preprint or omitted.","major_comments":[{"comment":"The step 'it has been shown in [9] that a Carleson sequence {z_i} satisfying the hypotheses has the property that {z_i^r} is also a Carleson sequence' is load-bearing for Theorem 3.1, yet the paper immediately notes that [9] proves this only for r∈N and gives no generalization. The map z↦z^r for non-integer r is not a Möbius transformation, is not injective for r>1, and requires a branch choice, so preservation of the Carleson/interpolating property is not a trivial consequence of the integer case. This lemma is used to apply Theorem 1.2 to J_r, and it is also used in the alternate proof of Theorem 4.1 in Section 5.1. Please provide a complete proof or a precise reference for the real-power case; if the lemma is unavailable, Theorem 3.1 must be restricted to r∈N.","section":"Section 3.2, proof of Theorem 3.1"},{"comment":"The necessity direction of Theorem 4.1 is delegated to a calculation said to be 'nearly identical' to [10]. Since [10] is a 2026 arXiv preprint and not a published reference, and since the block-diagonal setting requires checking the off-diagonal and nilpotent-block contributions, the paper should include the complete argument. In particular, the proof that L=∞ contradicts Besselness and the proof that 0∈σ_p(J) forces λ_j=j for 0≤j≤n_0−1 are only sketched. This is load-bearing for the 'only if' part of Theorem 4.1.","section":"Section 4.1, Proposition 4.5"},{"comment":"The final contradiction in Proposition 5.6 assumes that a Müntz-divergent subsequence {J^{μ_k}g} is 'complete but not minimal'. Proposition 5.3, however, only establishes completeness; minimality is not proved or cited. Since a Riesz sequence is minimal, this missing assertion is load-bearing for the proof of Theorem 1.4. Please either prove the nonminimality, cite a precise statement, or replace the argument.","section":"Section 5.2, Proposition 5.6"}],"minor_comments":[{"comment":"The Carleson sequence property is stated only in prose in the Introduction. It should be formalized as a numbered display in Section 1.1. Also, Theorem 1.2 and related statements should explicitly say that the eigenvalues z_i are distinct; Proposition 2.3 and Proposition 3.3 assume this, while the statements of the main theorems do not.","section":"Section 1.1 / Section 3.1"},{"comment":"The claim that '{J^{rk}f}_{k=K} is in fact a frame for ⊕_{i≥1}V_i' should be justified by noting that J^{rK} is invertible on that subspace, so {J^{r(k+K)}f}_{k≥0} is the image of the frame {J^{rk}f}_{k≥0} under an invertible operator. As written, the sentence 'Since J is invertible ... the preceding argument shows' is terse and could be misread as the false statement that deleting finitely many elements from a frame leaves a frame for the same space.","section":"Section 3.2, proof of Theorem 3.1, z_0=0 case"},{"comment":"The passage from Corollary 2.6 to the uniform entrywise estimate 'for all i,r≥M' should explicitly mention that the convergence is uniform because z_i z_r ≥ z_M² and z_M²→1. This is a minor clarification, but it would remove a potential concern about uniformity.","section":"Section 4.2, proof of Theorem 4.1"},{"comment":"The reference to the 'resolved Feichtinger conjecture' should cite a specific source (e.g. Marcus–Spielman–Srivastava) rather than only referring to the conjecture by name.","section":"Section 5.2, Proposition 5.6"}],"recommendation":"major_revision","confidential_remarks":"The main gap is localized and likely fixable: the authors should supply a full proof of the real-power Carleson lemma used in Theorem 3.1, and complete the delegated necessity argument in Proposition 4.5. The primary proof of Theorem 4.1 in Section 4.2 does not depend on the real-power lemma; only the alternate proof does. The self-citation [14] is appropriate and not circular. If the authors supply the missing proof in an appendix, I would be willing to accept a revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Start with the punchline: the central classification here is real and clean, and the paper deserves a serious referee. The one thing to know is a genuine gap in the proof of Theorem 3.1, where the advertised redundancy results rest on an unproved assertion about real powers of Carleson sequences.\n\nWhat is actually new: the class of block diagonal Carleson frames, and Theorem 1.2’s complete classification of singly generated frames for uniformly bounded Jordan-block direct sums. That is a genuine extension of the diagonal case, and the proof route is good: canonical vector, commutant of infinite Jordan matrices, and the Nikolskii–Vasyunin interpolation theorem. The C^m Müntz theorem and the derivative Karamata lemma are useful and properly proved. The natural-density criterion in Theorem 4.1 is a real strengthening, and the completeness results in Section 5 are interesting.\n\nThe soft spot is exactly where the stress-test note lands, though it does not land as hard as stated. In the proof of Theorem 3.1, the authors assert that if {z_i} is Carleson and {|z_i|} is Carleson, then {z_i^r} is Carleson for every real r>0, citing [9] for integer powers and saying “the proof is easily generalizable.” No argument is given. That is not immediate: z^r is not Möbius, it can have branch issues, and for non-integer r it is not injective on D. I suspect the claim is true under the stated two-sequence hypothesis, but a referee cannot be asked to take that on faith. The paper needs a proof of that lemma or a source that contains the real-power statement.\n\nThe stress-test note goes too far in saying Theorem 4.1’s sufficiency also collapses. The primary proof of Theorem 4.1 in Section 4.2 compares S_Λ with L S_{N0} using Corollary 2.6, not Theorem 3.1. Theorem 3.1 is only used in the alternative proof in Section 5.1. So the natural-density theorem has a live proof path. Also, the delegation in Proposition 4.5 to [10] is milder than the note suggests: the essential argument is sketched in the text.\n\nThe citation pattern is fine, including the authors’ own [14], which is explicitly the diagonal special case and proof template. The paper is honest about what it borrows and what it adds.\n\nBottom line: this is a well-constructed paper with one patchable gap. Send it to a serious referee, with the real-power lemma as the main request. It should be published after that is fixed.","headline":"A solid extension of Carleson frames to nonnormal block-diagonal generators, with one unproved real-power sequence lemma that should be filled before publication.","tokens_in":22409,"tokens_out":5526,"would_cite":true,"duration_ms":49175,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42C15","47A99"],"pacs":[],"model":"deepseek-v4-flash","headline":"For block-diagonal Jordan operators, orbit frames are exactly those with Carleson eigenvalues and bounded generating coefficients.","keywords":["block diagonal Carleson frames","dynamical sampling","Carleson sequences","Jordan blocks","frame redundancy","natural density","Müntz condition","operator orbits"],"falsifier":"Find a Carleson sequence {z_i} ⊂ D with {|z_i|} Carleson such that, for some non-integer r > 0, the sequence of powers {z_i^r} fails the Carleson product condition inf_k ∏_{j≠k} |(z_j^r − z_k^r)/(1 − \\bar{z_j^r} z_k^r)| > 0. Such an example would invalidate the key step of Theorem 3.1's proof. Alternatively, test Theorem 1.2 directly: with Carleson eigenvalues and a vector g of the stated coefficient form but sup |c_{i,j}| = ∞, the orbit should fail to be a frame; if it is a frame, the classification is false.","tokens_in":21567,"feed_emoji":"🧮","tokens_out":5651,"duration_ms":45964,"temperature":0.7,"pith_summary":"This paper classifies a broad class of singly generated dynamical frames in which the generating operator is a direct sum of Jordan blocks of uniformly bounded size, called block diagonal Carleson frames. The main theorem says that the orbit {J^k g} is a frame for ℓ²(N₀) exactly when the eigenvalues form a Carleson sequence and the generating vector has the form Σ c_{i,j}√(1−|z_i|²)δ_{i,j} with a uniformly nonzero leading coefficient and uniformly bounded entries. This is a complete classification for genuinely non-normal generating operators, and it reduces to the known Carleson-frame theorem when every block has size one. The paper also shows that these frames inherit the strong redundancy of scalar Carleson frames: decimating the orbit by powers J^{rk} preserves the frame property for all r > 0, and for nonnegative spectrum the frame property of a sampled orbit {J^{λ_k}g} is governed purely by the natural density of the sampling set Λ.","feed_headline":"Carleson eigenvalues completely classify block-diagonal frame orbits","feed_subtitle":"New theorem reduces the frame property to spectral distribution and generator coefficients, extending Carleson frames to nonnormal operators","key_machinery":"The proof rests on the canonical generating vector f = Σ_i √(1−|z_i|²) δ_{i,0}, whose orbit has synthesis operator equal to a derivative-evaluation map at the spectral points: {J^k f} is a frame precisely when the points form a Carleson sequence, by a classical interpolation theorem for Blaschke products with multiplicities. All other admissible generators are obtained by applying the commutant of J, which consists of block-diagonal lower-triangular Toeplitz matrices, to f; invertibility of that commutant is what forces the leading coefficients to be bounded away from zero. For the redundancy theorems, powers J^λ are defined by a binomial expansion, and frame behaviour of {J^{λ_k}g} is analy","core_discovery":"The central claim, Theorem 1.2, is: let J = ⊕_i J_{n_i}(z_i) with sup_i n_i < ∞, where J_{n_i}(z_i) is the Jordan block with z_i on the diagonal and 1−|z_i|² on the subdiagonal. Then {J^k g}_{k≥0} is a frame for ℓ²(N₀) if and only if {z_i} is a Carleson sequence and g = Σ_i Σ_j c_{i,j}√(1−|z_i|²) δ_{i,j} with inf_i |c_{i,0}| > 0 and sup_{i,j} |c_{i,j}| < ∞. This says the frame property is completely encoded by the spectral distribution and by the leading coefficients of the generator in each generalized eigenspace; the off-leading coefficients may vary but must remain bounded. The scalars c_{i,j} are exactly the parameters of the lower-triangular Toeplitz commutant of J.","pith_inferences":["Because the frame condition is expressed through simultaneous evaluation of derivatives, the classification is effectively a statement about stable Hermite interpolation at Carleson points; refining the paper's derivative-evaluation bounds would yield quantitative frame constants.","The density result for nonnegative spectrum suggests a sampling-theoretic reading: stable reconstruction depends only on the asymptotic sampling rate, so the same conclusion should plausibly hold for any sampling set with positive macroscopic lower density, as the paper's own remark hints.","The unproved step that z_i ↦ z_i^r preserves the Carleson property for non-integer r is testable independently: if it fails for some Carleson sequence with Carleson moduli, the decimation theorem's proof needs repair, though the theorem itself might survive."],"forward_implications":["Theorem 1.2 completely solves the dynamical sampling problem for uniformly bounded block-diagonal Jordan operators, a genuinely non-normal case.","When all blocks have size one, the new theorem recovers the known Carleson-frame classification, so the result is a strict extension of the diagonal theory.","Decimation: if {z_i} and {|z_i|} are both Carleson sequences, then {J^{rk}g} is a frame for every r > 0, with finitely many initial vectors added when 0 is an eigenvalue, so these frames tolerate very sparse regular subsampling.","For nonnegative spectrum, any sampling sequence with well-defined positive finite natural density yields a frame; local clustering or gaps in the sampling set are irrelevant to stability.","The Müntz divergence condition, though sufficient for completeness of sampled orbits, is not necessary; complete subsequences exist with convergent Müntz sums."],"fun_headline_variants":["Frame property reduces to eigenvalues and coefficient conditions","Block-diagonal Carleson frames: classification via spectrum and coefficients","Nonnormal operators: Carleson frames characterized by Jordan blocks","Eigenvalues and leading coefficients fully decide frame property"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that raising every spectral point to the power r preserves the Carleson property for all real r > 0 whenever both the points and their moduli form Carleson sequences; in the proof of Theorem 3.1 the paper cites this for integer exponents only and states without proof that it generalizes, and the decimation theorem rests on it.","fun_headline_variants_meta":{"raw":{"variants":["Frame property reduces to eigenvalues and coefficient conditions","Block-diagonal Carleson frames: classification via spectrum and coefficients","Nonnormal operators: Carleson frames characterized by Jordan blocks","Eigenvalues and leading coefficients fully decide frame property"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000264,"raw_usage":{"total_tokens":1408,"prompt_tokens":679,"completion_tokens":729,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":423,"completion_tokens_details":{"reasoning_tokens":663}},"tokens_in":423,"tokens_out":729,"duration_ms":6855,"temperature":1.0,"reasoning_tokens":663,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T15:15:46.943070+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a Carleson sequence {z_i} ⊂ D with {|z_i|} Carleson such that, for some non-integer r > 0, the sequence of powers {z_i^r} fails the Carleson product condition inf_k ∏_{j≠k} |(z_j^r − z_k^r)/(1 − \\bar{z_j^r} z_k^r)| > 0. Such an example would invalidate the key step of Theorem 3.1's proof. Alternatively, test Theorem 1.2 directly: with Carleson eigenvalues and a vector g of the stated coefficient form but sup |c_{i,j}| = ∞, the orbit should fail to be a frame; if it is a frame, the classification is false.","supporting_citations":[],"review_version":1}