{"id":"0c6db581-8a92-4b64-99f7-deb284bc7dab","arxiv_id":"2607.26904","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Spherical polynomials of degree N are controlled by their values on any closed set of positive (n-2+δ)-Hausdorff content for every δ in (0,1), yielding sharp heat observability on the sphere and lower-dimensional observability for super-quadratic Schrödinger operators on R^n.","lead":"The paper proves a Remez inequality for spherical polynomials controlled by values on fractal subsets of the sphere with arbitrarily small positive Hausdorff dimension excess, then uses it to get heat-equation observability from those sets. This improves prior sphere results that needed the fractal dimension to be close to full codimension one.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The central claim (Theorem 1.1 for every delta in (0,1)) rests on a clean reduction to a one-dimensional fractal Turán inequality that the authors cite from a concurrent preprint. That is a real external load, exactly as the reader flagged, but it is not an inconsistency or missing step inside the present argument. The geometric constructions (auxiliary set W with positive content and a density point, Riesz-capacity comparison, Grassmannian slicing that forces the arc to meet W) are standard and appear correctly executed; the spectral inequality (3.1) is then an immediate corollary, and the observability statements follow by the usual iteration. The appendix asymptotic (3.6) is classical and only needed for the radial-potential application. Because the paper honestly treats the concurrent lemma as an input and the internal logic is sound, the reader's ACCEPT (MODERATE confidence) needs no adjustment.","tokens_in":23333,"tokens_out":497,"duration_ms":9827,"concrete_test":"Independently verify that the constant and frequency range in (2.37) of arXiv:2607.17505 cover trigonometric polynomials with |m_q-m_1|=2N and sets A subset [0,1] of positive C_H^{delta/2} content (the exact regime used after (2.39)); if that external inequality holds as quoted, the Remez constant (1.5) and both observability theorems stand.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest-assumption call (external fractal Turán–Nazarov (2.37) from arXiv:2607.17505) is a genuine dependency, but it is not an internal soft spot of this manuscript: once (2.37) is granted, the reduction in Lemma 2.11 (density point W, capacity comparison, Grassmannian slice to an arc intersecting W, pull-back to a trigonometric polynomial of degree 2N) is standard and carefully written, and the subsequent spectral inequalities plus Lebeau–Robbiano iteration for Theorems 1.2–1.3 follow routinely. No hidden gap in the capacity-to-Hausdorff constants, the slicing measure, or the asymptotic (3.6) appears load-bearing for the stated claims on the sphere and for radial super-quadratic potentials.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper proves a fractal Remez inequality on S^{n-1}: for every closed M with positive (n-2+δ)-Hausdorff content (any δ∈(0,1)) and every spherical polynomial p of degree ≤N, the L^∞ norm of p on the sphere is controlled by its L^∞ norm on M, with an explicit constant of the form C_1((C_2/C_H^{n-2+δ}(M))^{2/δ} N^{4/δ-1})^{2N}. The proof reduces via an auxiliary density set W, Riesz-capacity comparison, Grassmannian slicing to a spherical arc meeting W, and pull-back to a trigonometric polynomial, to which a fractal Turán–Nazarov inequality is applied. This yields spectral inequalities and, via Lebeau–Robbiano iteration, heat observability from such fractal sets on the sphere (improving Burq–Moyano by removing the restriction that δ be close to 1). A further application gives lower-dimensional observability for the heat equation with radial super-quadratic potentials |x|^{2m} (m≥2) on R^n, observing on a conical annular set whose angular part has positive (n-2+δ)-content.","tokens_in":23349,"tokens_out":1279,"duration_ms":35027,"significance":"The main advance is a Remez inequality on the sphere valid for arbitrary δ∈(0,1) and the resulting sharp observability statements. Removing the “δ near 1” limitation of Burq–Moyano in the spherical setting is a clear improvement; the optimality remark via nodal sets of spherical harmonics is correctly noted. The super-quadratic application appears to be the first lower-dimensional observability result of this type for confining potentials beyond the harmonic case. The reduction is written with explicit constants and standard GMT tools (Mattila slicing, capacity–content comparisons). The dependence on the concurrent fractal Turán preprint is a genuine external dependency rather than an internal circularity; once that inequality is granted, the spherical and radial arguments are routine and carefully tracked.","major_comments":[{"comment":"Lemma 2.11, Step 3, display (2.37): the load-bearing one-dimensional fractal Turán–Nazarov inequality is taken from the authors’ concurrent preprint arXiv:2607.17505 [43]. The Remez constant in Theorem 1.1 and both observability theorems collapse if (2.37) fails for the frequency range |m_q-m_1|=2N or the Hausdorff content C_H^{δ/2}(A) used here. For journal publication the paper should either (i) include a self-contained proof of the needed special case of (2.37), or (ii) make the dependence fully explicit in the statements of Theorems 1.1–1.3 and ensure [43] is available (accepted or at least stably posted) before final acceptance.","section":"§2.3, Lemma 2.11, (2.37)"},{"comment":"Appendix A / Lemma 3.2, asymptotic (3.6): the eigenvalue asymptotic for the radial ODE is proved for even n>4 by adapting Titchmarsh/Langer expansions. The error-term bookkeeping (δ_k→0, the h(a)/√μ_k contribution, and the passage from (A.17) to (A.2)) is sketched rather than fully expanded. A short verification that the O(μ_k^{-1})+O(z^{-1}) remainders are absorbed uniformly in the angular momentum range ℓ≤C μ^{1+1/m} would make the bound on ℓ_max (and thus the Remez degree N) fully rigorous for all n.","section":"Appendix A, (3.6) and (A.14)–(A.17)"}],"minor_comments":[{"comment":"Title and running heads: “OBSERV ABILITY” appears with a spurious space in the manuscript header; correct to “OBSERVABILITY”.","section":"Title page"},{"comment":"Remark 1.4(ii): the claim that C_H^{n-2+δ}(M)>0 is optimal is correct via nodal sets, but a one-line reference to the standard bound dim_H{p=0}≤n-2 for nontrivial spherical polynomials would help non-specialist readers.","section":"Remark 1.4(ii)"},{"comment":"Lemma 2.6 / (2.19): the auxiliary radius r_0 is chosen relative to diam(M)/diam(S^{n-1}); a brief sentence clarifying that the final C_2 absorbs this geometric factor would improve readability.","section":"§2.2, Remark 2.7"},{"comment":"Corollary 3.1: the null-controllability statement is standard once observability is known; it could be shortened or moved to a remark to keep the focus on the new inequalities.","section":"§3.1, Corollary 3.1"},{"comment":"References: several arXiv numbers in the 2607 range appear (including the concurrent [43]); ensure final bibliographic data are updated upon acceptance.","section":"References"}],"recommendation":"minor_revision","confidential_remarks":"The central technical novelty is the spherical fractal Remez reduction; the observability applications are largely standard once the spectral inequality is in hand. The concurrent Turán preprint [43] is the only real external risk. If the editor can confirm that [43] is sound (or require the authors to include the needed 1-D case), the paper is suitable for the journal after minor revision. Scope fit with math.AP / control-theory venues is good."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing worth knowing: they get a Remez inequality on S^{n-1} that works for closed sets of positive (n−2+δ)-Hausdorff content for every δ in (0,1), not just δ near 1. That immediately gives heat observability on the sphere past Burq–Moyano’s restriction, and they push the same tool to a lower-dimensional observation set for −Δ+|x|^{2m} on R^n.\n\nWhat is actually new is the sphere Remez (Theorem 1.1) and the two observability theorems that ride on it. Dicke–Veselić handled positive measure; Burq–Moyano needed δ close to 1 via gradient smallness; the Euclidean fractal work (including their own earlier paper) does not cover the sphere. The reduction is the right one for this literature: density point and non-vanishing set W, Hausdorff-to-capacity comparison, Grassmannian slice to a spherical arc that meets W, pull-back to a trig polynomial of degree 2N, then fractal Turán. Constants are tracked; the spectral inequality and Lebeau–Robbiano step are routine and written carefully. They flag honestly that the general compact manifold case stays open, and the optimality remark via nodal sets of spherical harmonics is correct.\n\nThe soft spot is real but external: Step 3 of Lemma 2.11 leans on their concurrent fractal Turán–Nazarov inequality. If that fails in the frequency/Hausdorff range used here, the Remez constant and both observability theorems fall. That is a dependency chain, not circularity or a hidden hole in the capacity/slicing argument. The asymptotic for the radial eigenvalues (Appendix A) looks standard Titchmarsh-style work and is not load-bearing in a fragile way once granted. No free parameters, no data games, citation pattern is normal for the subfield (self-cites supply tools).\n\nThis is for people who do quantitative unique continuation and parabolic control. If you work on thick/fractal observation sets or confining potentials, read Theorems 1.1–1.3 and the reduction in §2. It deserves a serious referee; I would engage with it and expect it to clear peer review with the usual requests to clarify the Turán input and constant dependence.","headline":"Solid fractal Remez on the sphere for every δ∈(0,1), cleanly upgrading Burq–Moyano and giving a first lower-dimensional observability result for super-quadratic potentials; main load is an external concurrent Turán lemma, not an internal gap.","tokens_in":24138,"tokens_out":596,"would_cite":true,"duration_ms":18060,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35P99","35Q93","35K05"],"pacs":[],"model":"grok-4.5","headline":"Spherical polynomials of any degree are controlled by their values on fractal sets of every positive codimension less than one, yielding heat observability from those sets.","keywords":["Remez inequality","spherical polynomials","Hausdorff content","observability","heat equation","spectral inequality","fractal sets","super-quadratic potentials"],"falsifier":"Exhibit a sequence of spherical polynomials of degree N whose ratio of global maximum to maximum on a fixed set of positive (n-2+δ)-Hausdorff content grows faster than any constant times (C N^{4/δ-1})^{2N}, or show that the heat solution starting from an eigenfunction vanishing on such a set remains invisible in L^2 after time T.","tokens_in":24058,"feed_emoji":"○","tokens_out":1049,"duration_ms":18511,"temperature":0.7,"pith_summary":"The paper proves a Remez inequality on the sphere: the maximum of any spherical polynomial of degree at most N is bounded by a constant times its maximum on a closed fractal set M whose (n-2+δ)-Hausdorff content is positive, for every δ between 0 and 1. The constant grows like a power of N raised to 2N, with explicit dependence on the Hausdorff content of M. That bound is then fed into the Lebeau–Robbiano iteration to obtain observability inequalities for the heat equation on the sphere from the same fractal sets, improving earlier results that required δ close to 1. The same spherical estimate also yields a lower-dimensional observability inequality for the heat equation with super-quadratic confining potentials on Euclidean space, observing only on a conical shell whose angular cross-section has positive fractal content.","feed_headline":"Fractal sets of any positive thickness control heat on the sphere","feed_subtitle":"A Remez bound for every δ in (0,1) yields observability and null-controllability from lower-dimensional observation sets","key_machinery":"Fractal Remez inequality on the sphere (Theorem 1.1): after slicing M down to a spherical arc that meets an auxiliary set W where |p| is already comparable to its global maximum, the problem reduces to a one-dimensional fractal Turán inequality for the pulled-back trigonometric polynomial.","core_discovery":"For every δ∈(0,1) and every closed M⊂S^{n-1} with positive (n-2+δ)-Hausdorff content, every spherical polynomial p of degree ≤N satisfies sup|p| ≤ C_1 ((C_2/C_H^{n-2+δ}(M))^{2/δ} N^{4/δ-1})^{2N} sup_M |p|. This spectral inequality implies sharp heat observability from M on the sphere for all such δ, and a corresponding lower-dimensional observability result for -Δ+|x|^{2m} on R^n.","pith_inferences":["The slicing-plus-Turán strategy should extend verbatim to other compact rank-one symmetric spaces whose geodesics are circles.","If the underlying one-dimensional fractal Turán inequality can be sharpened, the Remez exponent 4/δ-1 would improve automatically.","The same angular Remez bound may give lower-dimensional observability for other radial magnetic or electric potentials whose eigenfunctions separate in spherical harmonics."],"forward_implications":["Heat null-controllability on the sphere holds with controls supported on any closed set of positive (n-2+δ)-Hausdorff content for every δ∈(0,1).","The same fractal sets are observable for the heat equation with any super-quadratic radial potential |x|^{2m}, m≥2.","The spectral inequality supplies an explicit Logvinenko–Sereda constant that tracks the precise dependence on the spectral parameter.","The dimensional threshold n-2 is sharp: zero sets of spherical harmonics have positive (n-2)-content yet fail observability."],"fun_headline_variants":["Fractal Remez bound controls heat on the sphere for any δ>0","Thin fractal sets observe the heat equation on spheres","Remez inequality on fractals yields sharp spherical heat observability","Positive Hausdorff content sets suffice for heat control on S^{n-1}","Fractal Remez on spheres sharpens lower-dimensional heat observability"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The one-dimensional fractal Turán bound on intervals of positive fractional Hausdorff content must hold with the precise power of the degree that is inserted into the spherical estimate.","fun_headline_variants_meta":{"raw":{"variants":["Fractal Remez bound controls heat on the sphere for any δ>0","Thin fractal sets observe the heat equation on spheres","Remez inequality on fractals yields sharp spherical heat observability","Positive Hausdorff content sets suffice for heat control on S^{n-1}","Fractal Remez on spheres sharpens lower-dimensional heat observability"]},"model":"grok-4.5","effort":"low","cost_usd":0.004841,"raw_usage":{"total_tokens":1466,"prompt_tokens":880,"num_sources_used":0,"completion_tokens":91,"cost_in_usd_ticks":48408000,"prompt_tokens_details":{"text_tokens":880,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":495,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":880,"tokens_out":91,"duration_ms":8128,"temperature":1.0,"reasoning_tokens":495,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-30T17:33:42.449806+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a sequence of spherical polynomials of degree N whose ratio of global maximum to maximum on a fixed set of positive (n-2+δ)-Hausdorff content grows faster than any constant times (C N^{4/δ-1})^{2N}, or show that the heat solution starting from an eigenfunction vanishing on such a set remains invisible in L^2 after time T.","supporting_citations":[],"review_version":1}