{"id":"2fdb5bd3-29c7-41a3-a7d5-94500cd709ef","arxiv_id":"2508.21214","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For harmonic functions in R^n, ε-smallness of the gradient on a set of Hausdorff dimension n-2+δ (any δ > 0) forces a C ε^α bound on the half-ball, reaching the sharp threshold and answering the Logunov-Malinnikova conjecture.","lead":"Gradients of harmonic functions obey a sharp unique-continuation law: if the gradient is ε-small on a set of Hausdorff dimension just above n-2, it is bounded by C ε^α on the whole ball. The paper proves this at the optimal dimensional threshold in every dimension, resolving a 2018 conjecture of Logunov and Malinnikova.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2's final step invokes Claim 1 after rescaling without verifying the small-capacity hypothesis; capacity scaling with ℓ^{-s} and the Frostman bound (4) point in the opposite direction.","rationale":"The reader's weakest_assumption is exactly the load-bearing concern: the transition from the minimal triadic cube P to the rescaled measure ν, and the assertion that Claim 1 applies, is asserted rather than derived. My reading of the scaling laws strengthens this concern: Riesz s-capacity is homogeneous of degree s, so the rescaling by ℓ(P)^{-1} multiplies capacity by ℓ(P)^{-s}; since Claim 2 makes A grow faster than K, this factor can be large. Moreover, the Frostman bound (4) implies that ν has finite s-energy with a bound independent of K, so Cap_s of the rescaled support is bounded below by a positive constant; this is incompatible with 'capacity small enough' for large C_n. I do not think this is a fatal disproof of the theorem—the gap might be repairable by adding a quantitative decay estimate in Lemma 1 or by a different argument—but as written Lemma 2, and hence Theorem 1, lack a complete proof. This matches the reader's CONDITIONAL verdict, so I recommend no change. The rest of the paper, including the recursive inequality and the base case via Naber–Valtorta, is structurally sound and not in question here.","tokens_in":13250,"tokens_out":22197,"duration_ms":235448,"concrete_test":"Re-derive the final paragraph of Lemma 2 with explicit quantitative tracking: (i) write Cap_{n-2+δ/4}(E') in terms of Cap_{n-2+δ/4}(E∩P) and ℓ(P) using the scaling law λ^s; (ii) extract from the proof of Lemma 1 a quantitative upper bound for Cap_{n-2+δ/4}(E∩P) as a function of A; (iii) check whether the resulting rescaled capacity is below Claim 1's threshold c/C_n. Independently, compute the s-energy of ν from the Frostman bound (4) and compare Cap_s(supp ν) with c/C_n. If Lemma 1 provides no quantitative decay, or if the energy estimate gives a positive lower bound for capacity, then Claim 1 cannot be applied and Lemma 2 is unproved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Lemma 2 (Section 3.1) is the only new ingredient behind Theorem 1, and its final paragraph is not supported. After choosing a minimal cube P with ℓ(P)=K(2A+1)^{-1}, the measure ν=µ(P)^{-1}µ|_P is rescaled to the unit cube, and the paper asserts that 'ν satisfies all the hypotheses of Claim 1 by construction.' One hypothesis of Claim 1 is Cap_{n-2+δ/4}(E') < c/C_n for the rescaled support E'. This is not verified. Riesz s-capacity scales by λ^s under dilation by λ, so rescaling P by ℓ(P)^{-1} multiplies capacity by ℓ(P)^{-s}. Since ℓ(P)=K/(2A+1) and Claim 2 forces A to grow faster than K, this factor can be enormous. Lemma 1 only says the original Cap_{n-2+δ/4}(E) can be made arbitrarily small by taking A large; it supplies no quantitative decay rate, so it does not imply the rescaled capacity is small. In fact, the minimality condition (4) is a Frostman bound with exponent α=n-2+δ/2 > s=n-2+δ/4. Any probability measure with ν(B(x,r)) ≤ C r^α has finite s-energy, hence Cap_s(supp ν) ≥ 1/I_s(ν) ≥ c(n,δ) > 0. For C_n large enough, c/C_n < c(n,δ), so Claim 1's small-capacity hypothesis is false—not satisfied by construction. The final instruction to 'take a descendant cube contained inside it' also assumes Claim 1's ball contains a lattice cube, i.e., has radius at least comparable to 1/K; Claim 1 provides no such lower bound. Since Lemma 2 feeds directly into Theorem 3 and then Theorem 1, this is a load-bearing gap in the central argument.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims Theorem 1: if u is harmonic in B_1 ⊂ R^n with sup_{B_1}|∇u| = 1 and E ⊂ B_{1/2} has positive (n−2+δ)-dimensional Hausdorff content for some δ > 0, then sup_{B_{1/2}}|∇u| ≤ C (sup_E |∇u|)^α with C, α depending only on n, δ, and the Hausdorff content of E. This is stated as an improvement over Logunov–Malinnikova's previous δ > 1−c_n restriction and reaches the sharp codimension-two threshold. The proof structure is: the hyperplane result of Malinnikova (Theorem 2) as the base case; Lemma 1, a hyperplane lemma with Riesz capacities; Lemma 2, the key new counting lemma for cubes near a hyperplane; Theorem 3, a bound on the number of bad subcubes; Proposition 2, a recursive inequality; and a double induction in the style of Logunov–Malinnikova, with the base case supplied by the Naber–Valtorta effective-critical-set estimate (Theorem 4).","tokens_in":13497,"tokens_out":46614,"duration_ms":431583,"significance":"If correct, Theorem 1 confirms the Logunov–Malinnikova conjecture for gradients of harmonic functions at the sharp threshold, and the δ = 0 obstruction (e.g., ∇(x_1 x_2) vanishes on a line in R^3) is clearly identified. The paper is well organized, uses appropriate external benchmarks ([Mal04], [NV17], [Log18a], [LM18b]), and the doubling-index apparatus is presented carefully. The main weakness is that the proof of Lemma 2, the sole new ingredient behind the main theorem, contains a serious gap in its final step. The architecture of the recursion and induction is standard and largely coherent, but acceptance hinges entirely on whether Lemma 2 can be proved.","major_comments":[{"comment":"The assertion that after rescaling eP to the unit cube, 'ν satisfies all the hypotheses of Claim 1 by construction' is unsupported. Claim 1 requires Cap_{n−2+δ/4}(E') to be small for the rescaled support E'. Riesz s-capacity scales by ℓ(P)^{-s} under dilation by ℓ(P)^{-1}; since ℓ(P)=K(2A+1)^{-1}, this introduces a factor ((2A+1)/K)^s, while Lemma 1 supplies no quantitative rate at which Cap_{n−2+δ/4}(E) tends to 0 as A→∞. Smallness of the original capacity therefore does not transfer to the rescaled set. Moreover, the minimality condition (4) is a Frostman bound of exponent n−2+δ/2 > n−2+δ/4; such a measure has finite (n−2+δ/4)-energy, so Cap_s(supp ν) is bounded below (up to lattice-dependent constants), making the asserted hypothesis at least as hard as the conclusion it is meant to produce. This gap is load-bearing: Lemma 2 is the only new input behind Theorem 3 and Theorem 1.","section":"§3.1, Lemma 2, final paragraph"},{"comment":"The instruction to 'take a descendant cube contained inside it' after applying Claim 1 is also unjustified. Claim 1 produces only a ball eB in the hyperplane with μ(eB) > C_n r(eB)^{n−2+δ/2}; it gives no lower bound on r(eB). To obtain a contradiction with the cube Frostman bound (4), one needs a lattice cube of side comparable to r(eB) contained in eB, which requires r(eB) ≥ c(2A+1)^{-1} (or cK^{-1} after rescaling). No such lower bound is supplied. Without it, Claim 1's ball and the bound (4) can coexist, and the claimed contradiction does not follow.","section":"§3.1, Lemma 2, final paragraph"}],"minor_comments":[{"comment":"The statement of Proposition 2 (coefficients A^{2−δ} and A^{−δ}) does not match the proof's displayed bounds (A^{n−2+δ/2} A^{−(n−2+δ)} and A^{n−2+δ/2} A^{−n}); also, the text switches from a (2A+1)-adic partition to an A-adic one without comment. The induction is plausible with any negative powers, but the formulas should be reconciled.","section":"§3.4, Proposition 2"},{"comment":"The displayed exponent '1 − (1−δ)/(1−δ/2)' in the last line of Claim 2 appears to be a miscalculation; direct algebra gives 1 + (1−δ)/(1−δ/2) = (2−3δ/2)/(1−δ/2). The claim still holds for δ ∈ (0,1) because the exponent is positive, but the displayed formula should be corrected.","section":"§3.1, Claim 2"},{"comment":"The proof's chain of L^2-average comparisons is difficult to follow (ratios over B(x,2r/(2±ε)) and adjacent balls). The argument is standard, but the exposition should be rewritten, and the reliance on [Fos24, Proposition 3] for the monotonicity of the L^2-based frequency should be stated explicitly.","section":"§2, Proposition 1"},{"comment":"The dependence of the final constants C, α on m (the Hausdorff-content lower bound) is never tracked through the base case, Lemma 5, and the induction. A short remark on where m enters would help the reader verify the claimed dependence.","section":"§1 / §3.3"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern about the final step of Lemma 2 is well founded: the capacity-scaling issue is real and is not addressed in the manuscript. The additional Frostman-based objection is less clear-cut, because the positive lower bound on Cap_s(supp ν) may decay with the lattice parameter A, but the onus is on the authors to prove the small-capacity hypothesis for the rescaled measure. Since Lemma 2 is the only new ingredient in the paper, acceptance should be conditional on a complete, correct proof of that lemma; the current final paragraph is a hand-wave over exactly the point at issue. I would ask the authors to supply the missing estimates or to restructure the contradiction, and I would also ask them to reconcile the statement and proof of Proposition 2."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe paper is a serious attempt at the Logunov-Malinnikova conjecture. The main theorem, if correct, is a major result: propagation of smallness for gradients of harmonic functions from sets of dimension n-2+δ for any δ>0, improving on the δ>1-c_n restriction. The overall architecture is sensible: base case via Naber-Valtorta, a recursive inequality, and a double induction, all in the spirit of Logunov's work. The exposition is clear and the use of Malinnikova's hyperplane result is a natural idea.\n\nBut the proof of Lemma 2, the one new ingredient, has a serious gap. After selecting the minimal triadic cube eP and rescaling it to the unit cube, the paper asserts that the rescaled measure ν satisfies all hypotheses of Claim 1 by construction. That is not true as written. Claim 1 asks for small (n-2+δ/4)-Riesz capacity. Under rescaling by ℓ(eP)^{-1}, capacity scales by ℓ(eP)^{-s}, and the minimality condition (4) actually forces the opposite: it is a Frostman bound with exponent α = n-2+δ/2 > s. Any probability measure satisfying ν(B(x,r)) ≤ C r^α has finite s-energy and hence positive s-capacity, bounded below by a dimensional constant. So the rescaled support cannot have arbitrarily small capacity, and the hypothesis of Claim 1 is not satisfied. The additional step of taking a descendant cube inside the ball from Claim 1 is also not justified—Claim 1 gives no lower bound on the ball's radius, so it may contain no lattice cube. Since Lemma 2 feeds into the induction step (Theorem 3), this is load-bearing. Without a repaired Lemma 2, Theorem 1 is unproven.\n\nThe other issues are minor: a self-citation for a standard L2-frequency monotonicity fact and some proofs delegated to Logunov's lemmas.\n\nI think the paper deserves a serious referee. The result is important, the strategy is plausible, and the gap is localized. Whether it is fixable depends on whether a different argument can avoid the scaling contradiction; maybe some extra hypothesis or a different selection of the cube is needed. A reader working on quantitative unique continuation will find the write-up useful even in preprint form, but it should not yet be accepted as a proof.\n\nMy recommendation: send it to peer review, but with a clear request to address the capacity scaling issue in Lemma 2. I would not cite the main theorem as established in my own work until the gap is closed.\n\nBest regards.","headline":"Claims the Logunov-Malinnikova conjecture for gradients, but the key lemma's final step appears broken: the rescaled measure cannot satisfy both small capacity and the Frostman bound the proof creates.","tokens_in":14225,"tokens_out":10999,"would_cite":false,"duration_ms":101770,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B60","31B05","35J05","28A78"],"pacs":[],"model":"deepseek-v4-flash","headline":"Smallness near codimension two propagates for harmonic gradients","keywords":["harmonic functions","propagation of smallness","doubling index","Hausdorff content","Riesz capacity","unique continuation","critical sets","codimension two"],"falsifier":"Compute the (n−2+δ/4)-Riesz capacity of a set E after scaling by the factor ℓ(P̃)^{-1}. If there is a family of configurations where the rescaled capacity stays bounded below while the original capacity tends to zero, then Claim 1 cannot be triggered and Lemma 2's contradiction collapses.","tokens_in":12913,"feed_emoji":"📐","tokens_out":8415,"duration_ms":72096,"temperature":0.7,"pith_summary":"The paper proves that if a harmonic function's gradient is small on a set that is almost of codimension two—specifically, a set with positive (n−2+δ)-dimensional Hausdorff content for any δ>0—then the gradient is also small (in a power sense) throughout the inner half-ball. This answers in the affirmative a conjecture raised in [LM18b], improving the earlier result there that required δ to exceed a small dimensional constant. The threshold is sharp: sets of exactly dimension n−2 need not propagate smallness, so the theorem reaches the best possible dimensional condition. If correct, it establishes the sharp version of quantitative unique continuation for harmonic gradients.","feed_headline":"Smallness near codimension two propagates for harmonic gradients","feed_subtitle":"A tiny gradient on any set of dimension above n−2 forces smallness on the whole ball.","key_machinery":"The doubling index N(Q) of a cube, the logarithmic growth of sup|∇u| when the ball is doubled. The new engine is an improved hyperplane lemma (Lemma 2): if the parent cube has bounded doubling index, then the number of high-doubling-index subcubes that meet a coordinate hyperplane is at most η(2A+1)^{n-2+δ}, for any δ>0. This is proved by combining a Riesz-capacity version of the hyperplane propagation result [Mal04] (Lemma 1) with a minimal-triadic-cube argument and a Frostman-type ball-concentration claim (Claim 1) to force a contradiction via a scale-invariant measure bound.","core_discovery":"The central discovery is Theorem 1: with sup_{B_1}|∇u|=1, if E⊂B_{1/2} has H^{n-2+δ}_∞(E)>m for some m,δ>0, then sup_{B_{1/2}}|∇u| ≤ C (sup_E |∇u|)^α, where C,α depend only on n,δ,m. Equivalently, the exponent is independent of the shape of E, and power-law decay follows from the mere dimensional content of the smallness set. The proof works by showing that if the doubling index of ∇u over B_1 is at most (1+c)N, then among a fine triadic partition of the unit cube, fewer than (1/2)(2A+1)^{n-2+δ} subcubes can have large doubling index. This bad-cube counting, together with a recursive inequality inherited from [LM18b], yields the propagation estimate for every δ>0.","pith_inferences":["If the scaling step in the proof of Lemma 2 is made fully rigorous, the same minimal-cube scheme may apply to other elliptic equations and to sets with dimensions arbitrarily close to n−2, potentially removing the analytic-coefficient restriction.","The sharpness of the threshold suggests that further improvement would require exploiting structure beyond Hausdorff dimension, such as rectifiability or porosity of the smallness set.","A testable extension is to replace the hyperplane with a general (n−2)-dimensional Lipschitz surface, which would need a quantitative version of [Mal04] on such surfaces.","Extracting the explicit dependence of α on n,δ,m would require tracking constants through the recursive inequality, a possible follow-up."],"forward_implications":["Resolves the conjecture from [LM18b] at the sharp codimension-two threshold for gradients of harmonic functions.","Yields quantitative unique continuation: a harmonic function whose gradient vanishes on such a set must be identically zero, with a power-law rate.","The recursive inequality gives the bound M(N,a) ≤ Ce^{-βa/N}, the engine behind the theorem.","The same arguments extend to solutions of div(A∇·)=0 with analytic coefficients, as noted in the paper.","For Lipschitz-coefficient operators, the method yields a weaker exponential-of-cube-root propagation (Corollary 3)."],"supporting_citations":[{"why":"supplies the prior propagation result with the δ>1−c_n restriction that Theorem 1 improves.","marker":"[LM18b]"},{"why":"provides the original hyperplane lemma and the bad-cube counting framework adapted here.","marker":"[Log18a]"},{"why":"gives propagation of smallness from positive Riesz capacity within a hyperplane, used as Theorem 2.","marker":"[Mal04]"},{"why":"supplies the effective-critical-set volume estimate that yields the base case.","marker":"[NV17]"},{"why":"defines Riesz capacity and Hausdorff content, which the arguments use.","marker":"[Mat95]"},{"why":"establishes the frequency/doubling-index monotonicity used throughout.","marker":"[GL86]"},{"why":"gives the L2-average monotonicity for gradients used in Proposition 1.","marker":"[Fos24]"}],"fun_headline_variants":["Harmonic gradient smallness propagates from any set of dimension n−2+δ","Sharp threshold: smallness on dimension n−2+δ controls harmonic gradients","Codimension-two barrier broken: harmonic gradient smallness propagates","Harmonic gradients: tiny values on thin sets force global smallness"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The proof of Lemma 2 assumes that after rescaling the minimal bad cube to unit size, the normalized measure satisfies the small-capacity hypothesis of Claim 1, and that Claim 1's conclusion applies to that rescaled measure; this step is asserted rather than derived, and if it fails, the bad-cube bound—and with it Theorem 1—has no proof.","fun_headline_variants_meta":{"raw":{"variants":["Harmonic gradient smallness propagates from any set of dimension n−2+δ","Sharp threshold: smallness on dimension n−2+δ controls harmonic gradients","Codimension-two barrier broken: harmonic gradient smallness propagates","Harmonic gradients: tiny values on thin sets force global smallness"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001491,"raw_usage":{"total_tokens":5837,"prompt_tokens":771,"completion_tokens":5066,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":515,"completion_tokens_details":{"reasoning_tokens":4986}},"tokens_in":515,"tokens_out":5066,"duration_ms":35499,"temperature":1.0,"reasoning_tokens":4986,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T14:32:16.860099+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the (n−2+δ/4)-Riesz capacity of a set E after scaling by the factor ℓ(P̃)^{-1}. If there is a family of configurations where the rescaled capacity stays bounded below while the original capacity tends to zero, then Claim 1 cannot be triggered and Lemma 2's contradiction collapses.","supporting_citations":[],"review_version":1}