{"id":"3092b97a-a90d-4c72-bd0b-bd67ac8d9f7e","arxiv_id":"2505.02981","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every convex curve in the plane admits an ℓ²L⁶ decoupling estimate with constant C_ε R^ε, uniformly over all such curves.","lead":"This paper proves that every convex curve in the plane satisfies a uniform 'decoupling' inequality, a tool for breaking oscillatory sums into smaller pieces, with only an epsilon loss in the scale parameter. The result extends a celebrated theorem for the parabola to curves with no regularity assumptions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The high/low iteration rests on unproved overlap-count bounds for frequency tubes/sumsets; for devil's-staircase curves these could be R^c with c independent of epsilon, which would break Theorem 3.2.","rationale":"The paper's central claim is a universal decoupling estimate for all convex curves, so the proof must work uniformly on low-regularity, non-self-similar examples. The high/low argument in §3–§4 is the only mechanism that achieves this, and its key orthogonality steps are precisely the ones the reader flags: High Lemma, Low Lemma, and Lemma 4.2. I read the full manuscript looking for a derivation of the overlap bounds and did not find one. The High Lemma's 'simple trigonometry' reduces the count to Lemma 2.21, but Lemma 2.21 bounds slope increments over four neighbouring boxes, not the number of boxes in a fixed angular window; the implication needs an additional argument, especially near exceptional boxes where the chain is only 'repaired' by Lemma 2.18. The Low Lemma's 'geometrically evident' O(R^{2δ})-overlap is even less supported: the boxes at scale k+1 have lengths λ_{k+1} that can be comparable to the perpendicular sagitta of the parent arc, so fattening by B(0,2λ_{k+1}) can make many descendants overlap unless an unproved separation property holds. Lemma 4.2 delegates the full sumset-overlap count to [21, Lemma 2.4], which was proved for the single-scale, well-spaced setting; the extension to the pigeonholed multi-scale setting with boxes created by exceptional iterations is not demonstrated. I also note the reverse Hölder upgrade from (2.51) to Proposition 2.15 is delegated to [12, §5]; this is a secondary self-containedness gap, but the overlap bounds are more load-bearing because they enter every iteration of the high/low argument. I do not dispute the novelty or the overall plausibility; the concern is that the proof's decisive geometric estimates are asserted rather than derived. The proposed test on the Cantor-function curve is a direct stress test: it is the canonical example where lengths vary across all dyadic scales and adjacent rectangles fail the comparability property (Example 1.9), so if the overlap bounds fail anywhere, they should fail there. Since my analysis does not move the reader's conditional verdict, I mark the verdict unchanged.","tokens_in":57881,"tokens_out":14617,"duration_ms":164687,"concrete_test":"On the Cantor-function curve of Example 1.9, take R=3^{2K}, ε=1/(10K), and the polygonal approximation of the curve at scale R^{-1}. Run the single-scale and multi-scale algorithm of §2.2 to generate T_1 and T_2 explicitly, then compute (a) the maximum number of sets 2R^{2δ}τ_2+B(0,2λ_2) containing a common frequency, and (b) for a pair τ_1,τ'_1 separated by K/4 boxes, the maximum overlap of R^{2δ}τ_2+R^{2δ}τ'_2+B(0,λ_2). If the maximum exceeds C R^{2δ} in case (a) or C R^{3ε+4δ} in case (b) for large K, the 'geometrically evident' assertions in §3.3 and Lemma 4.2 are false on a natural worst-case curve, and the universal estimate is not established. If the counts stay within those powers, the overlap concern is cleared for this model.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 3.2 depends on three quantitative overlap assertions that are not derived from the established lemmas. In the High Lemma (§3.3), the tubes R^{2δ}(τ_k−τ_k) outside B(0,λ_k R^{−ε}/2) are asserted to have overlap ≤ R^{ε+2δ} by 'simple trigonometry' from Lemma 2.21. In the Low Lemma and in Lemma 4.2, the sets R^{2δ}τ_{k+1}+B(0,2λ_{k+1}) and R^{2δ}τ_k+R^{2δ}τ'_k+B(0,λ_k) are asserted to have overlap O(R^{2δ}) or O(R^{3ε+4δ}) as 'geometrically evident' or via a 'minor modification' of [21, Lemma 2.4]. Lemma 2.21 only controls the total slope increment across four neighbouring boxes; it does not, by itself, bound how many boxes have tangent directions inside a window of width (λ_k R_k)^{-1}, nor does it control the directional spread of the descendants of a single level-1 box. For the devil's-staircase-type curves of §1.4 the level-1 boxes have lengths spread over R^{−1} to R^{−K}, and exceptional chains (Lemma 2.21, Case 2) can create many boxes with very small successive slope increments. If the true overlap of the high tubes or of the sumsets is R^c with c>0 independent of ε, the ℓ²L⁴ orthogonality that drives the high/low iteration fails and Theorem 3.2 — and hence Theorem 1.5 — is unsupported. The paper neither proves these overlap bounds nor fully localizes the cited arguments to the non-uniform, pigeonholed setting.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims a universal ℓ²L⁶ decoupling estimate for all convex curves in the plane with no regularity assumptions: for every ε>0 and every R≥1, there is an ideal partition J of the parameter interval and a constant C_ε, depending only on ε, such that ‖f‖_{L⁶} ≤ C_ε R^ε (Σ_J ‖f_J‖²_{L⁶})^{1/2} for all Schwartz f whose Fourier transform is supported in the R⁻¹-neighbourhood of any convex curve satisfying (1.4). The proof adapts the high/low argument of Guth–Maldague–Wang [12] to the non-smooth setting: it constructs a multi-scale partition of admissible curves, pigeonholes in the box lengths at every scale, applies wave-packet pruning, proves High and Low Lemmas, and combines them through a weak-type/broad-narrow iteration. The final passage from the L⁶-branch estimate (2.51) to Proposition 2.15 is delegated to [12, §5]. Section 5 discusses illustrative fractal examples.","tokens_in":58242,"tokens_out":12529,"duration_ms":138458,"significance":"If valid, this is a substantial advance: it extends Bourgain–Demeter decoupling from the parabola and C² curves of positive curvature to arbitrary convex curves, with a constant uniform over the whole class. The multi-scale construction of the ideal partition from the affine dimension is genuinely new, and the fact that the constant does not depend on the curve is a strong and clearly falsifiable claim. The paper also gives credit where due: the main external inputs, [12] and [21], are independent, and the author's own prior work [20] is used only for illustrative examples in Section 5. The proof is not circular: the partition is constructed from the curve's geometry and no constant is fitted to the target estimate. The main risk is not the overall strategy but several unproved geometric overlap assertions inside the High Lemma, Low Lemma, and Lemma 4.2, which are load-bearing for the orthogonality steps.","major_comments":[{"comment":"The sentence 'Geometrically, it is evident that the sets 2R^{2δ}·τ_{k+1}+B(0,2λ_{k+1}) are only O(R^{2δ})-overlapping' is used as the sole input that converts local L²-orthogonality (Lemma A.3) into the factor R^{2δ} in the estimate |g^lo_k| ≤ C_lo R^{2δ} g_{k+1}. This assertion is not proved, and it is not a formal consequence of the stated direction-separation lemma. By (2.44), the lengths of sibling boxes at level k+1 can vary by a factor R^ε, and by Lemma 2.13 a fixed parent has up to O(R^{ε/2}) children; hence many τ_{k+1} centres can lie within a ball of radius 2λ_{k+1}. Lemma 2.21 only controls the slope increment across four neighbouring boxes, so the required overlap bound is exactly the kind of geometric fact that must be written down for the pigeonholed, non-uniform setting. Please supply a proof of the overlap bound, or state and prove a variant with the exact quantitative loss it produces.","section":"§3.3, Low Lemma"},{"comment":"The High Lemma's orthogonality step asserts: 'By an application of Lemma 2.21, and simple trigonometry, it follows (see Figure 1) that # {τ̄^∘_k ∋ ξ} ≤ R^{ε+2δ} for all ξ.' This is the step that converts the high-frequency part into the ℓ⁴L⁴ expression ‖g^hi_k‖²_{L²} ≤ C R^{2ε} Σ ‖f_{k+1,τ_k}‖⁴_{L⁴}. Lemma 2.21 as stated bounds the increment of γ'_R between the first and fourth of four neighbouring boxes; it does not, by itself, bound the number of boxes whose dual tubes contain a fixed point. The counting argument must be made explicit, including the exceptional-box case of Lemma 2.21, because this is the precise point where unusual convex curves (e.g. devil's-staircase curves) could produce many boxes with almost parallel directions. Please write out the argument so that the R^{ε+2δ} factor is verifiable.","section":"§3.3, High Lemma"},{"comment":"The proof of Lemma 4.2 delegates the key overlap estimate to a 'minor modification' of [21, Lemma 2.4] and concludes that the sumsets R^{2δ}·τ_k + R^{2δ}·τ'_k + B(0,λ_k) are O(R^{3ε+4δ})-overlapping. The cited lemma is proved for the single-scale Seeger–Ziesler decomposition, not for the multi-scale pigeonholed collections T^Λ_k used here. In the present setting, boxes at a fixed level arise from different level-1 ancestors, have lengths spread by powers of R^ε, and can be linked by exceptional chains (Lemma 2.18) that force a box to share its interval with an earlier typical ancestor. This is precisely the situation in which the directional spread of descendants of a fixed level-1 box is not controlled by Lemma 2.21 alone. The overlap bound is load-bearing for the local bilinear square-function estimate and hence for the proof of (4.1) for all k≥1. Please provide a self-contained proof in the present setting, or state the exact modification of [21, Lemma 2.4] and verify all its hypotheses.","section":"§4.1, Lemma 4.2"},{"comment":"The paper states that Proposition 2.15 follows from (2.51) by a reverse Hölder argument whose details, 'mutatis mutandis, can be found in [12, §5]'. This step is not a cosmetic repetition: here the canonical boxes have variable aspect ratios and the partition depends on ε and R, whereas [12, §5] is written for the uniform parabolic boxes. The reverse Hölder inequality converts an averaged L⁶ estimate into the desired ℓ²L⁶ decoupling, so any mistake in its constants or hypotheses would invalidate Theorem 1.5 even if the high/low iteration is correct. Please either include the reverse Hölder lemma with its proof in the present variable-box setting, or give a precise statement of the version in [12] and verify that all hypotheses are satisfied when the partition is the ideal partition constructed in §2.3.","section":"§2.4, deduction of Proposition 2.15 from (2.51)"}],"minor_comments":[{"comment":"The manuscript consistently writes 'Schwarz function' where 'Schwartz function' is standard; please correct this spelling globally.","section":"Throughout"},{"comment":"The symbol N is first used for 1/ε (the number of scales) and is then redefined to denote an arbitrary integer ≤1/ε; the paper acknowledges this, but the notation remains confusing in the statements of Theorem 2.20 and Proposition 2.15. Please use two distinct symbols (e.g. N_0 and N).","section":"§2.4"},{"comment":"The proof refers to Figure 1 as the justification for the tube-overlap count, but the figure is not described quantitatively in the text; please add labels defining τ̄^∘_k, λ_k, R^{2δ}, and the relevant angular window, so that the 'simple trigonometry' can be checked without guessing.","section":"§3.3, High Lemma"},{"comment":"The bound #J_p ≤ C₁ N(Γ_p,R_p^{-1}) log(2+R_p) is obtained by combining (2.41) or (2.15) with [21, Lemma 2.3 (iii)]; this is a reasonable citation, but the sentence says 'using (2.41) ... to the left typical boxes' without explaining the role of the right and exceptional boxes. Please spell out the short argument.","section":"§2.3, Verifying (J2)"},{"comment":"The appendix proves the essential Fourier-support bound with an error O(R^{-100(k+1)}), while the main text uses the uniform error O(R^{-100}) in Lemmas 3.18 and 3.20. The difference is harmless, but the exponents should be reconciled for readability.","section":"Appendix A.1"},{"comment":"The sentence about the Hausdorff dimension of the set of points of non-differentiability of f_μ is correct as cited, but the exact statement needed for the example (the dimension of the non-differentiability set of γ'_μ) is not stated; please make the implication explicit or add the relevant reference.","section":"§1.4, Example 1.8"}],"recommendation":"major_revision","confidential_remarks":"The central claim is plausible and the high/low strategy is well chosen; the missing overlap estimates in the High Lemma, Low Lemma, and Lemma 4.2 are exactly where the low-regularity geometry can behave in unexpected ways, so they are not routine details. I found no circularity: the construction is not fitted to the target constant, and the cited external results are independent. The paper is not ready for acceptance as written, but the gaps appear fixable within the manuscript's scope, which is why I recommend major revision rather than rejection. Please have the revised version refereed by someone familiar with the original Seeger–Ziesler biorthogonality argument, since the manuscript's main unresolved point is the validity of the overlap count in the multi-scale pigeonholed setting."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear —\\n\\nThe headline: this is a real advance. Theorem 1.5 removes regularity assumptions entirely and gives a curve-independent R^ε constant for ℓ²L⁶ decoupling, which answers Thiele's question. No prior work covered arbitrary convex curves with a uniform constant, so the main theorem is new, not a repackaging. The adaptation of the Guth–Maldague–Wang high/low scheme is careful: the ideal partition construction, the multi-scale pigeonholing, and the rescaling lemmas are worked out in genuinely useful detail. Constants depend only on ε, and there is no circularity; the cited external inputs are independent.\\n\\nThe soft spots are real and load-bearing. Three geometric overlap estimates are asserted rather than proved. In the High Lemma, the bound # {τ_k^◦ ∋ ξ} ≤ R^{ε+2δ} is called 'simple trigonometry' from Lemma 2.21, but that lemma controls slope increments across four neighbouring boxes, not the number of descendant boxes whose direction differences pass through a point. In the Low Lemma, the O(R^{2δ}) overlap of 2R^{2δ}τ_{k+1}+B(0,2λ_{k+1}) is called 'geometrically evident'—which it is not for devil's-staircase-type curves with exceptional chains. In Lemma 4.2, the finite-overlap of sumsets is delegated to a 'minor modification' of [21, Lemma 2.4], but the modification has to work in the non-uniform, pigeonholed setting, and that is precisely where the cited argument does not obviously apply.\\n\\nThe stress-test worry about exceptional chains is legitimate: if any of these overlap counts is actually R^c with c>0 independent of ε, the ℓ²L⁴ orthogonality fails and Theorem 3.2—and hence Theorem 1.5—is unsupported. I am not saying the central claim is false; the architecture is sound and the result is very plausible. But the paper currently asks the reader to accept three quantitative geometric assertions on faith, and they sit exactly where the low-regularity difficulty lives. A referee report should require full derivations or precise localized citations for these three bounds before publication. The reverse Hölder step from (2.51) to Proposition 2.15 is also delegated to [12, §5]; that one is more standard and less concerning, but the needed modifications should at least be sketched.\\n\\nBottom line: the paper deserves a serious referee and likely acceptance after substantive revision. I would cite it and would bring it to reading group, mainly to test whether the overlap assertions can be proved. My recommendation: send it to review, with instructions to make the overlap bounds a required revision rather than a cosmetic request.\\n\\nBest,","headline":"Genuine new result—uniform R^ε decoupling for all convex curves—but the proof leans on three unproved overlap assertions that a referee should require before acceptance.","tokens_in":58770,"tokens_out":1621,"would_cite":true,"duration_ms":21469,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every convex plane curve, an ℓ²L⁶ decoupling holds uniformly with R^ε loss.","keywords":["decoupling","convex curves","high/low argument","ideal partition","affine dimension","ℓ²L⁶ estimates","low-regularity curves","Fourier restriction"],"falsifier":"For the Cantor-staircase curve of Example 1.9 at $R=3^{2K}$, compute the maximum number of sumset rectangles $\\tau_k+\\tau'_k$ that contain a common point when $\\tau'_k$ is an exceptional box. If this multiplicity grows faster than $R^{C\\epsilon}$ for every fixed $C$, then the local bilinear square-function estimate cannot hold with the stated $R^\\epsilon$ loss, and the proof's overlap premise would be refuted; if it remains $R^{O(\\epsilon)}$, the premise is verified.","tokens_in":109,"feed_emoji":"📐","tokens_out":11708,"duration_ms":219567,"temperature":0.7,"pith_summary":"This paper establishes a uniform $\\ell^2 L^6$ decoupling theorem for arbitrary convex curves in the plane. For every $\\epsilon > 0$ and every $R \\geq 1$, the graph of any convex function with slope between 0 and 1 carries an 'ideal partition' into canonical boxes such that any function Fourier-supported in the $R^{-1}$-neighbourhood of the curve satisfies an $L^6$ estimate with constant $C_\\epsilon R^\\epsilon$, where $C_\\epsilon$ depends only on $\\epsilon$ and not on the curve. This extends the classical parabolic decoupling theorem to curves that may have no curvature, no self-similarity, and no smoothness beyond convexity. The proof uses the high/low argument over a coarse sequence of scales, with a multi-scale algorithm that builds the partition and a dyadic pigeonholing that controls box lengths at every scale. The result matters because it shows that the essential cancellation behind parabolic decoupling is a feature of convexity itself, not of special algebraic or fractal structure.","feed_headline":"All convex plane curves obey one decoupling estimate","feed_subtitle":"The proof extends the parabola's ℓ²L⁶ decoupling to curves with no curvature assumptions, uniformly in shape.","key_machinery":"The load-bearing object is the ideal partition $\\mathcal{J}$ of the parameter interval, whose pieces satisfy $(b-a)(\\gamma'_L(b)-\\gamma'_R(a)) \\leq 2R^{-1}$, whose number is at most $c_\\epsilon R^\\epsilon N(\\Gamma,R^{-1})$, and whose lengths are at least $R^{-1}$; each piece corresponds to a canonical box of size roughly $|J| \\times R^{-1}$. The proof's engine is the high/low argument: the square function $g_k = \\sum_{\\tau_k} |f_{k+1,\\tau_k}|^2 * \\omega_{\\tau_k}$ is split into a low-frequency part $g_k^{\\mathrm{lo}}$ (frequencies near the origin) and a high-frequency part $g_k^{\\mathrm{hi}}$; the High Lemma gives an $\\ell^4 L^4$ orthogonality bound for $g_k^{\\mathrm{hi}}$ using direction separation and bounded overlap of tubes, and the Low Lemma shows $g_k^{\\mathrm{lo}}$ is pointwise controlled by the next finer square function $g_{k+1}$ up to an $R^{2\\delta}$ factor. Iterating over the $O(1/\\epsilon)$ coarse scales and pigeonholing lengths at each scale converts the high/low split into the desired $\\ell^2 L^6$ decoupling.","core_discovery":"The paper's central discovery is that the parabola's $\\ell^2 L^6$ decoupling bound is universal across all convex curves: for each $\\epsilon$ there is a constant $C_\\epsilon$ such that $\\|f\\|_{L^6(\\mathbb{R}^2)} \\leq C_\\epsilon R^\\epsilon (\\sum_{J \\in \\mathcal{J}} \\|f_J\\|_{L^6(\\mathbb{R}^2)}^2)^{1/2}$ for every convex curve $\\Gamma$ satisfying the slope condition, where $\\mathcal{J}$ is an ideal partition chosen for $\\Gamma$, $\\epsilon$, and $R$. The constant does not depend on $\\Gamma$. The theorem treats curves whose curvature may fail to exist on sets of Hausdorff dimension arbitrarily close to 1, so the proof cannot use non-vanishing curvature or parabolic rescaling. Instead, it constructs the partition by a multi-scale algorithm adapted to the affine geometry of each curve, then runs the high/low iteration to reduce the $L^6$ norm to an $\\ell^2$ sum over the finest boxes.","pith_inferences":["If the same ideal-partition/high-low scheme is stable under affine transformations, the theorem should extend to convex hypersurfaces in higher dimensions, where the role of 'convex' would be played by surfaces with no curvature regularity; the paper does not pursue this.","The overlap bound for exceptional boxes (Lemma 2.21, Case 2) is the natural place to probe uniformity: a careful computation of sumset overlaps for the Cantor-staircase curve at $R=3^{2K}$ would either confirm the geometrical assertion or reveal a need to modify the partition.","Since the theorem's constant is independent of the curve, it suggests that compactness arguments over families of convex curves could yield quantitative information about affine dimension, or that affine dimension controls only the box count and not the decoupling exponent within the high/low framework.","The proof yields $R^{\\epsilon}$ rather than polylogarithmic losses; a natural next step is to refine the coarse-scale pigeonholing near exceptional scales to recover a log-power bound for curves without self-similarity, as the high/low method does for the parabola."],"forward_implications":["By interpolation, the same partition yields $\\ell^2 L^q$ decoupling for every $2 \\leq q \\leq 6$ with $C_\\epsilon R^\\epsilon$ loss, so the uniform bound covers the full classical range.","For $\\gamma(t)=t^2$ the theorem recovers the parabolic $\\ell^2 L^6$ decoupling theorem; for piecewise convex polynomial curves it recovers the polynomial-curve decoupling theorem of the cited work [22], with a constant depending on the number of pieces.","The estimate applies to curves such as graphs of integrals of devil's staircases, whose curvature is undefined on a set of Hausdorff dimension arbitrarily close to 1, even though those curves lack smoothness and self-similarity.","Because the iteration uses only $O(1/\\epsilon)$ scales independent of $R$, the proof does not require any self-similarity or homogeneity of the curve, so the mechanism is available for curves with non-uniform behaviour across scales.","The $\\epsilon$-dependence of the partition is stated to be harmless for applications; if removed, the theorem becomes a one-parameter family of partitions independent of $\\epsilon$."],"supporting_citations":[{"why":"Proves the baseline $\\ell^2 L^6$ decoupling theorem for the parabola, which this paper generalizes to all convex curves.","marker":"[2]"},{"why":"Introduces the high/low argument for parabolic decoupling; the present proof adapts that scheme.","marker":"[12]"},{"why":"Defines affine dimension and ideal partitions for convex curves and supplies the biorthogonality estimate used in the local bilinear square-function lemma.","marker":"[21]"},{"why":"Supplies the broad/narrow inequality used to pass from broad-set bounds to the full theorem.","marker":"[3]"},{"why":"Gives the ellipsoid containment used to construct spatial weights and pruning functions in the argument.","marker":"[16]"}],"fun_headline_variants":["Universal decoupling for all convex plane curves","One ℓ²L⁶ decoupling estimate fits every convex curve","Convex curves, no regularity: a single decoupling bound","Uniform ℓ²L⁶ decoupling: no curvature assumptions needed","All convex curves satisfy the same decoupling estimate"],"cache_read_input_tokens":60800,"weakest_assumption_plain":"The proof assumes that after pigeonholing by box lengths, the high-frequency tubes and the sums of adjacent boxes overlap only $R^{O(\\epsilon)}$ times for every convex curve, including exceptional boxes; this geometric overlap bound is stated as evident or left to a minor modification of the cited work [21], and if it failed the $\\ell^2 L^4$ orthogonality driving the high/low iteration would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Universal decoupling for all convex plane curves","One ℓ²L⁶ decoupling estimate fits every convex curve","Convex curves, no regularity: a single decoupling bound","Uniform ℓ²L⁶ decoupling: no curvature assumptions needed","All convex curves satisfy the same decoupling estimate"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000235,"raw_usage":{"total_tokens":1422,"prompt_tokens":792,"completion_tokens":630,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":408,"completion_tokens_details":{"reasoning_tokens":547}},"tokens_in":408,"tokens_out":630,"duration_ms":7753,"temperature":1.0,"reasoning_tokens":547,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:39:38.472034+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the Cantor-staircase curve of Example 1.9 at $R=3^{2K}$, compute the maximum number of sumset rectangles $\\tau_k+\\tau'_k$ that contain a common point when $\\tau'_k$ is an exceptional box. If this multiplicity grows faster than $R^{C\\epsilon}$ for every fixed $C$, then the local bilinear square-function estimate cannot hold with the stated $R^\\epsilon$ loss, and the proof's overlap premise would be refuted; if it remains $R^{O(\\epsilon)}$, the premise is verified.","supporting_citations":[{"cited_title":"Improved decoupling for the parabola","cited_arxiv_id":null,"evidence_quote":"Introduces the high/low argument for parabolic decoupling; the present proof adapts that scheme."},{"cited_title":"Riesz means associated with convex domains in the plane","cited_arxiv_id":null,"evidence_quote":"Defines affine dimension and ideal partitions for convex curves and supplies the biorthogonality estimate used in the local bilinear square-function lemma."},{"cited_title":"Bounds on oscillatory integral operators based on multilinear estimates","cited_arxiv_id":null,"evidence_quote":"Supplies the broad/narrow inequality used to pass from broad-set bounds to the full theorem."},{"cited_title":"Extremum problems with inequalities as subsidiary conditions","cited_arxiv_id":null,"evidence_quote":"Gives the ellipsoid containment used to construct spatial weights and pruning functions in the argument."}],"review_version":1}