{"id":"beba755d-f1e8-4257-a407-8392ff7ebefd","arxiv_id":"2608.01446","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"On compact quotients of stratified groups, the number of δ-deep nodal domains and coarse zero-set features of sums of sub-Laplacian eigenfunctions grows polynomially in the eigenvalue bound, with arbitrarily small exponent loss.","lead":"This paper proves coarse topological bounds on the nodal sets of linear combinations of sub-Laplacian eigenfunctions on compact nilmanifolds. If correct, it gives the first sub-Riemannian analogues of coarse Courant and Bézout theorems and a roadmap toward the full equiregular conjecture.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central bound hinges on uniform, ball-local Lu–Wheeden polynomial approximation (Thm 5.3); if its constant depends on center or requires global Sobolev regularity, the covering argument in Thm 1.5 collapses. Otherwise no internal gap found.","rationale":"The reader's weakest assumption is exactly the dependence on the Lu–Wheeden polynomial approximation theorem (Theorem 5.3), and I agree that this is the most load-bearing point. I inspected the main chain from Theorem 1.5 to Theorems 1.1 and 1.3 and found no internal fatal gap: the subadditivity, stability, duality, and spectral estimates are applied consistently, and the removal of the +b_d(N) term in (36) is justified by the easy case δ > C_0||F||. Minor issues exist but are not load-bearing: the interval formula in Corollary 5.4 for S = -√P is written with reversed endpoints (the correct bar is [-b^2,-a^2)), but the cardinality statement still follows from a homeomorphic reparametrization; and the d=n-1 case silently assumes the same δ/(C_0||F||) ≤ 1 case split as d=0, but the complementary case is trivial. The only serious risk is that Theorem 5.3 as cited might not provide the uniform, ball-local approximation used in Section 6. Since the theorem is external and not reproduced, a targeted check of its hypotheses is the right way to settle this. The verdict should remain ACCEPT/UNCHANGED because the cited result is established and the paper's own proof is otherwise coherent.","tokens_in":34714,"tokens_out":32529,"duration_ms":295521,"concrete_test":"Verify the exact statement of [38, Theorem 5.11] (or [39, Theorem 5.11]): check that (i) the constant is independent of the center g and (ii) the L^p norm on the right is taken over the ball B_cc(g,r) itself, not a global norm. If the source theorem is stated for functions on the whole group, repeat the Section 6 covering argument with F extended by a partition of unity and confirm the error remains O(r^{k-Q/p}) with a uniform constant; if this fails for some k>Q/2, the proof of Theorem 1.5 requires an additional lemma.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1 rests on Theorem 1.5, whose proof (Section 6) uses Theorem 5.3 to approximate F on every ball B_cc(g_i, C_2 r) in a cover of a fundamental domain by an element of S_{k-1}(G). The barcode estimate requires the approximation error to be uniformly C r^{k-Q/p} ||F||_{W^{k,p}_X} over all centers and all radii in the cover. The manuscript states Theorem 5.3 in exactly this uniform, ball-local form and cites [38,39], but does not reproduce the statement or proof. If the source theorem has a constant depending on the center, or requires f to be compactly supported or globally defined, then Corollary 5.5 and the covering estimate (33) need an additional extension argument that is not supplied. This is the single point where the stratified-group structure does the essential work; the paper itself flags the absence of an equiregular analogue as the main obstacle to Conjecture 1.13. The cited theorem is established, so the risk is low, but this dependence is external and not machine-checked.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proves coarse Courant- and B\\'ezout-type bounds for finite linear combinations of eigenfunctions of sub-Laplacians on compact nilmanifolds associated with stratified groups. The main analytic result is Theorem 1.5, a persistence-barcode estimate for \\(\\pm|F|\\) in homological degrees \\(0\\) and \\(n-1\\) in terms of the anisotropic Sobolev norm \\(\\|F\\|_{W^{k,p}_X}\\), with exponent \\(Qp/(kp-Q)\\). Through subelliptic coercivity (Theorem 4.17) and the spectral bound \\(\\|(1+P)^{k/2}f\\|_{L^2}\\le(1+\\lambda)^{k/2}\\|f\\|_{L^2}\\), Theorems 1.1 and 1.3 follow, giving \\((\\lambda+1)^{Q/2(1+\\epsilon_k)}\\delta^{-2\\epsilon_k}\\) for \\(d\\in\\{0,n-1\\}\\). The proof of Theorem 1.5 combines ball-box geometry, a Vitali-Wiener covering, persistence subadditivity/stability, and the Lu-Wheeden Morrey-Sobolev polynomial approximation on Carnot balls. The paper also proves two non-optimal bounds in the general equiregular setting and formulates a conjecture with an explicit programme.","tokens_in":35009,"tokens_out":13513,"duration_ms":124321,"significance":"These appear to be the first results establishing the coarse Courant--Herrmann-type bound in the sub-Riemannian nilmanifold case, with an exponent arbitrarily close to the conjectured \\(Q/2\\). The method is structurally transparent: the spectral statement is reduced to a barcode estimate for anisotropic Sobolev functions, and the only place where the stratified-group structure is essential is the ball-local polynomial approximation. The paper is explicit about its limitations---homological degrees only \\(0\\) and \\(n-1\\), a slightly worse exponent, and the absence of an equiregular analogue of Theorem 5.3. If the cited Lu-Wheeden theorem indeed holds in the uniform, ball-local form stated here, the proofs are internally consistent and the paper is a solid contribution to the subject.","major_comments":[],"minor_comments":[{"comment":"Theorem 5.3 is the single most load-bearing external input: the covering argument in the proof of Theorem 1.5 requires a constant independent of the center \\(g\\), the radius \\(r\\), and the chosen ball in the cover. The paper cites [38,39] and the first author's thesis [52], but does not reproduce the statement. Since a center-dependent constant would invalidate estimate (33), please either quote the precise theorem from the cited sources or add a short appendix containing the statement and the uniformity claim.","section":"§5.3, Theorem 5.3"},{"comment":"The sentence after Corollary 1.10 says that applying the corollary to \\(D=P^\\kappa\\) yields Theorems 1.1 and 1.3 only for even \\(k>Q/2\\). However, the actual proof in Section 7 applies Theorem 1.5 directly with Theorem 4.17 and proves the statement for every integer \\(k>Q/2\\). Please adjust the introductory remark so that it does not understate the range of \\(k\\) covered.","section":"§1.5 and §7"},{"comment":"In the trivial case at the start of the proof, the condition \\(C_0\\|F\\|_{W^{k,p}_X}<\\delta\\) can be weakened to \\(\\le\\delta\\), since any finite bar has length at most \\(\\max|F|-\\min|F|\\le\\delta\\). Also, the notation in (32), \\(N_{0,\\delta}(\\pm|F|_{\\pi(B_\\infty(g_i,r))}|)\\), is slightly garbled; the intended meaning is clear but should be written cleanly.","section":"§6, proof of Theorem 1.5"},{"comment":"The passage from degree \\(0\\) to degree \\(n-1\\) uses Proposition 2.20, which assumes \\(X\\) is a closed, smooth, orientable manifold. Nilmanifolds of stratified groups are indeed orientable, but this fact is not stated where the \\(d=n-1\\) case is treated. Adding one sentence would make the argument fully self-contained.","section":"§2.3 and §6"}],"recommendation":"minor_revision","confidential_remarks":"The only substantial verification risk is the external dependence on the Lu-Wheeden theorem in exactly the uniform ball-local form stated in Theorem 5.3. In my reading the dependence is legitimate and the authors have correctly identified the lack of an equiregular analogue as the main obstacle. The paper is well within the scope of the journal and the central claims are credible."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is the real thing. It proves the first coarse Courant and Bézout bounds for linear combinations of eigenfunctions of sub-Laplacians on compact nilmanifolds, with the expected exponent up to a factor (1+ε_k). The engine is a barcode estimate for anisotropic Sobolev norms (Theorem 1.5), and the deduction to the spectral results goes through standard subelliptic coercivity and persistence inequalities. I checked the main chain and found no gap. The paper also extends to maximally hypoelliptic operators and handles homological degree n−1 via the duality in Proposition 2.20, which is a nice bonus.\n\nWhat is genuinely new: the combination of topological persistence with anisotropic Sobolev theory and local polynomial approximation on stratified groups is not a routine translation of the Riemannian argument. The Riemannian case [14] uses dyadic cube partitions; here the covering argument uses homogeneous balls and the ball-box theorem. Theorem 1.5 is the real content, and its proof is carefully written.\n\nSoft spots, in proportion. The main external dependence is Theorem 5.3 (Lu–Wheeden Morrey–Sobolev approximation). The stress-test concern about whether the constant is uniform in the ball center is fair in principle, but the theorem is stated in exactly the uniform ball-local form and it is a published result; the paper cites [38,39] and [52] for it. I would not call this a gap, just a dependency that is not re-proved. If I were refereeing, I'd ask the authors to state the precise version they use explicitly in the proof rather than citing a chain. The other limitations are explicitly acknowledged in Section 1.6: the exponent Q/2(1+ε) is not optimal, and intermediate homological degrees are out of reach of this method. That is honest, not a hidden flaw.\n\nCitation pattern: heavy self-citation is expected here; [14] is the Riemannian precursor and this paper is a direct extension. I don't see post-hoc fitting. The spectral step is the standard estimate ||f||_{W^{k,2}_X} ≤ C||(1+P)^{k/2}f||_2.\n\nWho it's for: people working on nodal geometry, sub-Riemannian spectral theory, and topological persistence in analysis. It deserves a serious referee; the proofs are long but the structure is clear. My recommendation: engage with it, send it to a specialist on subelliptic PDE and a topologist who knows persistence. Minor revision, not a desk reject.","headline":"First coarse Courant and Bézout bounds for sub-Laplacians on nilmanifolds; the proof is careful and the main external dependence is a published, uniformly stated approximation theorem, not a hidden gap.","tokens_in":35505,"tokens_out":1871,"would_cite":true,"duration_ms":19465,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35H20","35P15","53C17","58J50","55N31"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves coarse, persistence-based versions of Courant's nodal domain theorem and Bézout's theorem for linear combinations of sub-Laplacian eigenfunctions on compact nilmanifolds, with exponents arbitrarily close to the conjectured","keywords":["coarse nodal count","topological persistence","sub-Riemannian geometry","Carnot groups","sub-Laplacian eigenfunctions","Courant-type theorems","anisotropic Sobolev spaces"],"falsifier":"On the Heisenberg nilmanifold, take a fixed smooth F (e.g., the projection of sin(2πx)cos(2πy) with $L^{2}$ normalization) and compute the degree-0 bar count N_{0,δ}(|F|) as δ→0. If it grows faster than $δ^{{-Qp/(kp−Q)}}$ for some kp>Q while the anisotropic $W^{{k,p}}$_X norm stays bounded, Theorem 1.5's exponent is wrong. Similarly, a family of eigenfunction combinations with m_{0,δ} exceeding the $δ^{{-2ε_k}}$(λ+1)^{Q/2(1+ε_k)} growth would refute Theorem 1.1.","tokens_in":34621,"feed_emoji":"","tokens_out":10312,"duration_ms":90997,"temperature":0.7,"pith_summary":"This paper extends the coarse (persistence-based) Courant and Bézout bounds, previously known on Riemannian manifolds, to sub-Riemannian settings. On compact nilmanifolds—quotients of Carnot groups by lattices—it proves that the number of δ-deep nodal domains (and higher-degree coarse counts) of an $L^{2}$-normalized linear combination of sub-Laplacian eigenfunctions with eigenvalues ≤ λ grows at most like $δ^{{-2ε_k}}$(λ+1)^{Q/2(1+ε_k)}, where Q is the homogeneous dimension and ε_k can be made arbitrarily small by taking k large. The same method yields a coarse Bézout bound for the zero sets of such combinations, and all results extend to general maximally hypoelliptic operators. The proof combines topological persistence with anisotropic Sobolev theory of bracket-generating vector fields; a conjectured optimal bound for general equiregular sub-Riemannian manifolds is stated and a programme for proving it outlined, with the missing ingredient identified as an equiregular analogue of the polynomial approximation theorem.","feed_headline":"Nodal counts bounded near optimal on sub-Riemannian nilmanifolds","feed_subtitle":"First coarse Courant and Bézout bounds for eigenfunction sums, with exponent within ε of the homogeneous dimension.","key_machinery":"The principal objects are the coarse nodal count m_{d,δ}(F) = rank( Ȟ_d({|F|>δ}) → Ȟ_d({F≠0}) ) and the coarse zero count z_{d,δ}(F) = rank( Ȟ_d({F=0}) → Ȟ_d({|F|≤δ}) ), which for d=0 count δ-deep nodal domains and components of a δ-thickening that touch the zero set. These are controlled through the persistence barcode of ±|F|, whose bar-counting invariant N_{d,δ} obeys subadditivity, stability, and a duality between degree d and n−1−d. The new analytic input is the Morrey–Sobolev approximation theorem on Carnot balls (Theorem 5.3): a $C^{0}$ approximation of each f_i by polynomials of homogeneous degree k−1 with error C $r^{{k−Q/p}}$||X^k f||_{L^p}, which lets each ball in a cover be replaced","core_discovery":"On a closed sub-Riemannian nilmanifold N obtained as a lattice quotient of a stratified (Carnot) group, the authors prove a quantitative bound for the coarse topology of nodal sets of vector-valued functions F=(f_1,...,f_l) whose components lie in the spectral subspace F_λ of the sub-Laplacian. The central estimate (Theorem 1.5) is an anisotropic Sobolev inequality for persistence barcodes: for any smooth map F and any k,p with kp>Q, the number of bars of length >δ in the degree-0 (and degree n−1) barcode of ±|F| is bounded by C(||F||_{$W^{{k,p}}$_X}/δ)^{Qp/(kp−Q)} plus the Betti number. From this, Theorem 1.1 gives, for every k>Q/2 and d∈{0,n−1}, the coarse Courant bound m_{d,δ}(F) ≤ C δ^{-2ε_k","pith_inferences":["The δ-dependence δ^{-2ε_k} weakens as k grows, suggesting that in the conjectured limit the coarse count is governed by a fixed power of δ independent of the spectral window; a sharp version would make the δ and λ factors nearly independent.","The same barcode-plus-approximation strategy could be applied to solutions of other hypoelliptic equations or to functions with merely measurable anisotropic regularity, yielding analogue bounds without spectral structure.","The duality between degree 0 and degree n−1 is a free corollary of persistence duality; extending to intermediate degrees will require controlling the topology of intersections of the covering balls, a problem of independent interest in quantitative topology.","If the equiregular polynomial approximation is ever established, the rest of the programme is largely a technical extension, meaning the conjecture's hardest part is analytic (approximation theory), not topological."],"forward_implications":["On nilmanifolds, the coarse Courant count of δ-deep nodal domains for L^2-normalized combinations of eigenfunctions with eigenvalues ≤λ is at most C δ^{-2ε_k}(λ+1)^{Q/2(1+ε_k)}, with ε_k arbitrarily small for large k.","The same proof yields the coarse Bézout count of intersections of l nodal sets, with the same growth.","Both bounds extend from sub-Laplacians to any maximally hypoelliptic operator of order r>Q/2, with exponent 2Q/(2r−Q).","On general closed equiregular sub-Riemannian manifolds, the combination of the Riemannian result with subelliptic estimates gives a bound with exponent sn/2 instead of Q/2, confirming the need for genuinely sub-Riemannian methods.","The programme identifies three concrete gaps—an equiregular polynomial approximation, a multiscale subdivision, and covers with controlled intersections—each of which would carry the result from nilmanifolds to the general case."],"supporting_citations":[{"why":"Supplies the persistence framework: subadditivity, stability, duality, and the Riemannian Sobolev barcode estimate that the present results generalize.","marker":"[14]"},{"why":"Supplies the Morrey–Sobolev polynomial approximation theorem on Carnot balls, the key new input for the covering argument.","marker":"[38]"},{"why":"Simultaneous approximation variant of the same theorem, also used for the polynomial approximation step.","marker":"[39]"},{"why":"Establishes the anisotropic Sobolev theory and subelliptic estimates for bracket-generating vector fields, used to connect barcodes to eigenfunctions.","marker":"[49]"},{"why":"Provides the subelliptic coercivity estimate for powers of the sub-Laplacian (Theorem 4.17), the bridge to λ-growth.","marker":"[55]"},{"why":"Supplies the Vitali–Wiener covering lemma controlling the number of Carnot balls in the covering argument.","marker":"[25]"},{"why":"Structure theorem for q-tame persistence modules, justifying the barcode decomposition used throughout.","marker":"[50]"},{"why":"Finiteness of the bar-counting invariant for moderate modules, used in Lemma 2.14.","marker":"[4]"}],"fun_headline_variants":["Coarse nodal bounds near optimal for sub-Riemannian eigenfunctions","Sub-Riemannian Courant and Bézout bounds near optimal","Coarse Courant-Bézout bounds for eigenfunction sums on nilmanifolds","Near-optimal coarse Courant bounds in sub-Riemannian geometry","Coarse nodal count bounds near-optimal on Carnot nilmanifolds"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"Everything rests on the Morrey–Sobolev approximation theorem (Theorem 5.3): on every Carnot ball, a smooth function can be uniformly approximated by a stratified polynomial of controlled degree with error proportional to the ball radius raised to k−Q/p; if this approximation cannot be made uniform over the covering balls, the barcode estimate and the nodal bounds collapse.","fun_headline_variants_meta":{"raw":{"variants":["Coarse nodal bounds near optimal for sub-Riemannian eigenfunctions","Sub-Riemannian Courant and Bézout bounds near optimal","Coarse Courant-Bézout bounds for eigenfunction sums on nilmanifolds","Near-optimal coarse Courant bounds in sub-Riemannian geometry","Coarse nodal count bounds near-optimal on Carnot nilmanifolds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001713,"raw_usage":{"total_tokens":6594,"prompt_tokens":703,"completion_tokens":5891,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":447,"completion_tokens_details":{"reasoning_tokens":5794}},"tokens_in":447,"tokens_out":5891,"duration_ms":39085,"temperature":1.0,"reasoning_tokens":5794,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T00:10:25.215786+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On the Heisenberg nilmanifold, take a fixed smooth F (e.g., the projection of sin(2πx)cos(2πy) with $L^{2}$ normalization) and compute the degree-0 bar count N_{0,δ}(|F|) as δ→0. If it grows faster than $δ^{{-Qp/(kp−Q)}}$ for some kp>Q while the anisotropic $W^{{k,p}}$_X norm stays bounded, Theorem 1.5's exponent is wrong. Similarly, a family of eigenfunction combinations with m_{0,δ} exceeding the $δ^{{-2ε_k}}$(λ+1)^{Q/2(1+ε_k)} growth would refute Theorem 1.1.","supporting_citations":[{"cited_title":"Buhovsky, J","cited_arxiv_id":null,"evidence_quote":"Supplies the persistence framework: subadditivity, stability, duality, and the Riemannian Sobolev barcode estimate that the present results generalize."},{"cited_title":"Levitin, D","cited_arxiv_id":null,"evidence_quote":"Supplies the Morrey–Sobolev polynomial approximation theorem on Carnot balls, the key new input for the covering argument."},{"cited_title":"Lu and R","cited_arxiv_id":null,"evidence_quote":"Simultaneous approximation variant of the same theorem, also used for the polynomial approximation step."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the anisotropic Sobolev theory and subelliptic estimates for bracket-generating vector fields, used to connect barcodes to eigenfunctions."},{"cited_title":"Stojisavljevi´ c,Harmonic functions with highly intersecting zero sets","cited_arxiv_id":null,"evidence_quote":"Provides the subelliptic coercivity estimate for powers of the sub-Laplacian (Theorem 4.17), the bridge to λ-growth."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Vitali–Wiener covering lemma controlling the number of Carnot balls in the covering argument."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Structure theorem for q-tame persistence modules, justifying the barcode decomposition used throughout."},{"cited_title":"Bauer, A","cited_arxiv_id":null,"evidence_quote":"Finiteness of the bar-counting invariant for moderate modules, used in Lemma 2.14."}],"review_version":1}