{"id":"abd5a514-65de-4beb-b889-c1a464b2432c","arxiv_id":"2608.01673","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Harmonic functions on Z^d with u(0) nonzero have at least (10^(-10)/d) n^2 nonzero values in each n-cube, and this n^2 order is sharp in dimension three.","lead":"This paper proves that any nonzero harmonic function on Z^3 has at least c·n^2 nonzero lattice points in every n-box, giving the sharp growth rate for sparse supports in three dimensions. It also establishes sharp Zariski-dimension bounds for supports of lattice eigenfunctions in all dimensions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2.6 rests on a false Macaulay-successor identity; the promised consecutive plateau need not exist, so the curve-carrier lemma and the quadratic support bound are not established.","rationale":"The reader correctly identifies Lemma 2.6 as the load-bearing premise. My reading isolates a concrete algebraic error inside that lemma: the proof manufactures the needed consecutive plateau h_s = h_{s+1} through a false Macaulay-successor identity. The main theorem is not disproved, but the current proof has a false inequality at the exact point where the plateau is produced, so the Bigatti--Geramita--Migliore theorem is not applicable as written. Since the paper provides no machine-checked proof and no independent verification, the conditional verdict is appropriate; the condition should now be specifically that the plateau step in Lemma 2.6 be repaired or replaced. If it cannot be repaired, the central quadratic lower bound lacks support.","tokens_in":29690,"tokens_out":18128,"duration_ms":177032,"concrete_test":"Compute the standard Macaulay successor of 1 at degree 5 according to [20, Section 2]; if it is 6 rather than 1, then the identity used for (A.3) is false. To test whether the gap is repairable, run a search in Macaulay2 for reduced point sets Z in P^d whose first-difference h-vector satisfies h_j <= 160m for j in [D/3, 2D/3] and H_Z(D-1) <= 20Dm, with no two consecutive equal h-values; an explicit example would show that the plateau argument in Lemma 2.6 cannot be obtained from the stated hypotheses and that the proof must be revised before the quadratic bound can be accepted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Lemma 2.6 (Appendix A.1), the transition (A.3), h_{j+1} <= h_j for j >= j_0, is derived from the assertion that for an integer a with 0 <= a <= j, its j-th Macaulay successor is a^{<j>} = a. This is not the standard Macaulay successor used in [20, Section 2]. For example, a = 1 and j = 5: the unique Macaulay expansion is 1 = binom(5,5), so 1^{<5>} = binom(6,5) = 6, not 1. In general a^{<j>} can be much larger than a even when a <= j. Thus Macaulay's growth inequality h_{j+1} <= h_j^{<j>} does not imply h_{j+1} <= h_j. A positive integer sequence bounded by 160m over more than 160m steps can alternate 1,2,1,2,... and have no consecutive equal terms; the pigeonhole sentence following (A.3), concerning 'at most 160m-1 strict decreases,' is therefore invalid because strict increases are not bounded. Consequently the construction of s with h_s = h_{s+1} = e is unsupported, the Bigatti--Geramita--Migliore plateau theorem (Theorem A.1) cannot be invoked, and the containment of Z in a reduced curve of degree at most 160m does not follow. Lemma 2.6 is exactly the load-bearing curve-carrier step used through the shell estimate and Proposition 3.5 to prove the quadratic lower bound.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the minimal local support size of nonzero real harmonic functions on Z^d, and more generally of eigenfunctions of the discrete adjacency operator. Its main theorem (Theorem 1.1) asserts that for every d ≥ 3 every real harmonic function with u(0) ≠ 0 satisfies |supp(u) ∩ Q_n^{(d)}| ≥ (10^{-10}/d) n^2, that the order n^2 is sharp in dimension three, and that for d ≥ 17 a superquadratic exponent can be obtained from support-only assumptions. The proof combines moment identities and Hilbert-function bounds to force sparse supports onto low-degree algebraic curves, a thin-shell selection, and two combinatorial estimates for supportive sets. The paper also proves sharp Zariski-dimension bounds for full supports in Theorem 1.3 and gives explicit sparse product constructions.","tokens_in":30031,"tokens_out":8751,"duration_ms":83073,"significance":"If the proof is correct, the paper settles the three-dimensional support-growth problem with the sharp exponent n^2, removes the logarithmic loss in the exact-support estimate of Li and Zhang, and improves Krymskii's support-dimension exponents at all finite scales. The explicit constructions, the detailed auxiliary lemmas, and the clean Zariski-dimension theorem are genuine strengths. However, the central quadratic bound rests on Lemma 2.6, whose proof in Appendix A.1 contains an invalid Macaulay-successor identity; without a correct proof of that lemma the main theorem and Corollary 1.2 are not established.","major_comments":[{"comment":"The proof of Lemma 2.6 asserts that for an integer a with 0 ≤ a ≤ j, its j-th Macaulay successor satisfies a^{<j>} = a, and uses this to pass from Macaulay's inequality h_{j+1} ≤ h_j^{<j>} to the monotonicity h_{j+1} ≤ h_j. This identity is false: for j = 5 and a = 1, the unique Macaulay expansion is 1 = binom(5,5), so 1^{<5>} = binom(6,5) = 6, not 1. Consequently the monotonicity (A.3) is unsupported, and the subsequent pigeonhole argument producing s with h_s = h_{s+1} = e is invalid; a positive integer sequence bounded by 160m over more than 160m steps can alternate 1,2,1,2,... and have no consecutive equal terms. The invocation of the Bigatti–Geramita–Migliore plateau theorem (Theorem A.1) therefore does not follow, and Lemma 2.6 is not proved. Since Lemma 2.6 is the load-bearing curve-carrier step used in §3.2 to reach Proposition 3.5 and the quadratic lower bound, Theorem 1.1 and Corollary 1.2 are not established by the present proof. The authors must either correct this argument or supply a genuinely different proof of the plateau and of Lemma 2.6.","section":"Appendix A.1, Eq. (A.3)"}],"minor_comments":[{"comment":"The running title contains typographical artifacts: 'SP ARSE SUPPOR TS' should read 'SPARSE SUPPORTS', and 'LA TTICE' should read 'LATTICE'.","section":"Title page and headers"},{"comment":"The caption contains the typo 'fromed' instead of 'formed'.","section":"Figure 4 caption"},{"comment":"The sentence 'the construction in Proposition 2.1 have the same order' should read 'the construction in Proposition 2.1 has the same order'.","section":"Section 2.1, text after Proposition 2.1"},{"comment":"The phrase 'its root against the value d is larger than two exactly when d > 16' is correct but would be clearer as 'exactly when d ≥ 17'.","section":"Section 4.1, after Eq. (4.3)"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is not established because the curve-carrier lemma in Appendix A.1 rests on a false Macaulay-successor identity. The support-only arguments in Section 4 and the Zariski-dimension theorem in Section 5 appear to be independent of that lemma, and I did not find comparable defects in them. My recommendation assumes the authors can repair or replace Lemma 2.6; given that the appendix was produced with generative-AI assistance, a fully independent verification of the algebraic-geometric arguments would be appropriate before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick read: the paper is worth a referee, but the proof of its headline theorem is not sound. The curve-carrier lemma used to prove the quadratic lower bound contains a concrete false step, so Theorem 1.1's quadratic claim is unproved as written.\n\nWhat's good. The paper has real content beyond the main theorem. The sparse product constructions are explicit and give exact support counts. The moment identities and the positive-definiteness argument for independence are clean. The support-only results (Theorems 4.5 and 4.8) appear to stand on their own and improve Krymskii's bound; I did not find an analogous error there. The Zariski-dimension theorem has a separate proof, again seemingly independent of Lemma 2.6. The AI disclosure is unusually honest and does not by itself bother me; no formal proof assistant output is supplied, but that is not a defect per se.\n\nThe soft spot is real. Lemma 2.6 is the load-bearing step, and Appendix A.1 proves it by invoking the BGM plateau theorem. To reach a plateau, the proof claims that for 0≤a≤j the jth Macaulay successor satisfies a^{<j>}=a, and therefore h_{j+1}≤h_j^{<j>} gives h_{j+1}≤h_j. That identity is false: for a=1, j=5, the Macaulay expansion is 1=binom(5,5), so 1^{<5>}=binom(6,5)=6. Since h_j can jump up, the pigeonhole argument that finds consecutive equal values h_s=h_{s+1} is invalid. BGM cannot be applied, and the containment of the support in a curve of controlled degree is not established. Consequently the quadratic bound for all d≥3, including the sharp 3D result and the removal of the logarithmic loss in Li-Zhang, is not proved by this manuscript.\n\nI want to be clear about proportion: this is not a minor typo. It is the central mechanism of Section 3. But it is also confined to one lemma; the supportive-set packing and Zariski-dimension portions do not seem to use this faulty step, and they may well be correct. The paper should be sent to a referee who knows Hilbert functions and Cayley-Bacharach theory, because there is enough serious mathematics here that the quadratic argument might be repairable and the other results deserve scrutiny. I would not cite the quadratic theorem in its current form, and I would tell the authors to fix the successor identity before resubmission.","headline":"A broad, ambitious paper with real new ideas and several likely sound results, but the proof of the main quadratic theorem has a concrete error in Lemma 2.6 and the headline result is not established.","tokens_in":30551,"tokens_out":4959,"would_cite":false,"duration_ms":47585,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42B37","39A14","39A22","14N05"],"pacs":[],"model":"deepseek-v4-flash","headline":"A nonzero real harmonic function on $\\mathbb{Z}^d$ must have at least $c_d n^2$ supported points in every $n$-cube.","keywords":["discrete harmonic functions","lattice eigenfunctions","support growth","quantitative unique continuation","Zariski dimension","Cayley-Bacharach property","Hilbert function","multiscale packing"],"falsifier":"Exhibit, for some $d\\ge 3$, a harmonic function $u$ with $u(0)\\ne 0$ and a sequence $n_k\\to\\infty$ such that $|\\mathrm{supp}(u)\\cap Q_{n_k}^{(d)}|\\le C n_k^{2-\\varepsilon}$; the theorem asserts no such function exists. A more targeted check is to construct a finite $Z\\subset\\mathbb{C}^d$ satisfying $CB(t)$ and the Hilbert-rank bound but lying on no curve of degree at most $160m$, which would falsify Lemma 2.6 directly.","tokens_in":29506,"feed_emoji":"📐","tokens_out":11228,"duration_ms":85789,"temperature":0.7,"pith_summary":"The paper proves that real harmonic functions on the integer lattice $\\mathbb{Z}^d$ cannot be sparse: for every $d\\ge 3$, any harmonic $u$ with $u(0)\\ne 0$ has at least $(10^{-10}/d)n^2$ supported points in the cube $\\{-n,\\ldots,n\\}^d$ for every $n\\ge 1$. In dimension three this quadratic growth is sharp, settling the finite-scale support-growth problem there and removing a logarithmic loss from the previous best three-dimensional bound. In high dimensions the paper's support-only arguments push the exponent above $2$ for $d\\ge 17$ and within $4+o(1)$ of $\\log_2 d$. The same framework yields sharp lower bounds on the Zariski dimension of the full support of a lattice eigenfunction. The governing idea is algebraic rigidity: a local support that is too thin must lie on a low-degree algebraic curve, and the directional structure of the lattice equation rules such curves out.","feed_headline":"Lattice harmonics can't be sparse: support grows ~n^2","feed_subtitle":"Sharp in dimension 3, the new bound removes a logarithmic gap and pushes high-dimensional growth past n^2.","key_machinery":"The argument is carried by three interlocking mechanisms. First, the constant-coefficient operator $P=A_d-2d$ yields moment identities: any translate $v$ of $u$ satisfies $P^{m+1}(qv)=0$ when $\\deg q\\le m$, so on each local support $E(a,R)$ there is a linear dependence with nonzero coefficients, giving the Cayley–Bacharach property $CB(R-1)$ and, from independent translates, a Hilbert-rank bound $H_E(D-1)\\le |E|-|F|$, where $H_E(s)$ is the rank of polynomial evaluations of degree at most $s$ on $E$. Second, a curve-carrier lemma converts these two facts into geometry: a finite set satisfying $CB(t)$ with Hilbert rank at most $20Dm$ lies on a reduced algebraic curve of degree at most $160m$, via the extremal Hilbert-function plateau theorem and elimination of isolated points. Third, a directional port rule — evaluating the equation at a zero neighbor forces a supported point among the displacements $e_i,2e_i,e_i\\pm e_j$ — contradicts the existence of such a curve through Bézout intersection bounds and a component-cycle dimension argument. A separate support-only strand uses multiscale packing recurrences and endpoint-collision counts for long port words to deliver the superquadratic exponents in high dimensions.","core_discovery":"The central claim is a quantitative rigidity theorem for lattice eigenfunctions. For every $d\\ge 3$, the counting function $m_d(n)$ — the minimum number of support points a nonzero real harmonic function with $u(0)\\ne 0$ must have in the cube $Q_n^{(d)}$ — satisfies $m_d(n)\\ge 10^{-10}n^2/d$ for all $n\\ge 1$, and in dimension three this order is optimal: the explicit product of an alternating-line function with an exponential factor is harmonic and has exactly $(2n+1)^2$ support points in $Q_n^{(3)}$. The same theorem gives $m_d(n)\\gtrsim n^{\\Theta_d}$ with $\\Theta_d>2$ for $d\\ge 17$ and $\\Theta_d\\ge \\log_2 d-4-O(\\log d/d)$ as $d\\to\\infty$, while a sparse product construction caps every $m_d(n)$ by $(2n+1)^{\\lfloor d/2\\rfloor+1}$. Separately, the Zariski closure of the support of any nonzero eigenfunction has dimension at least $\\lceil d/2\\rceil$, and at least $\\lfloor d/2\\rfloor+1$ when the eigenvalue is nonzero; both bounds are attained by the constructed products. The paper also proves support-only growth estimates that hold for every supportive set, at every center and every radius.","pith_inferences":["The curve-carrier mechanism is not obviously confined to the zero-potential equation: a similar degree-lowering identity for bounded-potential operators would let the same curve contradiction remove the logarithmic loss for variable potentials, a case the paper explicitly does not treat.","The sharp Zariski bound plus the gap between $\\Theta_d$ and $\\lfloor d/2\\rfloor+1$ suggests that closing Conjecture 1.4 needs a quantitative version of the component-cycle argument, since the paper itself notes Zariski density carries no uniform finite-scale information.","The multiscale packing and endpoint-collision exponents are natural targets for improvement; the upper constructions point toward $\\lfloor d/2\\rfloor+1$ as the true finite-scale exponent in every dimension, as the paper conjectures."],"forward_implications":["In dimension three the support-growth problem is closed: $m_3(n)\\asymp n^2$, with the logarithmic loss of the previous estimate removed and the quadratic order shown sharp.","Every dimension $d\\ge 3$ gets a uniform quadratic floor for exact support in finite cubes; for $d\\ge 17$ the bound is superquadratic, with exponent $\\Theta_d\\ge \\log_2 d-4-o(1)$.","The support-only estimates apply to every supportive set, hence to supports of discrete Schrödinger solutions over any field, giving $|X\\cap Q_N(x)|\\ge c_{d,k}N^{\\alpha_{d,k}}$ at every center and every radius.","The Zariski dimension of the support closure of a nonzero eigenfunction is at least $\\lceil d/2\\rceil$ (and $\\lfloor d/2\\rfloor+1$ for nonzero eigenvalue), and both bounds are attained by explicit lattice products."],"supporting_citations":[{"why":"Supplies the two-dimensional quadratic support estimate for the base case and the alternating-line seed used in the sparse product constructions.","marker":"[3]"},{"why":"Provides the three-dimensional support-count bound with a logarithmic loss that the paper removes in the zero-potential case.","marker":"[18]"},{"why":"Introduces supportive sets and the discrete-dimension framework, and gives the endpoint-word construction that Section 4 refines at finite scales.","marker":"[14]"},{"why":"Supplies the extremal Hilbert-function plateau theorem that carries the curve-carrier lemma.","marker":"[2]"},{"why":"Provides the plateau-theorem formulation and Hilbert-function background used in the curve-carrier proof.","marker":"[20]"},{"why":"Supplies the Laurent-polynomial injectivity criterion used in the even-dimensional equality case of the Zariski-dimension theorem.","marker":"[21]"},{"why":"Supplies the classical zero-divisor result underlying the injectivity lemma.","marker":"[19]"}],"fun_headline_variants":["Sparse lattice eigenfunctions? No: support must be ~n^2","Sharp n^2 support bound for harmonic functions on lattices","For d≥17, lattice harmonic support grows > n^2","Zariski closure of eigenfunction support has dimension ≥ d/2"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the curve-carrier lemma: a finite set of lattice points that satisfies the polynomial-interpolation condition inherited from the harmonic equation and has small Hilbert rank must lie on a reduced algebraic curve of degree at most $160m$; if that geometric reduction fails, the quadratic lower bound does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Sparse lattice eigenfunctions? No: support must be ~n^2","Sharp n^2 support bound for harmonic functions on lattices","For d≥17, lattice harmonic support grows > n^2","Zariski closure of eigenfunction support has dimension ≥ d/2"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001577,"raw_usage":{"total_tokens":6344,"prompt_tokens":1049,"completion_tokens":5295,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":665,"completion_tokens_details":{"reasoning_tokens":5218}},"tokens_in":665,"tokens_out":5295,"duration_ms":35969,"temperature":1.0,"reasoning_tokens":5218,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:06:49.938458+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit, for some $d\\ge 3$, a harmonic function $u$ with $u(0)\\ne 0$ and a sequence $n_k\\to\\infty$ such that $|\\mathrm{supp}(u)\\cap Q_{n_k}^{(d)}|\\le C n_k^{2-\\varepsilon}$; the theorem asserts no such function exists. A more targeted check is to construct a finite $Z\\subset\\mathbb{C}^d$ satisfying $CB(t)$ and the Hilbert-rank bound but lying on no curve of degree at most $160m$, which would falsify Lemma 2.6 directly.","supporting_citations":[{"cited_title":"Buhovsky, A","cited_arxiv_id":null,"evidence_quote":"Supplies the two-dimensional quadratic support estimate for the base case and the alternating-line seed used in the sparse product constructions."},{"cited_title":"Li and L","cited_arxiv_id":null,"evidence_quote":"Provides the three-dimensional support-count bound with a logarithmic loss that the paper removes in the zero-potential case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the extremal Hilbert-function plateau theorem that carries the curve-carrier lemma."},{"cited_title":"The geometry of Hilbert functions","cited_arxiv_id":"math/0502145","evidence_quote":"Provides the plateau-theorem formulation and Hilbert-function background used in the curve-carrier proof."},{"cited_title":"Nasehpour, On zero-divisors of semimodules and semialgebras , Georgian Math","cited_arxiv_id":null,"evidence_quote":"Supplies the Laurent-polynomial injectivity criterion used in the even-dimensional equality case of the Zariski-dimension theorem."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the classical zero-divisor result underlying the injectivity lemma."}],"review_version":2}