{"id":"bdfdde34-90a6-43b4-ae94-bb501f50a4fd","arxiv_id":"2509.06935","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New lower bounds for lattice sets avoiding subspheres and subspaces, including f_circ(n) ≥ 7n/12, via deletion-method counting of cyclic quadrilaterals and cospherical tuples.","lead":"This paper improves lower bounds for lattice points avoiding affine, linear, and spherical degeneracies, in particular showing that the no-four-on-a-circle grid problem has solutions of size at least 7n/12. The main new ingredient is an asymptotic count of cyclic quadrilaterals in an n by n grid, obtained by combining older incidence and number-theoretic bounds with a computer-assisted constant.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4.2's diameter transfer from Huxley-Konyagin is asserted, not proved; if the asymmetric-quadrilateral bound fails, Theorem 1.3's error term and Corollary 1.4's 7n/12 conclusion lose their margin.","rationale":"The reader identified Lemma 4.2 as the weakest assumption, and I agree: the central claim of an improved no-four-on-a-circle bound depends on an O(n^{4.62}) count of asymmetric cyclic quadrilaterals. The manuscript gives only a 'closer inspection' assertion rather than a proof. I considered other potential issues—the Mathematica simplification in Lemma 4.4 and the computer-assisted interval for γ in Lemma 4.5—but these are secondary: even if the constant γ were slightly off, Corollary 1.4 could survive, whereas a failure of Lemma 4.2 by a constant factor would directly destroy the margin in the deletion argument. The paper is internally consistent and the rest of the counting (collinear tuples, spherical bounds) appears coherent. The concern is therefore best addressed as a condition on the proof of Lemma 4.2, not as an indication that the theorem is false. Hence the reader's CONDITIONAL verdict should stand unchanged.","tokens_in":18850,"tokens_out":22042,"duration_ms":237246,"concrete_test":"Obtain Huxley-Konyagin (Acta Arith. 138, 2009) and check whether their bound is proved for polygons whose vertex set has diameter O(R) or for circumradius R. If the proof counts equivalence classes of diameter-bounded quadrilaterals, Lemma 4.2 is immediate. If it genuinely counts circumradius-bounded classes, verify the paper's claim that the proof 'only uses distances' by locating the distance bound in their argument; a circumradius-to-diameter transfer is not automatic because almost-collinear integer points can have diameter n and circumradius ≫n. In the latter case, compute the asymmetric cyclic quadrilateral count in [n]^2 for n up to 40 via an exact enumeration over rational circles, and fit log(count) vs log n; if the fitted exponent is ≥5, Theorem 1.3 is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's main new constant relies on splitting cyclic quadrilaterals into isosceles trapezia (Lemma 4.4, γn^5) plus asymmetric quadrilaterals, whose count is bounded by Lemma 4.2 as O(n^{4+18/29+ε}). This Lemma 4.2 is the least secure step. A bound on equivalence classes with circumradius ≤ R does not automatically imply the same bound for diameter ≤ R: four almost-collinear integer points can have small diameter and arbitrarily large circumradius. The one-sentence assertion that Huxley-Konyagin's proof 'only uses the O(R) bound on the distances between the vertices' is not demonstrated in the manuscript. If the true asymmetric count were c n^5 with c > 0, the deletion-method constant in Corollary 1.4 requires the total hyperedge coefficient to be < 0.53134. The paper's isosceles-trapezia plus collinear coefficients sum to about 0.51983, leaving a margin of only about 0.0115. Thus even a small positive asymmetric main-term coefficient would invalidate the 7n/12 bound. The computer-assisted constants in Lemmas 4.4 and 4.5 are also hard to verify from the manuscript, but the unproved transfer in Lemma 4.2 is the most load-bearing: it is the only analytic bridge between an external theorem and the paper's headline improvement over Thiele.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies extremal problems on subsets of [n]^d avoiding affine, linear, and spherical degeneracies. For affine and linear degeneracies, Theorems 1.1 and 1.2 improve earlier bounds of Sudakov–Tomon and Lefmann by applying Spencer's deletion lemma to counts of rank-deficient matrices obtained from Katznelson's theorem. For spherical degeneracies, Theorem 1.5 and Corollary 1.6 bound S(n,d) and improve on Suk–White via Lund's incidence bound and lattice-point estimates on spheres. The central new result is Theorem 1.3, which asserts that the number of cyclic quadrilaterals in [n]^2 is γ n^5 + O(n^{4+18/29+ε}) with an explicit constant γ, leading to Corollary 1.4: f_circ(n) ≥ 7n/12, improving Thiele's n/4 lower bound. The proof splits cyclic quadrilaterals into isosceles trapezia, counted in Lemma 4.4, and asymmetric quadrilaterals, bounded via Huxley–Konyagin in Lemma 4.2.","tokens_in":19204,"tokens_out":10333,"duration_ms":120398,"significance":"If the main claims hold, this is a substantial contribution: it gives the first asymptotic count of cyclic quadrilaterals in a square lattice with an explicit constant, improves a twenty-year-old lower bound by a concrete factor, and extends the power of the deletion method to new counting problems. The paper is commendably parameter-free: the constant γ is defined by an explicit convergent sum, not fitted to any target conclusion. The higher-dimensional spherical bound and the clean use of external incidence results are also valuable. The main caveats are that the Huxley–Konyagin transfer in Lemma 4.2 is asserted rather than proved, and the computer-assisted summation in Lemma 4.4 is not auditable from the manuscript. These are local, fixable gaps, but they are load-bearing for the headline theorem.","major_comments":[{"comment":"This lemma is the only bound on asymmetric cyclic quadrilaterals and is therefore load-bearing for Theorem 1.3 and Corollary 1.4. The proof is the one-sentence claim that a closer inspection of Huxley–Konyagin's proof yields a diameter version because their proof 'only uses the O(R) bound on the distances between the vertices'. The cited theorem, as described, bounds equivalence classes with circumradius at most R, and a diameter bound does not automatically imply a circumradius bound: four almost-collinear lattice points can have small diameter and arbitrarily large circumradius. If the true asymmetric count has any positive n^5 coefficient, the margin in Corollary 1.4 collapses (c=0.51983 vs the required c<0.53134). Please supply a complete proof of Lemma 4.2 or a precise citation to the exact equations in [26] that establish the diameter version.","section":"§4.2, Lemma 4.2"},{"comment":"The computation of the constant γ hinges on the formula for f(a,b). The manuscript states that after plugging the area formula (13) into the sum, 'the above expression simplifies to f(a,b)·m^5 + ...', with the simplification performed in Mathematica and no derivation shown. A single algebraic error in f(a,b) would change γ and could invalidate the 7n/12 constant in Corollary 1.4. The authors should include the summation in an appendix, or provide the Mathematica code/algebraic steps so the simplification can be verified by a referee or reader.","section":"§4.2, Lemma 4.4"},{"comment":"The definition of S(n,d) in the introduction and in Theorem 1.5 explicitly treats hyperplanes as degenerate spheres. However, the upper-bound proof partitions tuples by their 'spherical span' and appears to handle only ordinary (finite-radius) spheres. The hyperplane contribution is first mentioned after the proof, when deriving Corollary 1.6. As written, the proof of the upper bound in Theorem 1.5 is incomplete unless the hyperplane contribution is separately incorporated into the statement of Theorem 1.5. Please reconcile the definition, the proof, and the later use in Corollary 1.6.","section":"§3.3, Theorem 1.5"}],"minor_comments":[{"comment":"The rigorous numerical bound γ ∈ (0.35974,0.36017) is obtained by computing the partial sum s_5500 'to an accuracy of 10^-5' in Mathematica. For full reproducibility, specify the algorithm (interval arithmetic? rigorous error bounds?) and include the code or output.","section":"§4.2, Lemma 4.5"},{"comment":"The paper switches between ordered tuples and unordered quadrilaterals without always saying which is meant. For instance, Proposition 4.1 counts ordered r-tuples, while Corollary 1.4 speaks of unordered collinear quadruples. Please state the convention explicitly in each statement and check that the factors of r! are consistent.","section":"Proposition 4.1 / Lemma 4.4 / Corollary 1.4"},{"comment":"In the d=2 base case, the proof assumes the ellipse contains at least 5 points and then solves a 5×5 linear system. If the ellipse is degenerate (e.g., a line or a pair of lines), the conclusion is still true, but the proof as written does not handle these cases. This is a minor gap that can be fixed by a sentence.","section":"§3.1, Lemma 3.1"},{"comment":"The notation f_d,k,r(n) and g_d,k,r(n) is reused in the 'In particular' clauses with different meanings (f_d,k(n) and g_d,k(n)); this should be disambiguated, as the subscripts differ but the reader may be confused by the same letters.","section":"§1, Theorem 1.1"}],"recommendation":"major_revision","confidential_remarks":"I am positive about the paper's potential. The central derivation is transparent in structure and the main new count is interesting. The referee report focuses on the unproved Huxley–Konyagin transfer in Lemma 4.2; this is the key obstruction to accepting the paper as is. If the authors can supply a complete proof of Lemma 4.2 (or, failing that, revise the claims accordingly), I would be willing to accept. The Mathematica-assisted constant in Lemma 4.4 needs to be auditable; this is a verifiability issue rather than a suspicion of error. There is also a minor internal consistency issue in Theorem 1.5 regarding hyperplanes, but it seems easily fixed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the take: the paper gives a new lower-bound constant for the Erdős–Purdy no-four-on-a-circle problem (7n/12, up from Thiele's n/4) and extends the range where no-r-in-k-flat sets have Θ(n^{d-k}) size from r > dk to r > d+1. The latter is a clean consequence of Katznelson's matrix-counting theorem plus Spencer's deletion method; the former rests on a new asymptotic count of cyclic quadrilaterals in [n]^2, which is the real meat.\n\nWhat the paper does well: the counting of isosceles trapezia is careful and yields an explicit constant γ, computed rigorously with a Mathematica-assisted partial sum and a tail bound. The spherical counting in higher dimensions is a coherent use of Lund's incidence bound and honestly compares with the independent Dong–Xu result. The exposition is clear and the authors flag their own assumptions.\n\nThe soft spot is Lemma 4.2. The paper needs O(R^{4+18/29+ε}) asymmetric cyclic quadrilaterals in any set A of diameter R, citing Huxley–Konyagin's bound for circumradius ≤ R and asserting 'a closer inspection' shows the proof transfers. That is load-bearing: the margin between the total edge coefficient (collinear + isosceles, ≈0.5198) and the 7/12 threshold (≈0.5313) is thin. If the asymmetric count had a positive n^5 term, the improvement would collapse. The claim may be true—H-K's proof might only use pairwise distances—but the paper doesn't show it, and a referee needs the details. The stress-test's example of almost-collinear points with small diameter and large circumradius is the right thing to press on.\n\nMinor issues: Lemma 4.4's Mathematica simplification is omitted, and Lemma 4.5 is computer-assisted (though tail bounds are given). Those are verifiable in principle and less worrying.\n\nBottom line: this deserves a serious referee. The affine/linear results are solid, the cyclic-quadrilateral count is interesting in its own right, and the 7n/12 bound is a genuine improvement over Thiele if Lemma 4.2 holds. I'd send it to peer review with the explicit request that the gap in Lemma 4.2 be filled.","headline":"New 7n/12 bound for no-four-on-a-circle is a real step, but the proof's key transfer from Huxley–Konyagin is asserted, not demonstrated.","tokens_in":19684,"tokens_out":6972,"would_cite":true,"duration_ms":66317,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05D40","52C10","52C35"],"pacs":[],"model":"deepseek-v4-flash","headline":"For large n, an n×n grid contains at least 7n/12 points with no four on a common circle or line.","keywords":["no-four-on-a-circle","lattice cubes","affine degeneracies","spherical degeneracies","deletion method","cyclic quadrilaterals","integral matrices of fixed rank","point-flat incidences"],"falsifier":"Enumerate all asymmetric cyclic quadrilaterals in [n]^2 (quadruples that are not isosceles trapezia) for n as large as feasible and fit their growth; if their count is Θ(n^5) rather than O(n^{4+18/29+ε}), Lemma 4.2 is false and Theorem 1.3's claimed error term collapses. Independently, evaluating the constant γ by truncating the sum in (12) should place it in (0.35974, 0.36017); any reliable fit outside this interval would refute Corollary 1.4's constant.","tokens_in":18787,"feed_emoji":"⭕","tokens_out":9401,"duration_ms":89686,"temperature":0.7,"pith_summary":"The paper studies how many points one can choose from the integer cube [n]^d while avoiding geometric coincidences: r points in a k-dimensional affine or linear subspace, or d+2 points on a (d−1)-dimensional sphere. Its central new result is an asymptotic count of cyclic quadrilaterals in the square grid [n]^2: there are γ n^5, with γ between 0.35974 and 0.36017, up to an error term of order n^{4+18/29+ε}. Feeding this count into the deletion method gives a randomized construction of at least 7n/12 grid points with no four collinear or concyclic, improving the previous n/4 bound. The same counting-plus-deletion strategy gives new lower bounds for avoiding affine and linear degeneracies when 1<k<d−1, and improves the known bound for avoiding cospherical points in dimensions d≥4. The paper thereby replaces some algebraic constructions by probabilistic ones that reach the same orders of magnitude.","feed_headline":"7n/12 grid points can avoid any four on a circle","feed_subtitle":"An exact cyclic-quadrilateral count breaks the old n/4 record and lifts to higher-dimensional sphere bounds.","key_machinery":"The engine is the deletion method for r-uniform hypergraphs: if a hypergraph has v vertices and e edges, removing vertices from a random subhypergraph of minimum degree yields an independent set of size about v^{r/(r−1)}/e^{1/(r−1)}. Each extremal problem is therefore reduced to counting edges in the right hypergraph. For affine and linear degeneracies, the count is of d×r integer matrices of rank at most k, controlled by an asymptotic theorem on integral matrices of fixed rank. For cyclic quadrilaterals, the count splits into isosceles trapezia—counted exactly via a lattice-point lemma for convex polygons, producing the constant γ—and asymmetric quadrilaterals, bounded by a transferred esti","core_discovery":"On its own terms, the paper establishes that almost all cyclic quadrilaterals in the square lattice are symmetric, namely isosceles trapezia. It counts these explicitly, obtaining γ n^5 + O(n^4 log n) with the constant γ given by a convergent number-theoretic sum, and it combines this with a transferred number-theoretic estimate to show that the remaining asymmetric quadrilaterals are only O(n^{4+18/29+ε}). Summing gives Theorem 1.3: the number of cyclic quadrilaterals in [n]^2 is γ n^5 + O(n^{4+18/29+ε}). After subtracting collinear quadruples, the deletion method yields Corollary 1.4: for large n, one can choose at least 7n/12 points with no four collinear or concyclic. For the higher-dime","pith_inferences":["If the transferred diameter bound for asymmetric cyclic quadrilaterals can be proved in full, the only remaining gap to a fully unconditional Theorem 1.3 is the computer-assisted enclosure of γ; a purely analytic evaluation of the sum defining γ could push the 7n/12 constant higher.","The random-matrix analogy in the paper suggests a route to its Conjecture 5.1: if the dominant singularity events for the (d+2)×(d+2) matrix are zero rows or columns and equal rows or columns, then S(n,d)=O(n^{d^2+d}), nearly matching the lower bound n^{d^2+d−2}.","The linear-size probabilistic no-four-circle set indicates that extremal configurations need not be algebraic; extending a similar random construction to the no-three-in-line problem would speak to whether large no-three-in-line sets must reduce to an algebraic curve modulo some prime.","The same deletion-plus-counting pipeline could be applied to avoiding five or more concyclic points, using the already-derived counts of isosceles trapezia and of collinear r-tuples."],"forward_implications":["No-four-on-a-circle: for large n, f_circ(n) ≥ 7n/12, a constant-factor improvement over the previous n/4 lower bound, obtained by a probabilistic construction rather than an algebraic one.","No cospherical points: for every d≥3, f_sph(n,d) = Ω(n^{min{d,4}/(d+1) − c/log log n}), improving the previous lower bound for d≥4; for d=3,4 this is within a subpolynomial factor of the conjectured n^{d/(d+1)}.","Affine degeneracies: for 1<k<d−1 and r>d+1, f_aff(n,d,k,r) = Θ(n^{d−k}), matching the trivial upper bound and extending the previously known range r>dk.","Linear degeneracies: for k<d and r≥k+1, f_lin(n,d,k,r) is determined up to polylog factors in new regimes, including the case k=d−1 where it recovers the known n^{d/(d−1)} order.","The asymptotic count of cyclic quadrilaterals, γ n^5 with γ≈0.36, settles the order of that quantity and provides a benchmark for any future construction or upper bound."],"supporting_citations":[{"why":"Supplies the Gaussian-integer estimate that asymmetric cyclic quadrilaterals with vertices in Z^2 and bounded circumradius are O(R^{2+18/29+ε}); Lemma 4.2 transfers this to the diameter form used for the error term in Theorem 1.3.","marker":"[26]"},{"why":"Gives the asymptotic count of d×r integral matrices of fixed rank, which Propositions 1.8 and 1.9 use to count r-tuples in affine and linear k-spaces, yielding Theorems 1.1 and 1.2.","marker":"[29]"},{"why":"Provides the incidence bound for rich nondegenerate k-flats applied to the lifted point set, producing the upper bound on S(n,d) in Theorem 1.5.","marker":"[32]"},{"why":"The deletion-method lemma converts every edge count into the lower bounds on independent sets used throughout the paper.","marker":"[37]"},{"why":"The previous lower bound f_circ(n) > n/4 that Corollary 1.4 improves to 7n/12.","marker":"[43]"},{"why":"Gives the asymptotic count of collinear triples and the method extended to collinear r-tuples in Proposition 4.1, needed to subtract collinear quadruples in Corollary 1.4.","marker":"[24]"},{"why":"The previous lower bound for affine degeneracies in the range 1<k<d−1 that Theorem 1.1 improves.","marker":"[30]"},{"why":"Shows f_aff(n,d,k,r)=Θ(n^{d−k}) for r>dk; Theorem 1.1 extends this to r>d+1 and provides the baseline that is improved.","marker":"[38]"}],"fun_headline_variants":["Exact cyclic quad count yields 7n/12 no-four-on-circle bound","Symmetric quadrilaterals dominate: new record for no four on a circle","7n/12 points with no concyclic quadruple: exact count from new bound","Number theory plus incidence geometry break old circle-free bounds","Lattice points: 7n/12 avoid circles, best known bound yet"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The numerical conclusion 7n/12 rests on an unproved transfer: a known bound on asymmetric cyclic quadrilaterals under a bounded-circumradius condition is asserted, by 'a closer inspection', to hold under a bounded-diameter condition, and if the true asymmetric count in [n]^2 were larger than O(n^{4+18/29+ε}), the error term in Theorem 1.3 could exceed the margin needed for the constant 7/12.","fun_headline_variants_meta":{"raw":{"variants":["Exact cyclic quad count yields 7n/12 no-four-on-circle bound","Symmetric quadrilaterals dominate: new record for no four on a circle","7n/12 points with no concyclic quadruple: exact count from new bound","Number theory plus incidence geometry break old circle-free bounds","Lattice points: 7n/12 avoid circles, best known bound yet"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000719,"raw_usage":{"total_tokens":3042,"prompt_tokens":694,"completion_tokens":2348,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":438,"completion_tokens_details":{"reasoning_tokens":2248}},"tokens_in":438,"tokens_out":2348,"duration_ms":17198,"temperature":1.0,"reasoning_tokens":2248,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T22:52:02.875576+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate all asymmetric cyclic quadrilaterals in [n]^2 (quadruples that are not isosceles trapezia) for n as large as feasible and fit their growth; if their count is Θ(n^5) rather than O(n^{4+18/29+ε}), Lemma 4.2 is false and Theorem 1.3's claimed error term collapses. Independently, evaluating the constant γ by truncating the sum in (12) should place it in (0.35974, 0.36017); any reliable fit outside this interval would refute Corollary 1.4's constant.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Gaussian-integer estimate that asymmetric cyclic quadrilaterals with vertices in Z^2 and bounded circumradius are O(R^{2+18/29+ε}); Lemma 4.2 transfers this to the diameter form used for the error term in Theorem 1.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the asymptotic count of d×r integral matrices of fixed rank, which Propositions 1.8 and 1.9 use to count r-tuples in affine and linear k-spaces, yielding Theorems 1.1 and 1.2."},{"cited_title":"Lund, Two theorems on point-flat incidences,Computational Geometry92(2021), Paper No","cited_arxiv_id":null,"evidence_quote":"Provides the incidence bound for rich nondegenerate k-flats applied to the lifted point set, producing the upper bound on S(n,d) in Theorem 1.5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The deletion-method lemma converts every edge count into the lower bounds on independent sets used throughout the paper."},{"cited_title":"Thiele, The no-four-on-circle problem,J","cited_arxiv_id":null,"evidence_quote":"The previous lower bound f_circ(n) > n/4 that Corollary 1.4 improves to 7n/12."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the asymptotic count of collinear triples and the method extended to collinear r-tuples in Proposition 4.1, needed to subtract collinear quadruples in Corollary 1.4."},{"cited_title":"Lefmann, Extensions of the No-Three-In-Line problem, preprint, 2012","cited_arxiv_id":null,"evidence_quote":"The previous lower bound for affine degeneracies in the range 1<k<d−1 that Theorem 1.1 improves."},{"cited_title":"Sudakov and I","cited_arxiv_id":null,"evidence_quote":"Shows f_aff(n,d,k,r)=Θ(n^{d−k}) for r>dk; Theorem 1.1 extends this to r>d+1 and provides the baseline that is improved."}],"review_version":1}