{"id":"7f2b0877-82a6-49cb-875b-44a2a9745203","arxiv_id":"2506.18113","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every fixed d, there exists a subset of [n]^d of size n - o(n) with no d+2 points lying on a common sphere or hyperplane.","lead":"This paper constructs subsets of the d-dimensional grid of size n - o(n) containing no d+2 points on any sphere or hyperplane, improving all known lower bounds. The construction uses rational curves over finite fields, motivated by projective algebraic geometry.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Euclidean-to-F_p reduction in Theorem 2 is asserted and false as stated; degenerate spheres reduce to hyperplanes, so the paper needs an explicit case analysis, though the claim is salvageable.","rationale":"The central theorem, Theorem 2, appears correct, but the proof contains a genuine gap in the reduction from Euclidean spheres to F_p. The reader identified this as fragile, and the paper's sentence is literally false: a Euclidean sphere whose primitive integer equation has leading coefficient divisible by p reduces to an F_p hyperplane, not an F_p sphere. The gap is patchable because the construction also forbids d+2 coplanar points in F_p, and the remaining degenerate subcase cannot contain grid points. I do not see a flaw in the algebraic engine needed for Theorem 2: the first three bullets of Theorem 4 (identity, degree, independence) are valid, and the flawed vanish step affects only Theorem 3, as does the false reciprocal-sum identity in its proof. The reader's conditional verdict therefore remains appropriate: the paper needs revision, but the main asymptotic result is likely salvageable with an explicit reduction lemma and correction of the auxiliary claims. My agreement with the reader is partial because the reader's weakest assumption named the polynomial existence broadly, while the load-bearing issue for Theorem 2 is specifically the unproved and literally false transfer step, not the vanish property used only in Theorem 3.","tokens_in":7237,"tokens_out":37748,"duration_ms":373157,"concrete_test":"Derive the missing reduction case explicitly: let A(x_1^2+...+x_d^2)+B_1x_1+...+B_dx_d+C=0 be a primitive integer equation of a sphere containing d+2 grid points, with p dividing A. Show that the residual equation B_1x_1+...+B_dx_d+C=0 is a nontrivial F_p hyperplane and that the plane-incidence bound (deg P <= d+1, P nonzero) in the proof of Theorem 2 rules out d+2 points of the constructed set on it. If this case analysis succeeds, the gap is closed; if a counterexample appears for some d and p, Theorem 2 is in doubt.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing step in the proof of Theorem 2 is the transfer from F_p to the Euclidean grid. The paper asserts: 'Since every plane (resp. sphere) in [n]^d is a plane (resp. sphere) in F_p^d' (proof of Theorem 2, Section 3). This is false as stated. After clearing denominators, a Euclidean sphere has a primitive integer equation A(x_1^2+...+x_d^2)+B_1x_1+...+B_dx_d+C=0. If p does not divide A, dividing by A gives an F_p sphere of the form used in the proof. If p divides A, the reduction is a hyperplane, not a sphere; if all B_i are also 0 modulo p, then p cannot divide C for a primitive equation, so no grid point satisfies the reduced equation and the original sphere contains no grid points. Thus the conclusion is recoverable: the degenerate case falls under the already-proven no-(d+2)-coplanar property in F_p. But this case analysis is missing, so the central theorem is not fully proved as written. This is a more direct gap for Theorem 2 than the vanish failure in Theorem 4, which affects only Theorem 3.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the extremal function ex([n]^d; d+2), the largest subset of the d-dimensional grid with no d+2 points on a hyperplane or sphere. The main result, Theorem 2, claims ex([n]^d; d+2) >= n - o(n) for every d >= 2, which would asymptotically resolve the Brass-Moser-Pach problem and confirm the Suk-White conjecture in a strong form. The proof constructs an explicit rational curve over F_p whose image avoids d+2 coplanar and cospherical points, transfers it to the integer grid by a random translate, and uses a polynomial identity f_1^2+...+f_d^2=gh to control sphere and plane incidences. A secondary result, Theorem 3, claims ex([n]^d; d+1, d+2) >= n/(d+1)-o(n) using the same curve restricted to a smaller parameter set.","tokens_in":7446,"tokens_out":23797,"duration_ms":245018,"significance":"If the construction is fully correct, Theorem 2 is a striking improvement over the previous lower bounds of Thiele (n/4 for d=2) and Suk-White (Omega(n^{3/(d+1)-o(1)}) for d>=3), and it is tight up to the o(n) term against the trivial upper bound (d+1)n. The algebraic construction is elegant: the use of the identity f_1^2+...+f_d^2=gh is a genuinely new idea for controlling sphere incidences, and the self-intersection argument giving |S|>=p-d-2 is clean. The manuscript is self-contained in its polynomial construction and uses external results only as benchmarks or for prime distribution. The main concerns below are gaps in the written proof rather than in the overall strategy, and they appear repairable.","major_comments":[{"comment":"The sentence 'Since every plane (resp. sphere) in [n]^d is a plane (resp. sphere) in F_p^d' is false for spheres. A Euclidean sphere has a primitive integer equation A(x_1^2+...+x_d^2)+B_1x_1+...+B_dx_d+C=0. If p does not divide A, reducing modulo p gives an F_p sphere of the form used in the proof. But if p divides A, the reduction is a hyperplane (if some B_i or C is nonzero modulo p) or the empty equation (if all coefficients are 0 modulo p, which cannot happen for a primitive equation). In the hyperplane case the already-proved no-(d+2)-coplanar property of S in F_p applies, and in the empty case there are no grid points at all. This case analysis is missing, so the transfer step is not justified as written; the claim is salvageable exactly as described, but it is load-bearing for Theorem 2.","section":"Section 3, proof of Theorem 2"},{"comment":"The proof asserts that after the replacements f_i -> f_i+mu_i g and f_1 -> f_1+(mu_1+nu t)g, 'identity, degree, and independence are left intact' for the later chosen nu. This is not automatic. Since g is monic of degree d-1 and f_1 has leading coefficient 1, the coefficient of t^d in f_1+(mu_1+nu t)g is 1+nu, so the value nu=-1 would drop the degree of f_1. More generally, if t g lies in the span of f_1,...,f_d,g, the linear transformation from (f_1,...,f_d,g) to the new polynomials has determinant 1+nu c_1 for some c_1, so it can become singular for a specific nu. The proof must show that the unique nu making [t]h tilde =0 avoids these exceptional values, or must prove independence of the final family by a separate argument. Because the incidence proofs in Theorems 2 and 3 use the degree and independence bullets of Theorem 4, this gap is load-bearing.","section":"Section 2, paragraph after Eq. (1)"},{"comment":"The Vieta step is not justified. The proof says that if P has d+1 distinct zeros t_1,...,t_{d+1} in the domain, then t_1^{-1}+...+t_{d+1}^{-1}=0 in F_p, and 'This implies that such zeros cannot live in the domain of e-gamma simultaneously.' As a modular statement this is false: for d=2 and p=13 the domain is {2,3,4,5}, and 2^{-1}+3^{-1}+4^{-1}=7+9+10=26=0 mod 13. Thus the argument needs an additional ingredient, for example a property specific to the span of f_1,...,f_d,h, or a different choice of parameter set, to rule out d+1 zeros. This gap affects Theorem 3 rather than Theorem 2, but Theorem 3 is a stated result of the paper.","section":"Section 3, proof of Theorem 3"}],"minor_comments":[{"comment":"In the sentence 'If P has d+1 distinct zeros t_1,...,t_{d+1} in {1,2,...,n}', the symbol n is undefined and appears to be a typo; the zeros should range over the domain of e-gamma, which is a subset of F_p.","section":"Section 3, proof of Theorem 3"},{"comment":"The phrase 'the image of gamma' in the proof of Theorem 3 should refer to the image of e-gamma; as printed it repeats the notation from Theorem 2.","section":"Section 3, proof of Theorem 3"},{"comment":"In Claim 6, the displayed equation 'c_1a_{1,j}+...+c_da_{d,j} holds' is missing '=0'; the intended conclusion is that the row vectors of A are linearly dependent.","section":"Section 2, proof of Claim 6"},{"comment":"There is a typo 'Since h is is of higher degree' with a duplicated 'is'; also in the self-intersection paragraph of the proof of Theorem 2 the polynomial R is written as 'rho_1f_1+...+rho_df_d+rho f' but should be 'rho h'.","section":"Section 2, after the proof of Theorem 4"}],"recommendation":"major_revision","confidential_remarks":"The central construction is promising and the main theorem is likely correct, but the written proof has two load-bearing gaps: the Euclidean-to-F_p transfer for spheres and the unproved preservation of degree/independence in the vanish tweak. The Vieta issue in Theorem 3 is also a real gap, though it does not affect the main theorem. I would encourage the authors to add the missing case analysis and to make the algebraic tweak step fully rigorous before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main result is real: ex([n]^d; d+2) ≥ n - o(n) for every d ≥ 2. This is a major jump from Thiele's n/4 in the plane and from the Suk-White polynomial bound, and it asymptotically settles the Brass-Moser-Pach problem and confirms the Suk-White conjecture in a strong sense. The construction — a rational curve over F_p engineered to pass through many infinite points — is genuinely new, and the algebra in Section 2 is the right core. I verified the identity, degree, and independence properties; they hold as stated.\n\nThe write-up has three soft spots, none fatal to the main theorem. First, the transfer from F_p to the Euclidean grid in the proof of Theorem 2 asserts without proof that every Euclidean sphere is a sphere over F_p. That is false as written: after clearing denominators, if p divides the quadratic coefficient the reduction is a hyperplane, and if all linear terms vanish as well the sphere contains no grid points. The argument is salvageable because the no-d+2-coplanar property covers the degenerate hyperplane case, but the proof needs an explicit case split. Minor gap.\n\nSecond, the 'vanish' step in Theorem 4 fails for d=2. Claim 8 is false for the 2x2 matrix: both g1 and g2 lie in the span of f1 and f2, so the proposed choice g = g2 with g(0)=0 is unavailable. This affects only Theorem 3, not Theorem 2. The d≥3 case looks fine.\n\nThird, the proof of Theorem 3 has typographical issues ('[x]P' should be '[t]P', and the domain is written inconsistently). For the record, the reciprocal-sum step is actually correct: a zero linear coefficient forces the sum of reciprocals of the nonzero roots to vanish by Vieta, and for positive t_i in the domain that is a contradiction. I disagree with the reader who called this step false.\n\nVerdict: the paper deserves a serious referee and should be accepted after a modest revision. The main theorem is the news; the reduction case and the d=2 vanish gap need to be addressed, and the typos cleaned up. For an extremal combinatorics audience this is a must-cite result.","headline":"Strong new lower bound for no-d+2-cospherical grid subsets, sound at its core but with a few fixable gaps in the write-up.","tokens_in":7972,"tokens_out":10955,"would_cite":true,"duration_ms":99861,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52C10","05D99","11T06"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper constructs, for every dimension d≥2, a subset of the grid [n]^d of size n−o(n) with no d+2 points on a common sphere or hyperplane.","keywords":["cospherical points","no-four-on-circle","grid subsets","extremal combinatorics","rational curves","finite fields","polynomial method","hyperplane incidences"],"falsifier":"For $d=2$ and $p=13$, compute the curve from the paper's matrix with a square root $\\alpha$ of $-1$, list the $10$ image points in $F_{13}^2$, and test every line and every sphere equation for four of those points; a single hit would refute the construction in that case, and since the theorem quantifies over all $d$, any such hit would disprove the claimed $n-o(n)$ bound.","tokens_in":7039,"feed_emoji":"⚪","tokens_out":14158,"duration_ms":136091,"temperature":0.7,"pith_summary":"This paper establishes a lower bound of $n-o(n)$ for the largest subset of the $d$-dimensional grid $[n]^d$ with no $d+2$ points on a common sphere or hyperplane, for every dimension $d\\ge 2$. This resolves, up to the sublinear error term, a generalized no-four-on-a-circle problem that previously had only much smaller bounds: $n/4$ in the plane and a fractional power of $n$ in higher dimensions. Because a simple slicing argument gives an upper bound of $(d+1)n$, the result pins down the growth rate as linear. A companion construction shows that $n/(d+1)-o(n)$ points can be chosen with no $d+1$ points on a hyperplane and no $d+2$ on a sphere.","feed_headline":"A rational curve yields n−o(n) grid points with no d+2 on a sphere","feed_subtitle":"The lower bound is linear in n, matching the trivial upper bound up to a sublinear error.","key_machinery":"The load-bearing object is the polynomial identity $f_1^2+\\dots+f_d^2=gh$ over $F_p$, together with the degree data $(d,\\ldots,d,d-1,d+1)$ and the linear independence of $f_1,\\ldots,f_d,g,h$. This identity is what converts the sphere equation, quadratic in the coordinates, into a linear combination of $g$ and the coordinate functions after substitution along the curve; the result is a polynomial of degree at most $d+1$ whose roots are exactly the curve parameters lying on the sphere. The same substitution for a hyperplane gives a degree-$(d+1)$ polynomial. Projectively, the mechanism is that all spheres share the same points at infinity, and the choice of the $f_i$ makes the curve pass through enough of those points to cancel the quadratic contribution. For the refined coplanar statement, the extra 'nice' condition—zero linear coefficient on $f_1,\\ldots,f_d$ and $h$—lets Vieta's formula forbid $d+1$ simultaneous zeros in the sampled reciprocal domain.","core_discovery":"The central discovery is that a rational curve over a finite field can be engineered so that every sphere and every hyperplane pulls back to a low-degree polynomial, bounding the number of curve points on such a surface. The curve is $\\gamma(t)=(f_1(t)/h(t),\\ldots,f_d(t)/h(t))$ in $F_p^d$, where the polynomials satisfy $f_1^2+\\dots+f_d^2=gh$. Substituting the curve into the equation of a sphere turns the quadratic part into $g$, leaving a polynomial of degree at most $d+1$; the same happens for a hyperplane. Because $f_1,\\ldots,f_d,g,h$ are linearly independent and have degrees $(d,\\ldots,d,d-1,d+1)$, neither pullback can vanish identically, and a nonzero degree-$(d+1)$ polynomial has at most $d+1$ roots. The curve has at most one self-intersection, so its image has $p-d-2$ distinct points, and a random translate of this image inside $[n]^d$, with the prime $p$ chosen just above $n$, yields the $n-o(n)$ subset. A refined condition on the linear coefficients lets the same curve be sampled on reciprocals, giving $n/(d+1)-o(n)$ points with no $d+1$ coplanar points.","pith_inferences":["A natural next step, not taken in the paper, would be to determine the leading constant: the true maximum lies somewhere between $n$ and $(d+1)n$, and the rational-curve construction alone does not say which side is closer.","The same mechanism—a curve whose substitution makes a family of hypersurfaces pull back to low-degree polynomials—should apply to other configurations defined by quadrics that share points at infinity, such as ellipsoids, hyperboloids, or more general algebraic surfaces.","Because the $o(n)$ error comes from the gap between the grid size $n$ and the chosen prime $p$, any improvement in the prime-selection step would immediately improve the error term; the current bound inherits the $n/\\exp(c\\sqrt{\\log n})$ error from primes in arithmetic progressions.","The reciprocal-sampling trick that forbids $d+1$ coplanar points suggests an analogy with no-three-in-line problems, where restricting parameter reciprocals may be a general way to avoid linear configurations; this is not explored in the paper."],"forward_implications":["In the plane, the no-four-on-a-circle lower bound jumps from $n/4$ to $n-o(n)$, bringing it to within a constant factor of the trivial $3n$ upper bound.","For every $d\\ge 3$, the previously conjectured lower bound $n^{d/(d+1)}$ is confirmed in the much stronger linear form, so the extremal number is $\\Theta(n)$ in each fixed dimension.","The generalized grid problem is settled in growth rate: a linear lower bound of $n-o(n)$ matches the slicing upper bound $(d+1)n$ up to a sublinear factor.","The construction works uniformly against both spheres and hyperplanes, so no separate handling of the two forbidden configurations is needed.","The refined sampling on reciprocal parameters gives a positive-density subset with no $d+1$ coplanar points and no $d+2$ cospherical points, indicating the coplanar threshold can be pushed lower at the cost of density."],"supporting_citations":[{"why":"supplies the previous best lower bound and the conjecture that the main theorem confirms in strong form.","marker":"[14]"},{"why":"gives the earlier lower bounds via moment curves and sets the baseline that the rational-curve construction improves.","marker":"[15]"},{"why":"defines the planar no-four-on-circle problem and provides the earlier constant-factor bounds in dimension two.","marker":"[16]"},{"why":"poses the generalized extremal problem in higher dimensions that Theorem 2 asymptotically resolves.","marker":"[3]"},{"why":"provides the prime number theorem for arithmetic progressions used to choose a prime p=n+o(n) with p congruent to 1 modulo 4.","marker":"[12]"}],"fun_headline_variants":["Rational curve gives n−o(n) grid points with no d+2 on a sphere or hyperplane","Grid subset of size n−o(n) keeps any d+2 points off any sphere or hyperplane","Nearly full grid avoids any sphere or hyperplane containing d+2 points","Beats prior bounds: n−o(n) grid points with no d+2 on a sphere or hyperplane"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction collapses unless, for every dimension $d$ and infinitely many suitable primes, one can find the required polynomials over the finite field whose sum of squares factors as a product of two polynomials of the right degrees with the right independence; the paper constructs them explicitly, but it also assumes without proof that real spheres and planes behave the same way inside the finite field once a prime just above $n$ is chosen.","fun_headline_variants_meta":{"raw":{"variants":["Rational curve gives n−o(n) grid points with no d+2 on a sphere or hyperplane","Grid subset of size n−o(n) keeps any d+2 points off any sphere or hyperplane","Nearly full grid avoids any sphere or hyperplane containing d+2 points","Beats prior bounds: n−o(n) grid points with no d+2 on a sphere or hyperplane"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001553,"raw_usage":{"total_tokens":6233,"prompt_tokens":997,"completion_tokens":5236,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":613,"completion_tokens_details":{"reasoning_tokens":5134}},"tokens_in":613,"tokens_out":5236,"duration_ms":33003,"temperature":1.0,"reasoning_tokens":5134,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:01:08.149639+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $d=2$ and $p=13$, compute the curve from the paper's matrix with a square root $\\alpha$ of $-1$, list the $10$ image points in $F_{13}^2$, and test every line and every sphere equation for four of those points; a single hit would refute the construction in that case, and since the theorem quantifies over all $d$, any such hit would disprove the claimed $n-o(n)$ bound.","supporting_citations":[{"cited_title":"Suk and E","cited_arxiv_id":null,"evidence_quote":"supplies the previous best lower bound and the conjecture that the main theorem confirms in strong form."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the earlier lower bounds via moment curves and sets the baseline that the rational-curve construction improves."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"defines the planar no-four-on-circle problem and provides the earlier constant-factor bounds in dimension two."},{"cited_title":"Brass, W","cited_arxiv_id":null,"evidence_quote":"poses the generalized extremal problem in higher dimensions that Theorem 2 asymptotically resolves."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the prime number theorem for arithmetic progressions used to choose a prime p=n+o(n) with p congruent to 1 modulo 4."}],"review_version":2}