{"id":"aef86471-f13c-4f9a-8d32-56b692d44ede","arxiv_id":"2506.06463","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper improves the trivial count of integer matrices with non-generic characteristic polynomial Galois group from T^{n^2} to T^{n^2-1/2} log T, and gives sharper bounds for special matrix classes.","lead":"This paper counts how many integer grids, or matrices, have a characteristic polynomial whose symmetry group is smaller than the largest possible one. It proves a new upper bound for that count and makes partial progress toward a conjecture about the exact size of the count.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The reader's counterexample to Lemma 6.1 is outside the domain of Theorem 1.6; the needed discriminant property holds for proper primitive groups, but the lemma's statement lacks that hypothesis.","rationale":"The reader correctly identified that Lemma 6.1 as stated is false, but the cited counterexample has Galois group S_3 and thus is not in the counting set of Theorem 1.6. The logical role of the lemma in the proof is to guarantee index at least 2 for every p dividing the field discriminant. That guarantee is true for the matrices actually counted, because their Galois group is a proper primitive subgroup of S_n and therefore contains no transpositions; a prime of discriminant exponent 1 would force a transposition in the inertia group. So the reader's rejection overstates the damage: the theorem's proof is repairable by adding the missing proper-subgroup hypothesis and invoking Jordan's theorem. The paper should be revised to correct Lemma 6.1, but the central claim of Theorem 1.6 is likely sound. I therefore recommend conditional acceptance rather than outright rejection.","tokens_in":11528,"tokens_out":36098,"duration_ms":362979,"concrete_test":"Check the correction computationally: in LMFDB, obtain degree-5 primitive fields with Galois group 5T3 = AGL(1,5) (a proper primitive group) and inspect the prime factorization of their field discriminants; confirm every prime occurs with exponent at least 2. In parallel, verify that the counterexample x^3-x-1 has Galois group S_3 and therefore lies outside the set counted by L_n(T). If all proper primitive examples are squarefull, the corrected Lemma 6.1 holds and Theorem 1.6 stands; if a prime exponent 1 occurs, the proof would require a different argument.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The only substantive gap I find is the statement of Lemma 6.1. It asserts that D is squarefull for every primitive number field, and the cubic field x^3-x-1 (D=-23) refutes that. However, this counterexample does not enter Theorem 1.6: L_n(T) is a subset of M_n(T), so for counted matrices the Galois group G_A is a proper primitive subgroup of S_n, not S_n itself. For proper primitive groups, Jordan's theorem gives the missing hypothesis: such a group contains no transposition. If a prime p divided D only to the first power, the local inertia group at p would be generated by a transposition swapping the two roots that coincide modulo p and fixing the simple roots, contradicting Jordan's theorem. Hence p^2 divides D for every p|D in the actual application, and the proof's step 'ind(chi_A mod p) >= 2' is justified. The submitted text should either prove this directly or state Lemma 6.1 with the hypothesis G_A != S_n; as written, the proof cites an overstrong false lemma. This is a localized, easily repaired presentation error, not a fatal flaw in Theorem 1.6.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies $M_n(T)$, the number of integer $n\\times n$ matrices with entries bounded by $T$ whose characteristic polynomial has Galois group not equal to the full symmetric group $S_n$. The authors conjecture $M_n(T) \\asymp T^{n^2-n+1}\\log T$ and prove several bounds toward it. Theorem 1.3 uses the Cohen--Serre large sieve to obtain $M_n(T) \\ll T^{n^2-1/2}\\log T$. Theorem 1.4 bounds the number of matrices whose characteristic polynomial has a low-degree irreducible factor by $T^{n^2-(n-k)/(n+k-1)}$. Theorem 1.5 gives the conjectured exponent $T^{n^2-n+1}\\log T$ for a restricted class of matrices preserving a lattice of short vectors. Theorem 1.6, the most novel result, claims that the number of matrices whose characteristic polynomial defines a primitive number field of discriminant $\\ge T^2$ is $O_\\varepsilon(T^{n^2-1+\\varepsilon})$, using a geometric sieve with a double-discriminant construction. The proof of Theorem 1.6 relies on Lemma 6.1, which asserts that every primitive number field has squarefull discriminant.","tokens_in":11721,"tokens_out":28746,"duration_ms":271188,"significance":"If Theorem 1.6 holds, it is a significant step toward Conjecture 1.2, giving a geometric-sieve power saving for non-generic Galois groups of integer matrices. The paper is clearly written and makes good use of recent techniques from Bhargava's proof of van der Waerden's conjecture, as well as work of Katznelson, Reiner, and Gerstenhaber. Theorems 1.3--1.5 appear sound and are useful contributions. However, the pivotal Theorem 1.6 has a load-bearing gap: Lemma 6.1 is false as stated, and the proof of Theorem 1.6 depends on it through the inference that every prime divisor $p$ of the field discriminant has $p^2\\mid D$ and hence that the index of the characteristic polynomial mod $p$ is at least $2$. This is a correctness issue that must be repaired before the paper's main new claim is supported.","major_comments":[{"comment":"Lemma 6.1 states that if a prime $p$ divides the discriminant $D$ of a primitive number field, then $p^2\\mid D$. This is false as stated: the cubic field defined by $x^3-x-1$ is primitive and has $D=-23$, so $23\\mid D$ but $23^2\\nmid D$. The proof of Theorem 1.6 uses this lemma to conclude that $\\operatorname{ind}(\\chi_A\\bmod p)\\ge 2$ for every $p\\mid D$, which then yields the double-discriminant congruence $p\\mid DD(A)$. Since $L_n(T)\\subseteq M_n(T)$, for matrices counted by $L_n(T)$ the Galois group $G_A$ is a proper primitive subgroup of $S_n$; the intended version of the lemma should be stated with this hypothesis and proved using Jordan's theorem (a primitive subgroup containing a transposition is $S_n$) together with the local ramification fact that $p\\|D$ would force the inertia group at $p$ to be generated by a transposition. As written, the lemma is false and the proof of Theorem 1.6 is formally invalid. This is the central issue of the paper.","section":"Section 6, proof of Theorem 1.6 (use of Lemma 6.2)"}],"minor_comments":[{"comment":"The abstract contains a grammatical error: 'We study the number $M_n(T)$ be the number...' should be 'We study the number $M_n(T)$ of...'.","section":"Abstract"},{"comment":"The citation '[Bha25], remark following Proposition 11' should be checked: the stated claim does not appear to be a correct reflection of Bhargava's remark, and the authors should provide a self-contained proof of the corrected lemma under the proper hypotheses.","section":"Section 6, Lemma 6.1"},{"comment":"The sentence 'Note that the loss due to those $A$ for which $\\Psi_g>1$ is of low order' is vague; the authors should either quantify this loss or provide a reference for the equidistribution statement that justifies neglecting the multiple-counting.","section":"Section 4, proof of Theorem 1.4"},{"comment":"The display of the matrix shape in Example 5.1 is visually confusing because some rows and columns are merged; clarifying the entries with explicit zero and nonzero blocks would improve readability.","section":"Section 5, Example 5.1"}],"recommendation":"major_revision","confidential_remarks":"The paper's central Conjecture 1.2 is not proved, but the partial results are of interest. The main obstacle is the incorrect Lemma 6.1 and the associated gap in the proof of Theorem 1.6. The repair via Jordan's theorem appears feasible, but the authors must also address the related index condition for the double-discriminant argument, including a careful treatment of $p=2$. If these lemmas are corrected and proved, the paper would be a solid contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the reader's reject verdict is too harsh. The paper is stronger than the report suggests. The advertised Theorems 1.3–1.5 look correct; Theorem 1.6 is also essentially correct, but the proof as written cites a false lemma. The good news is the false lemma is not needed in its full generality, and the gap is easily repaired.\n\nThe genuinely new content: Conjecture 1.2 strengthening Rivin, Theorem 1.3 via Cohen–Serre, Theorems 1.4 and 1.5 via Fourier and combinatorial sieves, and Theorem 1.6 via a double-discriminant twist in the geometric sieve. The techniques are not in the cited literature. The paper is well-written and the context is clear.\n\nThe soft spot: Lemma 6.1 asserts every primitive number field discriminant is squarefull, which is false (x^3 - x - 1 has D = -23). The reader took this as fatal. But L_n(T) counts only matrices whose Galois group is a proper primitive subgroup of S_n; the cubic example has Galois group S_3, which is not proper, so it lies outside the domain. For proper primitive groups, Jordan's theorem says no transposition is present, so a repeated root modulo p cannot be a simple double root with all other roots simple; hence the index of chi_A mod p is never 1, and p|D implies index >= 2. That is exactly what the proof needs. The lemma should be restated with the 'proper primitive' hypothesis, or the proof should argue directly. This is a localized presentation error, not a load-bearing flaw.\n\nThere are minor issues: the proof of Theorem 1.4 uses the phrase 'we take it' for the inner product convention, which is fine but should be specified; some constant tracking is loose but standard. Overall the math is sound.\n\nWho is this for: arithmetic statisticians and number theorists working on Galois groups of random polynomials and matrices. It deserves a serious referee. I would send it to review with a request to fix Lemma 6.1.","headline":"A solid contribution with one easily patched lemma; the reject verdict is too harsh.","tokens_in":12342,"tokens_out":3533,"would_cite":true,"duration_ms":33679,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11R32","11R45","11R29"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves new upper bounds on the count of integer matrices whose characteristic polynomial has Galois group smaller than $S_n$, including an $O_\\varepsilon(T^{n^2-1+\\varepsilon})$ bound for primitive fields of large discriminant.","keywords":["random integer matrices","Galois groups","non-generic Galois group conjecture","large sieve","Fourier-analytic sieve","geometric sieve","characteristic polynomials","number field discriminants"],"falsifier":"Take $b=5$ and the companion matrix of $x^3-x-5$; it defines a primitive cubic field with discriminant $D=671=11\\cdot 61$, so $D\\ge 25$ and the matrix is counted by $L_3(T)$ for $T=5$, yet neither $11$ nor $61$ divides $D$ twice. Carrying out the congruence count with $C=671$ tests whether the double-discriminant conditions can still give the $O_\\varepsilon(T^{n^2-1+\\varepsilon})$ bound; if the count needs a squarefullness factor that Lemma 6.1 was supposed to supply, the proof of Theorem 1.6 fails as written.","tokens_in":11256,"feed_emoji":"🎲","tokens_out":19195,"duration_ms":167206,"temperature":0.7,"pith_summary":"This paper counts how often the characteristic polynomial of an integer matrix fails to have the full symmetric group as its Galois group. The paper conjectures that the number $M_n(T)$ of such $n\\times n$ matrices with entries in $[-T,T]$ satisfies $M_n(T) \\asymp T^{n^2-n+1}\\log T$, matching the known lower bound from matrices with an integer eigenvalue. It proves $M_n(T) \\ll T^{n^2-1/2}\\log T$ by the large sieve, and then uses Fourier and geometric sieves to sharpen this for three special classes: matrices with a low-degree irreducible factor in their characteristic polynomial, matrices preserving a lattice generated by short vectors, and matrices whose characteristic polynomial defines a primitive number field of discriminant at least $T^2$. The main new result is the last one, $L_n(T) \\ll_\\varepsilon T^{n^2-1+\\varepsilon}$, obtained with a double-discriminant construction. If the conjecture is right, the dominant exceptional matrices are the reducible ones, and the paper's bounds move the problem close to that target for several natural subfamilies.","feed_headline":"New bounds trim exceptional Galois groups of integer matrices","feed_subtitle":"Sieve methods push the count below $T^{n^2-1/2}\\log T$, approaching the conjectured main order.","key_machinery":"The carrying objects are four parallel sieve mechanisms. The large-sieve irreducibility theorem treats the characteristic polynomial as a family $F(x,Y_1,\\ldots,Y_{n^2})$ and bounds the matrices whose Galois group drops below $S_n$ by $O(T^{n^2-1/2}\\log T)$. The Fourier sieve for reducible matrices uses Poisson summation over $\\mathbb{F}_p$, with a selector $\\Psi_g$ that detects an irreducible degree-$k$ factor $g$ modulo $p$; the support of its Fourier transform lies on rank $\\le k$ matrices, and balancing the rank-$r$ contributions at $p\\asymp T^{(2n-1)/(n+k-1)}$ yields Theorem 1.4. The lattice argument uses reduced bases $v_1,\\ldots,v_k$ with lengths $\\ell_i$, the entry bound $|g_{ij}|\\ll T\\ell_j/\\ell_i$, and a pinch-point analysis to show the number of possible restrictions $G$ is at most $T^{kn-n+1}$, before dividing by the covolume of $\\Lambda^\\perp\\otimes \\mathbb{Z}^n$. The geometric sieve for primitive fields uses the index of a polynomial modulo $p$, $\\operatorname{ind} f=\\sum_i(e_i-1)\\deg f_i$; $p^k\\mid D$ forces $\\operatorname{ind}(\\chi_A\\bmod p)\\ge k$, the fraction of polynomials with index at least $k$ is $O(p^{-k})$, and finite-field equidistribution of characteristic polynomials transfers this to matrices. The double discriminant then makes $\\operatorname{disc}\\chi_A\\equiv DD(A)\\equiv 0\\pmod C$, with $C=\\prod_{p\\mid D}p$.","core_discovery":"The core claim is that non-generic Galois groups of integer matrices can be controlled by sieving the matrices according to how their characteristic polynomials factor modulo primes. Theorem 1.3 gives $M_n(T) \\ll T^{n^2-1/2}\\log T$. Theorem 1.4 bounds the count $R_{n,k}(T)$ of matrices whose characteristic polynomial has an irreducible factor of degree $k\\le n/2$ by $T^{n^2-(n-k)/(n+k-1)}$. Theorem 1.5 gives $S_{n,k}(T)\\ll T^{n^2-n+1}\\log T$ for matrices preserving a lattice generated by vectors of length at most $T$, with irreducible restriction and short orthogonal complement. Theorem 1.6 bounds the contribution of primitive number fields with discriminant $D\\ge T^2$ by $O_\\varepsilon(T^{n^2-1+\\varepsilon})$; its proof introduces the double discriminant $DD(A)=\\operatorname{disc}_t \\operatorname{disc}_x \\det(xI-tS-A)$ with $S=\\operatorname{Diag}(1,2,\\ldots,n)$, which is invariant under translation by multiples of $S$ and lets the sieve impose simultaneous congruence conditions modulo $C=\\prod_{p\\mid D} p$.","pith_inferences":["An extension left implicit: if the squarefull-discriminant lemma is replaced by a weaker index condition, the double-discriminant sieve would apply to all primitive fields, and to any matrix ensemble admitting a diagonal direction whose characteristic polynomial is separable modulo every large prime.","The short-vector restriction in Theorem 1.5 marks the current boundary of the method; progress on large-value estimates for matrix products would extend the $T^{n^2-n+1}\\log T$ bound to all reducible matrices with an irreducible factor of degree $k$.","A uniform bound of the shape $M_n(T;f)\\ll T^{n(n-1)/2+o(1)}$ for the number of matrices with a fixed characteristic polynomial $f$ would complete the small-discriminant half of the geometric sieve and imply Conjecture 1.2."],"forward_implications":["For every fixed $n$, $M_n(T)\\ll T^{n^2-1/2}\\log T$, improving the trivial count of all matrices, $T^{n^2}$, by a factor $T^{1/2}/\\log T$.","Reducible matrices with an irreducible factor of degree $k\\le n/2$ obey $R_{n,k}(T)\\ll T^{n^2-(n-k)/(n+k-1)}$, beating the large-sieve bound when $k\\le n/3$ and the previous bound when $k\\gg\\sqrt n$.","Matrices preserving a lattice generated by short vectors are counted at the conjectured main order $T^{n^2-n+1}\\log T$, the same order as matrices with an integer eigenvalue.","Primitive number fields with discriminant at least $T^2$ contribute at most $O_\\varepsilon(T^{n^2-1+\\varepsilon})$, so any asymptotic main term must come from reducible characteristic polynomials or small-discriminant fields.","The lower bound of Conjecture 1.2 is supplied by integer eigenvalues and already includes the factor $\\log T$, showing that the logarithm is essential rather than an artifact."],"supporting_citations":[{"why":"Supplies the geometric-sieve template, the squarefull-discriminant lemma, and the double-discriminant idea that Theorem 1.6 adapts to matrices.","marker":"[Bha25]"},{"why":"Supplies the geometric fact that a polynomial with index at least 2 modulo p is a singular point of the discriminant hypersurface, used to prove p divides the double discriminant.","marker":"[Bha23]"},{"why":"Supplies the asymptotic count of singular integer matrices in expanding boxes used for the lower bound, and the lattice-counting estimate used to sum $d(\\Lambda)^{-n}$.","marker":"[Kat93]"},{"why":"Formulates the large-sieve irreducibility theorem in the shape used for Theorem 1.3.","marker":"[Zyw10]"},{"why":"Gives the exact count of matrices over a finite field with a fixed characteristic polynomial, the equidistribution input for the sieve over primes.","marker":"[Rei61]"},{"why":"Gives the companion result on nilpotent matrices over finite fields used with the characteristic-polynomial count to control equidistribution.","marker":"[Ger61]"},{"why":"Provides the best uniform upper bound on the number of matrices with a given characteristic polynomial, used in the small-discriminant discussion.","marker":"[HOS24]"},{"why":"Gives the previous bound on reducible matrices that Theorem 1.4 improves for $k\\gg\\sqrt n$.","marker":"[OS24]"},{"why":"Supplies the Fourier-analytic sieve method that Theorem 1.4 adapts to matrix reducibility.","marker":"[AGLO+23]"}],"fun_headline_variants":["Sieve methods tighten bounds on exceptional Galois groups","New upper bounds for rare Galois groups of integer matrices","Counting integer matrices with non-generic Galois groups","Refined sieve cuts count of exceptional matrix Galois groups","Tighter anomaly bounds for Galois groups of random matrices"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Lemma 6.1 in Section 6, which asserts that every prime dividing the discriminant $D$ of a primitive number field $K_A$ divides $D$ to at least the second power (squarefull $D$), forcing the index of $\\chi_A$ modulo $p$ to be at least $2$ and making the sum over $D\\ge T^2$ converge; this assertion is false, since the primitive cubic field defined by $x^3-x-1$ has discriminant $-23$.","fun_headline_variants_meta":{"raw":{"variants":["Sieve methods tighten bounds on exceptional Galois groups","New upper bounds for rare Galois groups of integer matrices","Counting integer matrices with non-generic Galois groups","Refined sieve cuts count of exceptional matrix Galois groups","Tighter anomaly bounds for Galois groups of random matrices"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000322,"raw_usage":{"total_tokens":1804,"prompt_tokens":933,"completion_tokens":871,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":549,"completion_tokens_details":{"reasoning_tokens":792}},"tokens_in":549,"tokens_out":871,"duration_ms":8181,"temperature":1.0,"reasoning_tokens":792,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:58:33.258478+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $b=5$ and the companion matrix of $x^3-x-5$; it defines a primitive cubic field with discriminant $D=671=11\\cdot 61$, so $D\\ge 25$ and the matrix is counted by $L_3(T)$ for $T=5$, yet neither $11$ nor $61$ divides $D$ twice. Carrying out the congruence count with $C=671$ tests whether the double-discriminant conditions can still give the $O_\\varepsilon(T^{n^2-1+\\varepsilon})$ bound; if the count needs a squarefullness factor that Lemma 6.1 was supposed to supply, the proof of Theorem 1.6 fails as written.","supporting_citations":[],"review_version":1}