{"id":"681cd066-2bb7-4d09-b75e-137896c59fa9","arxiv_id":"2608.08509","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Sun's conjectures on truncated Jacobi-symbol determinants are proved by a unified diagonalization plus supersingular-elliptic-curve method.","lead":"The paper proves Sun's conjectures about determinants whose entries are Jacobi symbols: after deleting the indices 0, 1 and -1, these determinants vanish or have predictable prime divisibility. Its value is the unified method, which reduces each symbolic determinant to the supersingularity of a specific elliptic curve.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proofs of the exact-zero cases in Theorem 1.1(b), (f), and (h) assume square roots α of 2, 5, or 18 in F_p that do not exist for the primes in question; Sections 3.2–3.3 require an explicit F_{p^2} reworking before the argument is valid as written.","rationale":"The reader's weakest_assumption was the black-box import of Z.-H. Sun's congruences and Deuring's theorem. I agree those are load-bearing external inputs, but the more immediately checkable issue is the internal contradiction in Sections 3.2 and 3.3: the text asserts the existence of α∈F_p with α^2=d when, for the exact-zero residue classes, d is a quadratic nonresidue modulo p. This affects Theorem 1.1(b) for p≡13,19 (mod 24) and the exact-zero conclusions in Theorem 1.1(f) and (h). Direct computation for p=13 shows the involved coefficient does vanish, so the mathematical claim is likely correct and repairable by an explicit treatment over F_{p^2}; however, as written the proof contains a false premise in those cases. This reinforces the reader's CONDITIONAL verdict rather than overturning it: the paper needs a corrected presentation of the quadratic-extension cases, including the Deuring reduction over F_{p^2}, and ideally a fuller justification or precise citation for the Sun congruences. No circularity or evidence of a false theorem was found; the concern is about the soundness of the proof as written, not about the truth of the statements.","tokens_in":18150,"tokens_out":30768,"duration_ms":290671,"concrete_test":"For p=13, confirm that (2/13)=-1 and then directly expand (X^2+2X+2)^6 mod 13 to compute a_4=[X^4]; the three contributing binomial terms give 11+9+6=26≡0, so the coefficient vanishes despite α∉F_13. Then re-derive this vanishing by working over F_{169}: evaluate the Legendre-polynomial identity (7) at t∈F_{169} with α^2=2, and apply Deuring's theorem to the reduction of E^{(3)}_{T_0} over the residue field F_{169} (the residue field when p is inert in Q(√2)). If the F_{169} version reproduces a_4=a_8=0, the flaw is a presentational gap; if not, the proof of Theorem 1.1(b) for p≡13,19 is genuinely broken.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 3.2, the proof of Theorem 1.1(b) says 'Choose α∈F_p with α^2=2' and then derives a_r via equation (6). But the two residue classes where the exact conclusion {2,2}_p=0 is claimed, p≡13,19 (mod 24), satisfy (2/p)=-1, so no such α exists in F_p. In Section 3.3, the analogous instruction 'Choose α∈F_p with α^2=d' is impossible for (c,d)=(5,5) when p≡13,17 (mod 20), which is exactly the exact-zero case of Theorem 1.1(f), and for (c,d)=(8,18) when p≡13 (mod 24), one of the exact-zero cases in Theorem 1.1(h). Consequently, equations (6) and (10) are not legitimate as written for the primes on which the exact-zero conclusions depend. The later discussion of good reduction in Q(√2) or Q(√5) shows the authors are aware that the parameter T may live in a quadratic extension, but the Legendre-polynomial identities are not rewritten over F_{p^2}, and the application of Deuring's theorem to reductions over F_{p^2} is not spelled out. Direct computation for p=13 shows the coefficient a_4=[X^4](X^2+2X+2)^6 mod 13 does vanish, so the theorem's conclusion may be true, but the internal derivation in the manuscript is invalid for these primes unless the F_{p^2} repair is supplied. This is a separate and more concrete difficulty than the imported Sun congruences, because no citation can fix a false assertion about the existence of α in F_p.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a unified proof of several conjectures of Z.-W. Sun on truncated Jacobi-symbol determinants {c,d}_n. The method diagonalizes the nonzero-residue matrix M^×_p(c,d) over F_p, identifies its eigenvalues with the coefficients a_s of q(X)^{(p-1)/2}, and then proves the required coefficient vanishings by linking them to Hasse invariants of explicitly given CM elliptic curves. Deuring's reduction theorem converts inertness of p in the relevant CM field into supersingularity, which forces the coefficient to vanish; composite moduli are handled by a Chinese-remainder tensor-product argument. The main results are Theorem 1.1, covering Sun's conjectures in eight families, and Theorem 1.3, giving n^2-divisibility for the full nonzero-residue determinants (2,3)_n and (6,15)_n.","tokens_in":18546,"tokens_out":20844,"duration_ms":220966,"significance":"If the proof can be repaired, this is a substantial contribution: it resolves a substantial batch of Sun's conjectures through a single conceptual mechanism, connecting truncated Legendre-symbol determinants to supersingular reduction and complex multiplication. The linear algebra in Lemmas 2.1–2.3 is clean and explicit, the coefficient-to-Hasse-invariant translations are concrete, and the included Sage listings provide machine-checked verification of the CM data against Hilbert class polynomials. The low-degree coefficient arguments in §3.4 and the central-coefficient arguments in §3.1 are checkable and persuasive. The paper is also honest about its external inputs, namely Z.-H. Sun's congruences and Deuring's theorem, and it does not assume or fit any target result; the conjectures serve as inputs, not parameters.","major_comments":[{"comment":"The proofs of the exact-zero cases choose α∈F_p with α^2=d, but for the primes on which those conclusions depend this α does not exist. For (c,d)=(2,2), p≡13,19 (mod 24) satisfies (2/p)=-1; for (c,d)=(5,5), p≡13,17 (mod 20) satisfies (5/p)=-1; and for (c,d)=(8,18), p≡13 (mod 24) satisfies (18/p)=-1. Consequently equations (6) and (10) are not legitimate as written, and the identities a_r=0 on which Theorem 1.1(b), (f), and (h) rest are not established. The good-reduction discussion in §3.2 shows awareness that the parameter T may live in a quadratic extension, but the Legendre-polynomial and Hasse-invariant computation is not reworked over F_{p^2}. A repair must rewrite (6)/(10), Corollaries 3.5 and 3.8, and the Deuring step for reductions whose residue field is F_{p^2}. Direct computation for p=13 supports the truth of the conclusion, but the manuscript's internal derivation is invalid for these primes as it stands.","section":"§3.2, Eq. (6); §3.3, Eq. (10); proof of Theorem 1.1(b),(f),(h)"},{"comment":"Even after moving the computation to F_{p^2}, the implication 'E_t/F_{p^2} is supersingular ⇒ H_p^{(3)}(t)=0' is not immediate from Theorem 3.2 as stated. Theorem 3.2(1) is phrased for a curve over F_p, while the reduction E_t in the non-residue cases is defined over F_{p^2}. The missing bridge is the identity Ha_{p^2}(E)=H_p(E)^{p+1} for these cubics, which follows by extracting the coefficient of x^{p^2-1} in f^{(p^2-1)/2}=f^{(p-1)/2}(f^{(p-1)/2})^p. This identity would show that supersingularity over F_{p^2} forces the ordinary p-th Hasse invariant to vanish. The same issue affects the one-quarter coefficient in Corollary 3.8. The paper should state and prove this Hasse-invariant relation explicitly, since it is load-bearing for the exact-zero and p^2-divisibility conclusions.","section":"§3.2 after Eq. (9); Theorem 3.2"},{"comment":"The one-third and one-quarter coefficient vanishings import Z.-H. Sun's congruences without proof. As external published results this is acceptable in principle, but the application to t=-α^{-1} with α outside F_p is not covered by the stated hypotheses: those theorems are for t∈Z_(p), and the Legendre-symbol sums involve g_t(x) or f_t(x) evaluated in F_p. A citation cannot fix the mismatch. The authors should either verify that Corollaries 3.5 and 3.8 remain valid, as polynomial identities over F_p[T], at F_{p^2}-valued t, or supply a direct extension of the congruences to the quadratic residue field. Until this is done, parts (b), (f), (g), and (h) rest on black-box hypotheses that are not met at the primes used in the exact arguments.","section":"§3.2–§3.3, Theorems 3.4 and 3.7"}],"minor_comments":[{"comment":"The table is poorly formatted; in the (3,3) row the columns for Δ(E) and j(E) appear merged as '−2 433 0−3', and the factorization is not readable. Please reformat so that each column is unambiguous.","section":"Table 1"},{"comment":"The remark states additional families, such as {22,125}_p, {28,250}_p, and {126,2625}_p, with proofs omitted. If these are intended as results, they should be proved or explicitly labelled as conjectural; as written, an unproved assertion in a remark is not a contribution.","section":"Remark 3.11"},{"comment":"The acknowledgment mentions 'GPT-5.6 Sol' as an assistant. This is not a mathematical defect, but the paper should either specify precisely how the tool was used or omit the mention, to avoid ambiguity about the provenance of the mathematical arguments.","section":"Acknowledgments"},{"comment":"In the discussion before the proof of Theorem 1.1(a), the vector u_3=(0,1,-1)^T is described as a kernel vector 'over Z'; it would be clearer to note explicitly that its entries are integers and that it vanishes at the boundary coordinates 0,1,-1, which is what allows Lemma 2.4(ii) to be applied.","section":"§3.1, exceptional prime p=3"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is technically interesting and the intended mechanism is attractive, but the missing F_{p^2} treatment in §§3.2–3.3 is a genuine, load-bearing gap that affects several stated theorems. The dependence on two external Sun congruences for these same cases makes the repair non-cosmetic. The unproved families in Remark 3.11 should also be addressed. None of this suggests bad faith or circularity; the paper's core idea is sound enough to warrant a major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this paper gives the first unified proof of most of Sun's truncated Jacobi-symbol determinant conjectures, and the core mechanism — diagonalize the nonzero-residue matrix, read eigenvalues as coefficients of (X^2+cX+d)^{(p-1)/2}, identify those coefficients with Hasse invariants of CM elliptic curves that reduce to supersingular curves under Deuring's theorem — is real and reusable. Second, the exact-zero cases in Theorem 1.1(b), (f), and (h) have a concrete gap as written: the proofs choose alpha in F_p with alpha^2 = 2, 5, or 18 at primes where the Legendre symbol is -1, so no such alpha exists. The later discussion over Q(sqrt(2)) and Q(sqrt(5)) shows the authors know the parameter lives in a quadratic extension, but equations (6) and (10) are not reworked over F_{p^2}, and the Deuring reduction argument over F_{p^2} is not spelled out. Direct computation for p=13 suggests the conclusion is true, so this looks repairable, but it must be fixed before the paper is final.\n\nWhat is genuinely good: Section 2's linear algebra is clean; the kernel-vector constructions for divisibility and exact zero are careful; the composite-modulus CRT handling is nontrivial. The SageMath verification of CM data is exact and externally reproducible, which is real evidence. The imported congruences of Z.-H. Sun are black boxes, but they are published results and the dependence is stated. Minor issues: Remark 3.11 lists additional families without proofs; that is acceptable as a remark but should be labeled as speculative family examples rather than asserted consequences, and the one-sixth coefficient application is very terse. The references are appropriate for the determinant-and-CM literature.\n\nThe main theorems are likely true, the method is significant, and the central flaw is localized. I would send this to peer review with a request for major revision: rewrite the alpha-selection over F_{p^2} or choose a basis that avoids needing the square root in F_p, and clarify the status of Remark 3.11. After that, it should be a solid addition to the field.","headline":"A genuinely new unified proof of Sun's truncated-determinant conjectures, with a concrete and localized gap in the exact-zero cases over F_p that needs a quadratic-extension repair before the paper is final.","tokens_in":665,"tokens_out":1525,"would_cite":true,"duration_ms":42902,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11A07","11E25","11G05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Through exact diagonalization of the residue-class matrix and Deuring's reduction theorem, this paper proves Sun's conjectures on truncated Jacobi-symbol determinants: eight families of vanishings and divisibilities, plus n^2-divisibility…","keywords":["Jacobi-symbol determinants","truncated determinants","Legendre symbols","supersingular elliptic curves","complex multiplication","Hasse invariants","Deuring reduction theorem","Sun conjectures"],"falsifier":"Compute {2,2}_13 directly as an integer from the 10×10 matrix with entries (($j^{2}$+2jk+$2k^{2}$)/13), 2≤j,k≤11. Theorem 1.1(b) predicts the determinant is exactly 0; any nonzero value would disprove the claim. Similarly, {2,2}_23 is predicted to be divisible by 23, and the paper's own Remark 3.6 gives the value, so a direct check settling that divisibility would test the same chain.","tokens_in":17956,"feed_emoji":"🔢","tokens_out":11693,"duration_ms":107095,"temperature":0.7,"pith_summary":"The paper proves a large share of Zhi-Wei Sun's conjectures about truncated Jacobi-symbol determinants {c,d}_n = det[( ($j^{2}$+cjk+$dk^{2}$)/n )]_{2≤j,k≤n-2}. Theorem 1.1 settles eight conjecture families: exact vanishing or p-, $p^{2}$-, and $n^{2}$-divisibility for the parameter pairs (3,2), (2,2), (4,2), (8,8), (3,3), (42,-7), (21,112), (2,3), (6,15), (5,5), (10,9), and (8,18) under the stated congruence conditions on n, and Theorem 1.3 shows $n^{2}$ divides the full nonzero-residue determinants (2,3)_n and (6,15)_n for every odd n>3. The engine is a single reduction: diagonalizing the matrix indexed by F_p^× produces eigenvalues that are coefficients of q(X)^{(p-1)/2} with q(X)=$X^{2}$+cX+d, and the vanishing of those coefficients is shown to be the supersingularity of an explicit CM elliptic curve via Deuring's theorem. If the paper is right, scattered determinant identities become corollaries of one geometric criterion, and the same template generates further families.","feed_headline":"Supersingular curves prove Sun's determinant conjectures","feed_subtitle":"Eight families of Jacobi-symbol determinant congruences follow from one eigenvalue argument plus Deuring's theorem.","key_machinery":"The load-bearing object is the exact eigenvalue decomposition of M^×_p(c,d) on the power-vector basis, with eigenvalues -a_s, together with the symmetry a_{p-1-s}=$d^{{s-(p-1)/2}}$a_s. It converts every determinant statement into a statement about coefficients of q(X)^{(p-1)/2}. The coefficient vanishing is then carried by the Hasse invariant Hap(E)=[$x^{{p-1}}$]f(x)^{(p-1)/2} of an explicitly given elliptic curve: supersingular reduction implies the invariant is zero, and Deuring's reduction theorem supplies the supersingularity when p is inert in the curve's CM field. The final piece is the kernel-vector transfer: zero eigenvalues at reciprocal indices produce a vector in the kernel of the truncated matrix, and Lemma 2.4's tensor-product argument lifts these local kernels to composite moduli n.","core_discovery":"On its own terms, the paper's central claim is a chain of equivalences linking combinatorics to arithmetic geometry. For an odd prime p with p∤d, the (p-1)×(p-1) residue matrix M^×_p(c,d) is diagonalized by the power vectors w_s=(1^s,2^s,...,(p-1)^s)^T, whose eigenvalues are -a_s where q(X)^{(p-1)/2}=∑_{s=0}^{p-1}a_s X^s in F_p[X]; the coefficient symmetry a_{p-1-s}=$d^{{s-(p-1)/2}}$a_s doubles each vanishing. The truncated determinant {c,d}_p is then forced to vanish or be divisible by p when two reciprocal coefficients vanish, because the difference of the corresponding kernel vectors vanishes at the deleted indices 1 and -1; for exact zero over Z, a character-coset indicator vector and an absolute-value bound do the job. Each coefficient a_s is identified with a Hasse invariant of an explicit elliptic curve—$y^{2}$=x($x^{2}$+cx+d) for the middle coefficient, $E_T^{{(3)}}$ and $E_T^{{(4)}}$ for the one-third and one-quarter coefficients—and computation of j-invariants reveals complex multiplication by orders of discriminants -4, -8, -3, -28, -7, -24, -20, and -6. Deuring's reduction theorem then converts inertness of p in the CM field into supersingularity, which makes the Hasse invariant zero, and the Chinese Remainder Theorem lifts the resulting kernel vectors from primes to squarefree composite moduli. This is the proof of Theorem 1.1 and Theorem 1.3.","pith_inferences":["One could run the same three-step template systematically: enumerate coefficient degrees s for which a Legendre-polynomial congruence is known, solve the corresponding (c,d) parameter equations, compute the CM field of the resulting curve, and read off new determinant congruences; the one-sixth example in Remark 3.11 is the first such output.","For forms of higher degree than X^2+cX+d, the analogous eigenvalues would be coefficients of f(X)^{(p-1)/2} with f of higher degree, and the Hasse invariant would be replaced by the corresponding invariant of the Jacobian or abelian variety, using the generalized Deuring theorem cited at the end of the paper.","The exact-zero mechanism suggests a statistical picture: a pair (c,d) should exhibit vanishing whenever the associated CM curve is supersingular at all primes in a dense-enough set, so the set of moduli n with {c,d}_n=0 should be governed by the splitting behaviour of primes in the CM field."],"forward_implications":["For every odd n≡1 (mod 4) that is not a sum of two squares, {3,2}_n=0; this is part (a) of Theorem 1.1.","For the one-third family, {2,2}_p=0 for primes p≡13,19 (mod 24) and {2,2}_p≡0 (mod p) for p≡17,23 (mod 24), so the parity of the kernel-vector exponents decides between exact vanishing and mere p-divisibility.","For composite squarefree n, the CRT tensor-product lemma turns any local kernel modulo a prime divisor into a kernel of the truncated matrix modulo n, yielding n^2-divisibility for (2,3)_n and (6,15)_n and for {2,3}_n whenever n>3 and n not ±1 mod 12.","The same three-step template stated in Remark 3.11 predicts new explicit families, including p|{22,125}_p for p≡5,11 (mod 12) and p^2|{22,125}_p for p≡17,53 (mod 60)."],"supporting_citations":[{"why":"Source of the eight conjectures and the definition of the truncated determinant {c,d}_n that Theorem 1.1 resolves.","marker":"[11]"},{"why":"Defines the full and nonzero-residue determinants [c,d]_n and (c,d)_n; Conjecture 4.8(i) there is proved as Theorem 1.3.","marker":"[10]"},{"why":"Imported congruence P_⌊p/3⌋(t) ≡ -(3/p)Σ(.../p) used to link the one-third coefficient to the Hasse invariant of E_T^{(3)}; not proved in this paper.","marker":"[9]"},{"why":"Imported congruence for P_⌊p/4⌋(t) used to link the one-quarter coefficient to the Hasse invariant of E_T^{(4)}.","marker":"[7]"},{"why":"Contains the Deuring reduction theorem (Chapter 13, §4, Theorem 12) that converts inertness in the CM field into supersingularity of the reduced curve.","marker":"[5]"},{"why":"Supplies the standard facts on Hasse invariants, the trace formula t_q(E) ≡ Ha_q(E) mod p, supersingularity, and the Hasse bound used in Theorem 3.2.","marker":"[6]"},{"why":"A generalization of Deuring's reduction theorem cited as the basis for extending the argument to Jacobians of higher-genus curves in Remark 3.11.","marker":"[15]"}],"fun_headline_variants":["One elliptic curve trick settles Sun's determinant conjectures","Jacobi-symbol determinants fall to supersingularity","Deuring's theorem proves Sun's determinant conjectures","CM reduction settles truncated Jacobi-symbol conjectures","Supersingular reduction unifies Sun determinant proofs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof leans on two previously published congruences of Z.-H. Sun that identify the one-third and one-quarter Legendre-polynomial values with elliptic-curve Hasse invariants; if either congruence failed for a prime in the stated residue classes, the vanishing results in parts (b), (f), (g), and (h) of Theorem 1.1 would no longer follow.","fun_headline_variants_meta":{"raw":{"variants":["One elliptic curve trick settles Sun's determinant conjectures","Jacobi-symbol determinants fall to supersingularity","Deuring's theorem proves Sun's determinant conjectures","CM reduction settles truncated Jacobi-symbol conjectures","Supersingular reduction unifies Sun determinant proofs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000884,"raw_usage":{"total_tokens":3900,"prompt_tokens":1110,"completion_tokens":2790,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":726,"completion_tokens_details":{"reasoning_tokens":2713}},"tokens_in":726,"tokens_out":2790,"duration_ms":19023,"temperature":1.0,"reasoning_tokens":2713,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:36:18.984751+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute {2,2}_13 directly as an integer from the 10×10 matrix with entries (($j^{2}$+2jk+$2k^{2}$)/13), 2≤j,k≤11. Theorem 1.1(b) predicts the determinant is exactly 0; any nonzero value would disprove the claim. Similarly, {2,2}_23 is predicted to be divisible by 23, and the paper's own Remark 3.6 gives the value, so a direct check settling that divisibility would test the same chain.","supporting_citations":[{"cited_title":"Sun, Congruences involving 2k k 23k k m−k,Journal of Number Theory, 133 (2013), no","cited_arxiv_id":null,"evidence_quote":"Imported congruence P_⌊p/3⌋(t) ≡ -(3/p)Σ(.../p) used to link the one-third coefficient to the Hasse invariant of E_T^{(3)}; not proved in this paper."},{"cited_title":"Sun, Congruences concerning Legendre polynomials II,Journal of Number Theory, 133 (2013), no","cited_arxiv_id":null,"evidence_quote":"Imported congruence for P_⌊p/4⌋(t) used to link the one-quarter coefficient to the Hasse invariant of E_T^{(4)}."},{"cited_title":"Lang,Elliptic Functions, second ed., Graduate Texts in Mathematics 112, Springer-Verlag, 1987","cited_arxiv_id":null,"evidence_quote":"Contains the Deuring reduction theorem (Chapter 13, §4, Theorem 12) that converts inertness in the CM field into supersingularity of the reduced curve."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the standard facts on Hasse invariants, the trace formula t_q(E) ≡ Ha_q(E) mod p, supersingularity, and the Hasse bound used in Theorem 3.2."},{"cited_title":"Zaytsev, Generalization of Deuring reduction theorem, J","cited_arxiv_id":null,"evidence_quote":"A generalization of Deuring's reduction theorem cited as the basis for extending the argument to Jacobians of higher-genus curves in Remark 3.11."}],"review_version":1}