{"id":"1c7c734c-a80d-466b-8238-2fbb3ac1e63f","arxiv_id":"2605.19517","paper_version":3,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves explicit determinant formulas and a uniform congruence modulo p for Legendre symbol matrices, resolving parts of Sun's conjectures 4.8(i) and 4.10(i).","lead":"The paper proves two determinant evaluations for matrices whose entries are Legendre symbols modulo primes. This resolves the p congruent to 1 mod 4 case of one Sun conjecture and supplies a uniform modular congruence for another.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Vsemirnov factorisation applicability to four-indeterminate H and cofactor yielding exact claimed inverse package","rationale":"The reader's weakest_assumption pinpoints exactly this extension step as the load-bearing point in the abstract-described proof; the second result (uniform mod-p congruence) appears independent of it. Full text would be needed to confirm no hidden adjustments, but the concern is internal to the argument structure rather than external consensus.","tokens_in":1808,"tokens_out":334,"duration_ms":16505,"concrete_test":"For p=13 (≡1 mod 4), explicitly build the 6x6 matrix H using the four indeterminates and definitions of a'_p, b'_p, U, η; attempt to apply Vsemirnov factorisation, compute det H and the 2x2 inverse package via cofactor expansion, and check numerical agreement with the stated formulas (within the field F_p).","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The first result's explicit det H and 2x2 U^T H^{-1} U rest on Vsemirnov's factorisation of Chapman's matrix extending directly to the four-indeterminate matrix H = [χ(j-k+1)] with the stated indeterminates, plus an adjacent cofactor calculation that produces precisely the four-entry package without extra correction terms. This step is least secure because the original factorisation targets a different (determinate) matrix; the extension to indeterminates and the cofactor extraction could introduce unaccounted discrepancies that alter the claimed expressions involving leg(2/p), a'_p, b'_p.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves two determinant evaluations attached to Sun's conjectures on matrices of Legendre symbols. The first resolves the p≡1 mod 4 part of Conjecture 4.8(i) by reducing det H for the four-indeterminate matrix H=[χ(j-k+1)]_{0≤j,k<(p-1)/2} to the explicit formula det H=(2/p)(b'_p - a'_p) together with the 2×2 matrix U^T H^{-1} U, where U=(1,η) and η_j=χ(j); the proof invokes Vsemirnov's factorisation of Chapman's matrix followed by cofactor expansion. The second result supplies a uniform exact congruence modulo p for the determinant underlying Conjecture 4.10(i), valid for any ordered half-system (modulo sign) and all u,v∈F_p, whose standard specialisation recovers the asserted square class.","tokens_in":1961,"tokens_out":537,"duration_ms":14898,"significance":"If the derivations hold, the results furnish explicit closed-form evaluations and a uniform modular congruence that resolve previously conjectural statements, thereby advancing the study of Legendre-symbol matrices. The explicit 2×2 inverse package and the half-system refinement constitute concrete, falsifiable contributions that build directly on Vsemirnov's factorisation and Sun's earlier T(d,p) evaluation.","major_comments":[{"comment":"The central step of the first result (abstract and the section containing the proof of the four-indeterminate evaluation) rests on extending Vsemirnov's factorisation of the determinate Chapman matrix directly to the four-indeterminate H and then performing an adjacent cofactor calculation that is asserted to produce precisely the claimed four-entry package without extra correction terms. The manuscript must supply an explicit verification that the indeterminate extension introduces no discrepancies altering the expressions involving (2/p), a'_p and b'_p.","section":"Proof of first result (Vsemirnov factorisation + cofactor expansion)"}],"minor_comments":[{"comment":"Clarify the precise definitions and ranges of the auxiliary quantities a'_p and b'_p at their first appearance, as they are used in the displayed formulas for det H and the inverse package.","section":null},{"comment":"In the statement of the second result, make explicit the dependence (or independence) on the choice of ordered half-system so that the uniformity claim is immediately verifiable from the displayed congruence.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the detailed report and positive evaluation of the significance of the results. We address the single major comment below.","responses":[{"response":"We agree that making the extension explicit would improve clarity. Vsemirnov's factorisation is an identity in the ring of polynomials over the indeterminates, and the four-indeterminate matrix H is obtained by the same substitution pattern as the original Chapman matrix; the cofactor expansion therefore carries over verbatim with no additional correction terms. Nevertheless, to address the request directly we will insert a short verification paragraph immediately after the statement of the factorisation, confirming by direct comparison of the (1,1) and (1,2) minors that the factors (2/p), a'_p and b'_p remain unaltered. This addition will be included in the revised manuscript.","revision_made":"yes","referee_comment":"[Proof of first result (Vsemirnov factorisation + cofactor expansion)] The central step of the first result (abstract and the section containing the proof of the four-indeterminate evaluation) rests on extending Vsemirnov's factorisation of the determinate Chapman matrix directly to the four-indeterminate H and then performing an adjacent cofactor calculation that is asserted to produce precisely the claimed four-entry package without extra correction terms. The manuscript must supply an explicit verification that the indeterminate extension introduces no discrepancies altering the expressions involving (2/p), a'_p and b'_p."}],"tokens_in":1478,"tokens_out":316,"duration_ms":16735,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main point is that the authors prove two concrete determinant evaluations attached to Sun's conjectures. They resolve the p≡1 mod 4 part of conjecture 4.8(i) by showing det H = leg(2/p)(b'_p - a'_p) together with the explicit 2x2 package for U^T H^{-1} U, and they supply a uniform exact congruence modulo p for the determinant in 4.10(i) that holds for any ordered half-system and all u,v in F_p.\n\nWhat is new is the four-entry inverse package and the half-system refinement; the square-class statements already followed from Sun's earlier T(d,p) work, but the exact expressions and uniformity do not. The paper does this by reducing the four-indeterminate determinant to the adjacent minor via Vsemirnov's factorisation of Chapman's matrix followed by cofactor expansion. The approach stays direct and derives the formulas independently rather than by fitting.\n\nThe proofs cite the external results cleanly and avoid circularity. The only soft spot worth checking is whether Vsemirnov's factorisation extends without adjustment to the four-indeterminate matrix H and whether the cofactor step produces exactly the claimed entries. The abstract indicates they carried out that calculation, so the full text needs to confirm no extra correction terms appear.\n\nThis is narrow work in combinatorial number theory on character matrices. Specialists tracking Sun's conjectures or similar determinant identities will get direct use from the formulas. It shows honest engagement with the literature and delivers verifiable progress on the stated claims.\n\nIt deserves peer review to verify the factorisation extension and the uniformity argument.","headline":"This paper gives explicit determinant formulas that settle the p≡1 mod 4 case of Sun conjecture 4.8(i) and a uniform mod-p congruence refining 4.10(i).","tokens_in":2402,"tokens_out":417,"would_cite":false,"duration_ms":25116,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Two determinant evaluations resolve the p≡1 mod 4 case of one Sun conjecture on Legendre symbol matrices and give an exact modular refinement for the other.","keywords":["Legendre symbol","determinant evaluation","Sun conjectures","Chapman matrix","Vsemirnov factorisation","half-system","congruence modulo p","number theory"],"falsifier":"Explicit computation, for a small prime p ≡ 1 mod 4 such as p = 5 or p = 13, of the four-indeterminate determinant and direct comparison against the stated formula leg(2/p)(b'_p - a'_p) together with the displayed 2-by-2 inverse matrix.","tokens_in":2712,"feed_emoji":"","tokens_out":938,"duration_ms":34815,"temperature":0.7,"pith_summary":"The paper proves explicit formulas for two determinants whose entries are built from the Legendre symbol. For the first, when p is congruent to 1 modulo 4, the determinant of the four-indeterminate matrix H equals the Legendre symbol (2/p) times the difference b'_p minus a'_p, together with a closed 2-by-2 expression for the inverse applied to the all-ones vector and the symbol vector. This settles the corresponding part of Conjecture 4.8(i) by reducing to the inverse of an adjacent minor via factorisation of Chapman's matrix. The second result supplies a uniform exact congruence modulo p for the determinant appearing in Conjecture 4.10(i); the congruence holds for every ordered half-system (up to sign) and every pair u, v in the prime field, and specialises to the asserted square class.","feed_headline":"Determinant formulas settle two Sun conjectures on Legendre matrices","feed_subtitle":"Explicit value for p≡1 mod 4 and uniform mod-p congruence for any half-system confirm the matrix claims.","key_machinery":"The four-indeterminate matrix H built from Legendre symbols, reduced via Vsemirnov's factorisation of Chapman's matrix to a four-entry inverse package for the adjacent minor [χ(j-k+1)]_{0≤j,k<(p-1)/2}.","core_discovery":"We prove that det H = leg(2/p)(b'_p - a'_p) and that U^T H^{-1} U equals the displayed 2-by-2 matrix whose entries involve leg(2/p) and the ratio (b'_p - a'_p - 1)/(b'_p - a'_p), thereby resolving the p≡1 mod 4 part of Conjecture 4.8(i). The proof proceeds from Vsemirnov's factorisation of Chapman's matrix followed by an adjacent cofactor calculation. Separately, the determinant underlying Conjecture 4.10(i) satisfies an exact congruence modulo p that is valid for any ordered half-system modulo sign and for all u, v in F_p; its standard specialisation recovers the asserted square class, which is also a consequence of Sun's earlier evaluation of T(d,p).","pith_inferences":["The exact half-system refinement may permit uniform statements in other conjectures that involve matrices of character values.","Because the square-class claim follows both from this congruence and from Sun's earlier T(d,p) evaluation, the two approaches are consistent on that coarser statement.","The cofactor technique used here could be tested on other low-rank extensions of the same Legendre-symbol matrices."],"forward_implications":["The p≡1 mod 4 part of Conjecture 4.8(i) is resolved by the explicit value det H = leg(2/p)(b'_p - a'_p) and the accompanying 2-by-2 matrix for U^T H^{-1} U.","The determinant in Conjecture 4.10(i) obeys a uniform exact congruence modulo p for any ordered half-system (modulo sign) and every u, v in F_p.","Standard specialisation of the second congruence recovers the asserted square class of the determinant.","The half-system independence (up to sign) of the second congruence supplies a refinement beyond the square-class statement alone."],"fun_headline_variants":["Determinant evaluations settle two Sun Legendre conjectures","Determinant proofs resolve Sun conjectures on Legendre symbols","Sun Legendre matrix conjectures confirmed by det evaluations","Exact det formulas settle two Sun conjectures on Legendre symbols"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"Vsemirnov's factorisation of Chapman's matrix applies directly to the four-indeterminate matrix and the adjacent cofactor calculation produces the claimed four-entry inverse package without extra correction terms.","fun_headline_variants_meta":{"raw":{"variants":["Determinant evaluations settle two Sun Legendre conjectures","Determinant proofs resolve Sun conjectures on Legendre symbols","Sun Legendre matrix conjectures confirmed by det evaluations","Exact det formulas settle two Sun conjectures on Legendre symbols"]},"model":"grok-4.3","cost_usd":0.007974,"raw_usage":{"total_tokens":3702,"prompt_tokens":810,"num_sources_used":0,"completion_tokens":60,"cost_in_usd_ticks":79737000,"prompt_tokens_details":{"text_tokens":810,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2832,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":810,"tokens_out":60,"duration_ms":21501,"temperature":1.0,"reasoning_tokens":2832,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-30T18:27:37.461726+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Explicit computation, for a small prime p ≡ 1 mod 4 such as p = 5 or p = 13, of the four-indeterminate determinant and direct comparison against the stated formula leg(2/p)(b'_p - a'_p) together with the displayed 2-by-2 inverse matrix.","supporting_citations":[],"review_version":2}