{"id":"2138ddca-3ded-4949-9ceb-4e0bc1bfe69e","arxiv_id":"2608.09149","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper settles open questions on core quandle ranks and idempotents in quandle rings, and introduces zero-divisor graphs for these rings.","lead":"This paper answers several open questions about quandles and their rings, including a counterexample to the proposed rank formula for core quandles of dihedral groups, and shows that some quandle rings have only trivial idempotents. It also introduces zero-divisor graphs for quandle rings and constructs a polynomial that encodes a family of commutative quandles.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Even-n generation claim in Proposition 2.2 is asserted, not proved; it is true and easily verified, so the rank result stands but the exposition needs a fill-in.","rationale":"The reader's weakest-assumption identification is correct: the even-n part of Proposition 2.2 leaves the main generation step unproved. My own check shows the claim is true: the invariance of type and parity under left multiplication forces one representative from each of the four orbits, and the affine structure R_i*R_j=R_{2j-i} makes one even and one odd rotation (and similarly one even and one odd reflection) generate the full rotation and reflection sets. So the rank theorem itself is sound, but the manuscript should replace the sentence 'Theorem 2.1 (Case 2) implies...' with an explicit argument. The paper also contains a genuinely false statement in Corollary 7.2, since for p=5, n=2 the element S=x0+x1 satisfies S^2=2S, not S^2=S; however, this error does not bear on Proposition 2.2, which is the strongest claim under review. The reader's CONDITIONAL verdict remains appropriate: the central rank result is correct but needs proof repair, and the false corollary should be corrected, for example by requiring p|n as in Proposition 7.1.","tokens_in":30162,"tokens_out":17457,"duration_ms":166988,"concrete_test":"Compute, for n=4 and n=6, the full subquandle generated by {a^0, a^1, b, ab} in Core(D_n) using x*y = y x^{-1} y, and check membership of all 2n group elements; also confirm the invariance rule that x*y has the same type and same parity as x for every y. If any orbit representative is not generated, the upper bound rk(Core(D_n)) ≤ 4 in Proposition 2.2 fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Proposition 2.2 for even n is incomplete at exactly the point where the rank upper bound is established. After Theorem 2.1 gives four orbits O1,...,O4, the paper asserts without derivation that (i) no set of fewer than four elements can generate Core(D_n) because some orbit is omitted, and (ii) one element from each orbit generates the whole quandle. Both assertions are true, but neither is immediate from 'carefully examining' as written. The missing mechanism is that for Core(D_n), the type (rotation/reflection) and the parity of the left operand are invariant under the operation: for x=a^i, x*y is always a rotation congruent to i mod 2 as y ranges over all elements, and for x=a^i b, x*y is always a reflection congruent to i mod 2. Hence an element of a given orbit can only be produced as a left product with a generator from that same orbit, which makes four representatives necessary. For sufficiency, within the two rotation orbits the operation is R_i*R_j=R_{2j-i}; with representatives a^0 and a^t (t odd), the products a^0 * a^t = a^{2t}, a^t * a^0 = a^{-t}, and repeated doubling generate all even and odd rotations because gcd(2,n)=2; the same argument applies to the two reflection orbits. Thus rk(Core(D_n))=4 holds, but the paper should supply this verification rather than leave it as an assertion.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies core quandles of dihedral groups and algebraic properties of quandle rings. It claims a counterexample to the conjecture rk(Core(G)) = rk(G)+1 by proving rk(Core(D_n)) = rk(D_n)+2 for n>2, determines idempotents in several quandle rings (including solving two systems left open in earlier work), analyzes units and nilpotents in extended quandle rings, introduces zero-divisor graphs for extended quandle rings and observes a mirror symmetry in their in- and out-degrees, discusses prime/semi-prime quandle rings, and constructs a bivariate polynomial over Z_{2n+1}[Q] that realizes a commutative quandle of prime order.","tokens_in":30340,"tokens_out":21391,"duration_ms":200479,"significance":"If the main results stand, the paper makes a useful contribution: Proposition 2.2 gives a clean counterexample to a published conjecture, and Section 4 settles two previously open systems of equations on idempotents. The zero-divisor graph computations are concrete and checkable, and the mirror symmetry observation is a genuinely new phenomenon that could spur further work. The paper is also commendably explicit: it reports Gröbner bases, multiplication tables, and degree tables rather than only existence statements. However, the rank proof has a load-bearing gap in the even case, and Section 7 contains a false corollary and an unjustified ideal computation, so the paper is not yet acceptable in its current form.","major_comments":[{"comment":"The even-n case of the proof is incomplete at the point where the upper bound on rk(Core(D_n)) is established. After Theorem 2.1 gives four orbits, the text asserts, without derivation, that fewer than four elements cannot generate Core(D_n) and that any choice of one element from each orbit generates the whole quandle. Neither assertion is immediate from the displayed orbit analysis. A correct proof needs the invariance observation that the type (rotation/reflection) and the parity of the exponent of the left operand are preserved by the quandle operation, so no orbit can be omitted, and then a doubling argument (or an equivalent) showing that, for example, a^0 and a^t with t odd generate all rotations of both parities. The odd-n case also silently assumes that two suitably chosen rotations generate the full rotation orbit; this is true since 2 is invertible modulo odd n, but it should be stated. As written, the central rank theorem is not fully proved.","section":"§2, Proposition 2.2"},{"comment":"Corollary 7.2 is false as stated. Let Q be the trivial quandle of order 3 and take p=5. Writing S=x0+x1+x2, the trivial-quandle multiplication gives S^2=3S and more generally S^k=3^{k-1}S. Then S^3=9S≡4S, but (S^3)^2=S^6=3^5S≡3S, which is not 4S in Z_5. Hence S^3 is not an idempotent. The intended statement becomes true only with an additional hypothesis such as p|n, under which S^2=0 and the nth power is 0. The attribution 'which also appeared in [3]' should be checked against the exact statement in that reference.","section":"§7, Corollary 7.2"},{"comment":"The proof of Proposition 7.1 does not justify the assertion that the two-sided ideal generated by S=x0+...+x_{n-1} is exactly {αS : α∈Z_p}. The displayed identity verifies only products of the form S·(Σα_i x_i). For arbitrary A, the product A·S involves the row sums of the quandle multiplication table, which are not shown to be scalar multiples of S unless the quandle is Latin or has some additional balanced-row property. Consequently the square of a general element of the ideal is never computed, and the conclusion I^2=0 is not established for arbitrary quandles. The proposition may be salvageable by a different argument or by restricting the class of quandles, but the present proof is incomplete.","section":"§7, Proposition 7.1"}],"minor_comments":[{"comment":"The rational root candidates for the displayed one-variable polynomial in β4 are the integer divisors of 6, namely ±1, ±2, ±3, ±6, not ±1, ±2, ±3, ±4. The proof should either replace the list or explicitly check the missing candidates ±6.","section":"§4, C5 system after (4.4)"},{"comment":"The line 'in-degree = 2.2^{n-1}-1' should read 2^n-1, and the notation 'x0+x1 x2' in Remark 6.2 is missing commas and likely a missing plus sign.","section":"§6.1, Case 3"},{"comment":"In the displayed definition of L_{(X,2n)}(X), the product is written with 'i≠n−1'; it should be 'i≠2n'.","section":"§8, Lagrange basis"},{"comment":"There are numerous typos and inconsistent notations, including 'commutaitve', 'whcih', 'zero-divsor', 'Proporition', 'ε' versus 'ϵ', and the duplicated Propositions 1.8 and 3.3. A careful copyedit is needed.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The Section 7 issues are not merely cosmetic: Corollary 7.2 is false without an added hypothesis, and Proposition 7.1's proof is incomplete for non-Latin quandles. Both are fixable, but the authors should also re-examine whether the claimed attribution to [3] is accurate. For reproducibility, I would encourage the authors to include the Mathematica code or a verification script for the Gröbner basis computations, since the printed bases are large and no independent check is provided."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Val,\n\nHere's my read on arXiv:2608.09149. The paper is a genuine contribution to the quandle-ring literature. It settles two open questions from Bardakov–Elhamdadi on idempotents in Z[R5] and Z[C5] by explicit Gröbner basis computations, gives a counterexample to Bardakov–Fedoseev's rk(Core(G)) = rk(G)+1 conjecture, and opens up zero-divisor graphs for quandle rings. Those are real, new results. The Gröbner basis work is reproducible in principle, the orbit count for Core(D_n) is plausible, and the main line holds up.\n\nThe soft spots are real but localized. First, Corollary 7.2 is false as stated. It claims S^n is idempotent in Z_p[Q] for any prime p and any quandle of order n. Try p=5, n=2: S^2 = 2S, and (2S)^2 = 4S^2 = 8S ≡ 3S ≠ 2S in Z_5, so not idempotent. The missing hypothesis is likely p|n, as in Proposition 7.1, but that makes the statement trivial since S^n = 0. The authors need to fix this; it's not load-bearing, but a false corollary in print is embarrassing.\n\nSecond, the even-n case of Proposition 2.2 asserts, without proof, that one element from each of the four orbits generates all of Core(D_n). The stress-test note supplies the missing argument: a parity invariant forces four generators, and a doubling argument on the rotation orbits gives sufficiency. I verified it works. So the result is correct, but the proof as written has a gap that a referee should flag.\n\nThird, Section 6 is largely computational. The mirror symmetry in the zero-divisor graphs is proved for the trivial quandle case, but for the Joyce quandle it is observed from tables, not proven. The authors should explicitly separate proved from observed; otherwise readers may over-credit the generality.\n\nThe rest is fine. The unit and nilpotent characterizations are routine but correct. The Lagrange interpolation polynomial is a straightforward construction, correctly executed. Citation practice is honest; the open problems are attributed, and the props from [4] are cited rather than reproved, which is acceptable.\n\nBottom line: the paper deserves a serious specialist referee. The core results are sound; the fixes needed are exposition and one corollary, not new mathematics. I would send it out with a request for minor-to-moderate revision. I wouldn't bring it to our reading group unless you're already in quandle-land; it's not in my direct lane, though I'd cite the rank counterexample if I worked nearby.","headline":"Settles two open questions and gives a valid counterexample to the rank conjecture, but needs a corrected Corollary 7.2 and a filled gap in Proposition 2.2.","tokens_in":30967,"tokens_out":4539,"would_cite":false,"duration_ms":37658,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K12","17A01"],"pacs":[],"model":"deepseek-v4-flash","headline":"Core quandles of dihedral groups have rank 4, refuting the proposed formula rk(Core(G)) = rk(G)+1.","keywords":["Quandle","Core quandle","Quandle ring","Zero-divisor graph","Idempotent","Nilpotent","Nil clean ring","Prime ring"],"falsifier":"For n=4, compute the four orbits of Core(D_4) under the operation x*y = y $x^{{-1}}$ y, choose one representative from each orbit, and check whether the subquandle they generate has all eight elements of D_4; if it does not, the claimed rank 4 is too small.","tokens_in":1803,"feed_emoji":"🪢","tokens_out":3851,"duration_ms":129976,"temperature":0.7,"pith_summary":"The paper investigates core quandles of groups and the rings built from quandles. Its headline claim is that the core quandle of the dihedral group D_n of order 2n has rank 4 = rk(D_n)+2 for every n>2, because Core(D_n) splits into two orbits when n is odd and four when n is even; this disproves the conjecture that rk(Core(G)) always equals rk(G)+1. The paper also settles two open idempotent questions, showing that the integer quandle rings Z[R_5] and Z[C_5] have no nontrivial idempotents of augmentation 1, and gives examples showing the quandle ring of the core of an infinite field can carry nontrivial idempotents. It introduces zero-divisor graphs for extended quandle rings, computes them over F_2 for trivial, Joyce, and dihedral quandles, and proves a mirror symmetry in in-degrees and out-degrees. Additional results cover units, nilpotents, the non-nil-clean property, prime and semi-prime quandle rings, and a bivariate Lagrange interpolation polynomial that determines commutative quandles of prime order.","feed_headline":"Dihedral core quandles break a rank conjecture","feed_subtitle":"Their rank is 4, not the conjectured 3, and two idempotent questions are settled","key_machinery":"The load-bearing machinery is the explicit multiplication rule of Core(D_n): with elements written a^i b^j and operation x*y = y $x^{{-1}}$ y, the parity of n decides whether the map j ↦ 2j-i permutes residues modulo n (odd n) or only permutes them modulo n/2 (even n). That decides the number of orbits and lets the rank proof run on subquandle generation. For the ring-theoretic results, the paper works in the augmentation ideal with basis E_i = e_i - e_0, whose multiplication table (Lemma 3.5) converts idempotent equations into polynomial systems solved by Gröbner bases. For units, zero-divisors, and nilpotents it uses the extended quandle ring S = k[Q] ⊕ ke with e central, and characterises units and zero-divisors by solvability of a small linear system. The final construction uses bivariate Lagrange interpolation in (Z_{2n+1}[Q])[X,Y] to write a polynomial f whose values f(a,b) reproduce the commutative quandle operation e_{(n+1)(a+b)}.","core_discovery":"The paper's central discovery is a counterexample to a proposed rank formula. For n>2, rk(Core(D_n)) = 4 = rk(D_n)+2, so the answer to the question posed in [5]—whether rk(Core(G)) always equals rk(G)+1—is no: the core quandle can require two more generators than the group. The orbit structure, two orbits for odd n and four for even n, is what forces the extra generator. Alongside this, the paper proves that the only idempotents of augmentation 1 in Z[R_5] and Z[C_5] are the trivial basis elements, so those rings have no nontrivial idempotents of that augmentation, and it exhibits infinite-field core quandles with nontrivial idempotents, contradicting another conjecture. In the zero-divisor graphs of extended quandle rings over F_2 it finds a mirror symmetry: a vertex with in-degree i and out-degree o is paired with a vertex with in-degree o and out-degree i, and the paired elements add to a non-zero-divisor.","pith_inferences":["The mirror pairs in the paper all add to 1 in the extended ring, suggesting the general symmetry is the map u -> u+1; proving that for all finite fields would extend the F_2 computation to arbitrary characteristic.","The even-n rank proof's gap suggests a direct computer test of Core(D_4) and Core(D_6) would confirm or refute the orbit-generation step, and similar orbit methods could be tried on other two-generator groups to see how widely the rank jump rk(G)+2 occurs.","Because the polynomial systems for Z[R_5] and Z[C_5] were solved by Gröbner bases, the same pipeline can be applied to R_p and C_p for other primes to classify nontrivial idempotents systematically.","The Lagrange interpolation result connects quandle-ring-valued polynomials with local permutation polynomials and latin squares, a direction the paper mentions only in motivation; one testable extension is whether every finite latin quandle admits such a polynomial over its quandle ring."],"forward_implications":["The proposed formula rk(Core(G)) = rk(G)+1 is false for dihedral groups, so a correct general statement about core-quandle ranks must depend on more than the group's rank.","The quandle rings Z[R_5] and Z[C_5] have no nontrivial idempotents with augmentation 1, so idempotent-based knot invariants built from these rings must use augmentation-0 elements or other coefficient rings.","Since latin core quandles of odd-order groups have nontrivial idempotents without containing a trivial subquandle of order greater than 1, subquandle criteria cannot characterise nontrivial idempotents in quandle rings.","Over F_2, the zero-divisor graphs of extended quandle rings of trivial and Joyce quandles obey a mirror symmetry between in-degrees and out-degrees, and mirror pairs sum to a non-zero-divisor.","If p divides |Q| then Z_p[Q] is not semi-prime and not prime; moreover the bivariate polynomial f(X,Y) evaluates to exactly the elements of the commutative quandle on Z_{2n+1}."],"supporting_citations":[{"why":"Poses the conjecture rk(Core(G)) = rk(G)+1 that Proposition 2.2 refutes.","marker":"[5]"},{"why":"Supplies the unsolved polynomial systems for nontrivial idempotents in Z[R_5] and Z[C_5] that Section 4 solves.","marker":"[2]"},{"why":"Provides the subquandle-to-idempotent proposition used in Examples 3.1 and 3.4 to get nontrivial idempotents.","marker":"[4]"},{"why":"Identifies idempotents of trivial quandle rings, used in the nil-clean result and in the study of units.","marker":"[6]"},{"why":"Introduces quandle rings and the fact that the augmentation ideal squares to zero for trivial quandles, used in unit and nilpotent computations.","marker":"[7]"},{"why":"Shows core quandles of odd-order groups are latin, used to disprove the converse of the subquandle idempotent criterion.","marker":"[13]"}],"fun_headline_variants":["Core(D_n) rank 4 refutes rank conjecture","Idempotent questions settled for R5 and C5","Zero-divisor graphs exhibit mirror symmetry","Trivial quandle rings fail nil clean test","Polynomial identifies odd-order commutative quandles"],"cache_read_input_tokens":33024,"weakest_assumption_plain":"For even n, the rank-4 proof assumes, without a complete derivation, that one element chosen from each of the four orbits of Core(D_n) generates the entire quandle.","fun_headline_variants_meta":{"raw":{"variants":["Core(D_n) rank 4 refutes rank conjecture","Idempotent questions settled for R5 and C5","Zero-divisor graphs exhibit mirror symmetry","Trivial quandle rings fail nil clean test","Polynomial identifies odd-order commutative quandles"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000744,"raw_usage":{"total_tokens":3364,"prompt_tokens":1035,"completion_tokens":2329,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":651,"completion_tokens_details":{"reasoning_tokens":2256}},"tokens_in":651,"tokens_out":2329,"duration_ms":20732,"temperature":1.0,"reasoning_tokens":2256,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T22:39:11.366427+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For n=4, compute the four orbits of Core(D_4) under the operation x*y = y $x^{{-1}}$ y, choose one representative from each orbit, and check whether the subquandle they generate has all eight elements of D_4; if it does not, the claimed rank 4 is too small.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Poses the conjecture rk(Core(G)) = rk(G)+1 that Proposition 2.2 refutes."},{"cited_title":"Idempotents and Powers of Ideals in Quandle Rings","cited_arxiv_id":"2601.07057","evidence_quote":"Supplies the unsolved polynomial systems for nontrivial idempotents in Z[R_5] and Z[C_5] that Section 4 solves."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Identifies idempotents of trivial quandle rings, used in the nil-clean result and in the study of units."},{"cited_title":"Bardakov, I","cited_arxiv_id":null,"evidence_quote":"Introduces quandle rings and the fact that the augmentation ideal squares to zero for trivial quandles, used in unit and nilpotent computations."},{"cited_title":"Spaggiari and M","cited_arxiv_id":null,"evidence_quote":"Shows core quandles of odd-order groups are latin, used to disprove the converse of the subquandle idempotent criterion."}],"review_version":1}