{"id":"9ea017c4-7d8d-4175-9bbb-0d839efa0293","arxiv_id":"1908.08447","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Twelve open cases from Strassler's table of circulant weighing matrices are proven nonexistent, and all proper weight-16 circulant weighing matrices are classified.","lead":"This math paper proves that twelve types of circulant weighing matrices, square grids of 0, +1 and -1 entries with orthogonal rows, cannot exist for previously open parameter pairs. It also completes the classification of the smallest irreducible such matrices of weight 16.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The five computer-eliminated nonexistence claims rest on Algorithm 1's exhaustiveness, but no code, data, or logs are provided, so one implementation bug could invalidate them.","rationale":"The reader's weakest assumption names exactly this unverified backtracking exhaustiveness, and I agree it is the load-bearing point. The other issues, such as the Proposition 5.1 typo and the terse finite checks, are real but addressable and do not threaten the central claim as directly. No code, data, or logs accompany the search, so the five computer-based results are currently unverifiable. A successful independent reimplementation would settle the matter; until then, conditional acceptance is appropriate. I considered whether Theorem 6.2's classification could be a more serious flaw, but the paper cites [8] for the odd-order classification and Type II reduction, and the construction it gives is explicit enough to test; the main gap remains the computer exhaust.","tokens_in":12106,"tokens_out":25944,"duration_ms":254413,"concrete_test":"Obtain or independently re-implement the Algorithm 1 search for the five computer-eliminated cases, using a separate codebase and randomized orbit and box order. First validate the implementation against positive controls, such as CW(7,4) and CW(63,16), to confirm it finds known solutions; then run it to completion for CW(144,49), CW(152,49), CW(160,49), CW(104,81), and CW(160,81). If the independent search also reports zero solutions and reproduces the paper's node counts, the concern is resolved. If it finds any CW, the corresponding nonexistence claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's headline nonexistence results for CW(144,49), CW(152,49), CW(160,49), CW(104,81), and CW(160,81) are produced solely by the backtracking search described in Section 5. The text asserts 'As stated, Algorithm 1 will find all CW(n,k)' but does not prove that the enumeration of folded-equation solutions is complete, does not specify the orbit tables or branch order used for each parameter set, and ships neither source code nor logs. The reported 15-day, 2.4-billion-node count for CW(144,49) is not independently auditable. Because a false negative in any one of these five searches would reduce the claimed twelve resolved cases to seven, the central claim depends on an unverified computational exhaust. The hand proofs and Theorem 6.2 are less exposed: the hand proofs are checkable, aside from the obvious ICW_2(91,64) typo in Proposition 5.1, and Theorem 6.2 derives from an existing classification in [8]. The single load-bearing gap is therefore the integrity and completeness of Algorithm 1's search.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies circulant weighing matrices CW(n,k) with n ≤ 200 and k ≤ 100, using group-ring methods and multiplier theorems. It claims nonexistence for twelve open cases in Strassler's table: seven by hand using orbit tables and folding arguments, and five more (CW(144,49), CW(152,49), CW(160,49), CW(104,81), CW(160,81), plus CW(182,64) via an ICW search) by a backtracking computer search over orbit assignments. The paper also characterizes the proper CW(n,16), proving that their orders are exactly 21, 31, 63, and 14m for all m ≥ 2.","tokens_in":12220,"tokens_out":17602,"duration_ms":155914,"significance":"If the proofs are correct, the paper resolves twelve previously open cases and completes the classification of proper circulant weighing matrices of weight 16. The orbit-table method is a useful systematic framework, and the hand proofs in Sections 3–4 are largely checkable. The claimed computer-eliminated cases, however, are not independently verifiable from the manuscript alone, which limits the reproducibility of the headline results.","major_comments":[{"comment":"The nonexistence of CW(144,49), CW(152,49), CW(160,49), CW(104,81), CW(160,81), and CW(182,64) rests entirely on the unreleased backtracking implementation described in Section 5. The manuscript provides pseudocode and node counts (e.g., 2.4 billion for CW(144,49)) but no code, input orbit tables, branch order, or logs, and the assertion that \"Algorithm 1 will find all CW(n,k)\" is not accompanied by a formal completeness proof. A false negative in any of these searches would invalidate six of the twelve headline nonexistence claims, so this is a load-bearing gap. Please make the implementation and search logs available, or provide a machine-checkable certificate and a rigorous completeness argument for the exact search space used.","section":"Section 5, Algorithm 1 and Table 9"},{"comment":"The pruning condition `else if rj = 0` appears to test the row-sum vector r at the moment column j is completed; based on the initialization and on Table 1, this should be `cj`. If the executed code follows the printed pseudocode, the search may prune incorrectly and be incomplete; if it does not, the pseudocode does not describe the computation actually performed. This inconsistency must be corrected before the computational results can be assessed.","section":"Section 5, Algorithm 1 pseudocode"},{"comment":"The proof states that the first two Z13 solutions, (0,3,3,0,-3) and (0,4,1,1,-3), \"may be quickly eliminated\" because a column sum of -9 is impossible, and then for the third solution it checks only one representative of the 12 choices. Since this proposition is presented as a hand proof, these finite checks should be written out explicitly; as it stands, the elimination is asserted rather than demonstrated, leaving the argument incomplete at a point that is load-bearing for the nonexistence of CW(143,81).","section":"Section 3.5, Proposition 3.5"}],"minor_comments":[{"comment":"The proof line \"Since there is no ICW_2(91,81)\" contradicts Table 9, which records zero for ICW_2(91,64); the weight 81 should be 64.","section":"Section 5, Proposition 5.1"},{"comment":"The sentence \"The seven cases resolved above leave 27 open cases\" is arithmetically inconsistent: Sections 3–4 resolve six cases (CW(110,81), CW(154,81), CW(130,81), CW(143,81), CW(143,36), and CW(132,81)). The count should be 28 open cases, or a seventh resolved case should be identified.","section":"Section 5, opening paragraph"},{"comment":"The phrase \"N has order 36 ≡ 0 (mod 1)2\" contains a typographical error; it should read \"(mod 12)\".","section":"Section 3.4, Proposition 3.4"},{"comment":"The caption ending \"Entries cm are for all m such that cm ≥ k\" is garbled; please clarify the notation for infinite families such as 2m, 7 and 14m, 21, 31, 63.","section":"Table 11 caption"}],"recommendation":"major_revision","confidential_remarks":"The reader's concern about the unreleased computer search is the main obstacle to acceptance. The authors should be asked to release the code and logs for the six computer-assisted eliminations, or to provide a formal completeness proof for Algorithm 1 and the exact per-case inputs. The hand-proof gaps and typographical issues are localized and fillable. The paper's topic and the significance of resolving open cases in Strassler's table fit the journal's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper settles twelve open parameter pairs in the circulant weighing matrix table. Six of those are handled by hand-checkable multiplier/orbit arguments, and the classification of proper CW(n,16) of even order as exactly 14m (Theorem 6.2) is a genuinely new structural result. The update to the status table and the pointer to an online database are also useful.\n\nThe soft spot is exactly what the stress-test flags: Algorithm 1 is only pseudocode. The claimed exhaustive eliminations of CW(144,49), CW(152,49), CW(160,49), CW(104,81), and CW(160,81) come with node counts but no code, data, or logs. The same applies to the ICW_2(91,64) count behind Proposition 5.1, which also contains a typo (it says ICW_2(91,81)). One false negative in any of these searches would reduce the twelve claimed results, so the headline rests on unverified computation. This is not a mathematical flaw in the method—multiplier theory plus orbit exhaustion is sound—but it is a real reproducibility gap for a journal.\n\nSome hand-proof eliminations are also terse. In Proposition 3.5, the 'quickly eliminated' first two solutions and the 12 choices after Table 7 are asserted rather than shown. An editor should ask for those to be expanded. Minor, but fixable. I also noticed a count inconsistency: 'the seven cases resolved above' should be six by my count. Not a mathematical issue, just a slip.\n\nOverall, the paper deserves a serious referee. The hand proofs are mostly rigorous, the classification theorem is nice, and the incremental table progress is real. I would send it out with a request that the authors release code, logs, and exact branch/pruning parameters for Algorithm 1, fix the typo, and expand the terse eliminations. If the artifacts are not provided, the computer-dependent claims should be marked as unverified.","headline":"Solid incremental progress on Strassler's table; the hand proofs are the real contribution, but the computer-eliminated cases need code and logs before the headline twelve nonexistence claims are fully auditable.","tokens_in":12833,"tokens_out":6956,"would_cite":false,"duration_ms":62183,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05B20","05B10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Twelve parameter pairs long open in the small-order table for circulant weighing matrices are proven to admit no such matrix.","keywords":["circulant weighing matrices","nonexistence proofs","multiplier theorems","group rings","orbit tables","proper circulant weighing matrices","exhaustive search","open cases table"],"falsifier":"Exhibit a $0,\\pm1$ circulant matrix with $WW^T=kI_n$ for any of the twelve parameter pairs, for instance a $\\mathrm{CW}(144,49)$, and that nonexistence claim falls. For the classification, a proper $\\mathrm{CW}(n,16)$ with $n\\notin\\{21,31,63\\}\\cup\\{14m:m\\ge2\\}$ would refute Theorem 6.2. A reader can also rerun an independent implementation of Algorithm 1: finding any matrix among the five computer-eliminated cases would overturn them.","tokens_in":11804,"feed_emoji":"🧮","tokens_out":14673,"duration_ms":121568,"temperature":0.7,"pith_summary":"The paper's goal is to prove that twelve parameter pairs left open in the 2018 update of the standard small-order table for circulant weighing matrices—square matrices with entries $0,\\pm1$, constant row weight $k$, and mutually orthogonal cyclic shifts—admit no solution. These cases include $\\mathrm{CW}(110,81)$ and $\\mathrm{CW}(143,36)$, and closing them reduces the table's unresolved entries from 34 to 22 over the range $n\\le200$, $k\\le100$. The proofs convert a candidate matrix into a group-ring polynomial, use multiplier theorems to force the nonzero entries to be unions of orbits, and then solve a small system of intersection-number equations that admits no compatible assignment; for the five hardest cases the same orbit framework is settled by an exhaustive backtracking search rather than by hand. The paper also completes the classification of proper weight-$16$ circulant weighing matrices, with orders exactly $21$, $31$, $63$, and $14m$ for every $m\\ge2$. If correct, the small-parameter table is sharply narrowed and the known boundary between existent and nonexistent circulant weighing matrices is redrawn.","feed_headline":"Twelve open cases of circulant weighing matrices are impossible","feed_subtitle":"Hand proofs and exhaustive search eliminate 12 parameter pairs; proper weight-16 orders are 21, 31, 63, and 14m.","key_machinery":"The load-bearing machinery is the orbit table for the multiplier group acting on $\\mathbb{Z}_n\\cong\\mathbb{Z}_d\\times\\mathbb{Z}_m$. Each cell lists the full-group orbits that project onto a given orbit in each factor; assigning an orbit to the positive set $P$, the negative set $N$, or neither fixes row sums $r_i$ and column sums $c_j$, which Lemma 2.5 requires to satisfy $\\sum b_i=s$ and $\\sum b_i^2=k=s^2$. The equations are small enough to solve by hand or short exhaust, and the table then shows that every alleged solution lifts to something whose group-ring product $A(X)A(X^{-1})$ is not $k$. When the multiplier group is too small for hand work but the search space is still finite, Algorithm 1 performs the same assignment exhaustively by recursing over orbits with the three options 'skip, add to $P$, add to $N$' and pruning partial row and column sums.","core_discovery":"The central discovery is a set of twelve nonexistence theorems: no circulant weighing matrix exists for $\\mathrm{CW}(110,81)$, $\\mathrm{CW}(154,81)$, $\\mathrm{CW}(130,81)$, $\\mathrm{CW}(143,81)$, $\\mathrm{CW}(143,36)$, $\\mathrm{CW}(132,81)$, $\\mathrm{CW}(144,49)$, $\\mathrm{CW}(152,49)$, $\\mathrm{CW}(160,49)$, $\\mathrm{CW}(104,81)$, $\\mathrm{CW}(160,81)$, or $\\mathrm{CW}(182,64)$. The arguments model a putative matrix as $A(X)$ in the integer group ring of the cyclic group, where the condition is $A(X)A(X^{-1})=k$; multiplier theorems turn the support into a union of orbits, and folding by coprime divisors $d,m$ of $n$ yields equations $\\sum b_i=s$ and $\\sum b_i^2=k=s^2$ that the orbit structure must satisfy. Several of these cases are eliminated by hand from those equations, while the remaining five are eliminated by the paper's exhaustive orbit search. The classification result is that every proper $\\mathrm{CW}(n,16)$—one whose group-ring element is not a multiple of a smaller circulant weighing matrix—has order $21$, $31$, $63$, or $14m$ for some $m\\ge2$, and all of those orders occur.","pith_inferences":["The same orbit-table and search machinery could be exported to the remaining open cases that have a nontrivial multiplier group; the bottleneck there is not search size but the absence of a multiplier theorem strong enough to start the table, so new multiplier results would have the largest effect.","The weight-16 classification suggests a template for the next square weights: if the analogous complete classification for weight 25 were verified, the same Type II construction and finiteness results would determine exactly which orders are proper, and the same $14m$ pattern may reappear for other weights.","Because the five computer-eliminated cases come with no code, logs, or certificates, an independent reimplementation of Algorithm 1 is a direct way to test the paper's computational claims; the reported 2.4-billion-case run for $\\mathrm{CW}(144,49)$ gives a concrete scale for that check."],"forward_implications":["The 2018 open-case table for $n\\le200$, $k\\le100$ goes from 34 unresolved parameter pairs down to 22, with no existence entry disturbed.","The infinite family statement in Proposition 3.2 rules out every $\\mathrm{CW}(11m,81)$ with $\\gcd(33,m)=1$ and $3$ self-conjugate modulo $m$, not just the tabulated cases.","Every proper $\\mathrm{CW}(n,16)$ now has a known order: $21$, $31$, $63$, or $14m$ for $m\\ge2$, so orders outside this set are either improper or nonexistent.","For parameters with a sufficiently large multiplier group, Algorithm 1 is a decision procedure: it either produces a circulant weighing matrix or proves none exists, as done for $\\mathrm{CW}(144,49)$, $\\mathrm{CW}(152,49)$, $\\mathrm{CW}(160,49)$, $\\mathrm{CW}(104,81)$, and $\\mathrm{CW}(160,81)$."],"supporting_citations":[{"why":"Supplies the original small-order table of existence results that this paper's nonexistence proofs are measured against.","marker":"[21]"},{"why":"Gives the 2018 table version listing the 34 open cases that fix the paper's target parameters.","marker":"[22]"},{"why":"States the self-conjugate group-ring vanishing theorem used to force folded intersection numbers to be trivial.","marker":"[14]"},{"why":"Gives the prime-power multiplier theorem used to restrict positive and negative supports to multiplier orbits.","marker":"[3]"},{"why":"Gives the generalized multiplier theorem used when the weight is not a prime power, as for CW(143,36).","marker":"[5]"},{"why":"Supplies McFarland's multiplier theorem for integer circulant weighing matrices, the basis of the contracted-case reductions.","marker":"[18]"},{"why":"Classifies weight $2^{2t}$ circulant weighing matrices and supplies the Type II construction that underlies the even proper weight-16 orders.","marker":"[8]"},{"why":"Classifies proper CW(n,4), used in the proof of Theorem 6.2 to rule out alternative Type II constructions.","marker":"[11]"}],"fun_headline_variants":["Nonexistence proofs eliminate 12 open cases of circulant weighing matrices","Hand proofs and search close 12 circulant weighing matrix cases","Twelve circulant weighing matrix cases proven impossible","New theorems rule out 12 open circulant weighing cases"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that Algorithm 1's backtracking search is exhaustive and its implementation is free of false negatives, because the five cases $\\mathrm{CW}(144,49)$, $\\mathrm{CW}(152,49)$, $\\mathrm{CW}(160,49)$, $\\mathrm{CW}(104,81)$, and $\\mathrm{CW}(160,81)$ are settled only by that search, whose code, logs, and certificate outputs are not included in the paper.","fun_headline_variants_meta":{"raw":{"variants":["Nonexistence proofs eliminate 12 open cases of circulant weighing matrices","Hand proofs and search close 12 circulant weighing matrix cases","Twelve circulant weighing matrix cases proven impossible","New theorems rule out 12 open circulant weighing cases"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000864,"raw_usage":{"total_tokens":3764,"prompt_tokens":979,"completion_tokens":2785,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":595,"completion_tokens_details":{"reasoning_tokens":2716}},"tokens_in":595,"tokens_out":2785,"duration_ms":19758,"temperature":1.0,"reasoning_tokens":2716,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:39:56.865511+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a $0,\\pm1$ circulant matrix with $WW^T=kI_n$ for any of the twelve parameter pairs, for instance a $\\mathrm{CW}(144,49)$, and that nonexistence claim falls. For the classification, a proper $\\mathrm{CW}(n,16)$ with $n\\notin\\{21,31,63\\}\\cup\\{14m:m\\ge2\\}$ would refute Theorem 6.2. A reader can also rerun an independent implementation of Algorithm 1: finding any matrix among the five computer-eliminated cases would overturn them.","supporting_citations":[{"cited_title":"Strassler","cited_arxiv_id":null,"evidence_quote":"Supplies the original small-order table of existence results that this paper's nonexistence proofs are measured against."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the 2018 table version listing the 34 open cases that fix the paper's target parameters."},{"cited_title":"Jungnickel","cited_arxiv_id":null,"evidence_quote":"States the self-conjugate group-ring vanishing theorem used to force folded intersection numbers to be trivial."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the prime-power multiplier theorem used to restrict positive and negative supports to multiplier orbits."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the generalized multiplier theorem used when the weight is not a prime power, as for CW(143,36)."},{"cited_title":"McFarland","cited_arxiv_id":null,"evidence_quote":"Supplies McFarland's multiplier theorem for integer circulant weighing matrices, the basis of the contracted-case reductions."},{"cited_title":"Arasu, K.H","cited_arxiv_id":null,"evidence_quote":"Classifies weight $2^{2t}$ circulant weighing matrices and supplies the Type II construction that underlies the even proper weight-16 orders."},{"cited_title":"Eades and R.M","cited_arxiv_id":null,"evidence_quote":"Classifies proper CW(n,4), used in the proof of Theorem 6.2 to rule out alternative Type II constructions."}],"review_version":1}