{"id":"9976daf4-ee61-4b0f-b3a5-34e4ca6b7b41","arxiv_id":"2504.15980","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New Scarpis-type constructions produce Butson matrices of orders n(n-1) and n(n/2-1) from smaller Butson matrices.","lead":"A mathematics paper gives new recipes for building Butson matrices, grids of complex roots of unity whose rows are perpendicular, from smaller Butson matrices. The recipes generalize a 1898 construction and may create new infinite families, but their reach depends on a disputed assumption about sets of Latin squares.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The constructions require complete sets of LSESC, which the paper itself proves are conjugate to MOLS; self-cited Theorem 2.6 of [5] would therefore produce a complete set of MOLS of order 6, contradicting the known fact that no such set exists.","rationale":"The reader and I identify the same weakest assumption. The construction theorems are conditional: given a complete tensor set of LSESC, the block matrices are shown to have orthogonal rows, and the key cancellations in Theorem 1, including the single-coincidence use of the LSESC property, are consistent. The severity comes from the universal-existence claim. The paper itself establishes LSESC iff MOLS and N'(n)=N(n), then invokes [5, Thm 2.6] to get complete LSESC for every order. For order 6 this implies the existence of 5 MOLS of order 6, which is known false and is even flagged by the paper as outside the verified range. Thus the unconditional or Corollary 3 version of the central claim is unsupported. This does not invalidate the conditional constructions for cases where the LSESC input genuinely exists, such as prime-power orders, but it demands restating the theorems as conditional and deleting or correcting Corollary 3. The reader's CONDITIONAL verdict is the right level; no adjustment is needed. There is also an apparent order-formula issue in Corollary 3 under the natural reading n=2(2r+1), but the LSESC existence issue is the load-bearing defect.","tokens_in":14943,"tokens_out":13736,"duration_ms":119700,"concrete_test":"Attempt to construct, by following Theorem 2.6 of [5], a complete set of 5 LSESC of order 6, for example by direct backtracking search over Latin squares of order 6, and then apply the paper's own conjugation (interchange symbol and row indices) to the resulting squares. If no such set is found, or if conjugation fails to produce 5 MOLS, the universal-existence premise fails; this directly invalidates Corollary 3 at r=3 and reduces Theorems 1-2 to conditional statements for orders whose LSESC input is actually supplied.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2 states N'(n)=N(n) and says LSESC sets are conjugate to MOLS, then notes that complete MOLS of order n exist only conjecturally for prime powers and are known not to exist for order 6. The unconditional applicability of Theorems 1 and 2 depends entirely on the existence of complete tensor LSESC sets of orders n-1 and n/2-1. The only support for existence for every required order is the self-cited Theorem 2.6 of [5], which is said to construct n-1 LSESC of every order n. If that theorem were true for n=6, conjugacy would give a complete set of 5 MOLS of order 6, contradicting the paper's own statement that the conjecture is verified for orders at most 6 and the known result N(6)=1. This is an internal inconsistency, not merely a departure from consensus. In particular, Theorem 2 with n=14 requires a complete tensor LSESC set of order 6, so Corollary 3 cannot hold for all r under the natural reading n=2(2r+1). The conditional theorems themselves appear row-orthogonal, but the advertised universal scope and Corollary 3 do not follow.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes two conditional constructions of Butson matrices. Theorem 1 constructs a BH(m,(n-1)n) matrix from a BH(m,n) matrix, given a complete tensor set of LSESC of order n-1. Theorem 2 constructs a BH(m,(n/2-1)n) matrix from a BH(m,n) matrix in A(1,2)(m,n), given a complete tensor set of LSESC of order n/2-1. Two-input versions and applications to Hadamard matrices are also given. The proofs verify row orthogonality directly. The paper further claims that these constructions apply for every n because the author's earlier work [5] supplies complete sets of LSESC of every order, and it states Corollary 3 asserting the existence of certain matrices from Fourier inputs for all r.","tokens_in":15195,"tokens_out":26208,"duration_ms":210042,"significance":"Conditional on the existence of the required LSESC inputs, the constructions appear algebraically sound and provide a new, potentially useful method for enlarging Butson orders, with explicit counts of output matrices. The Hadamard variants and the use of Fourier matrices as inputs are of interest. However, the paper's universal claims rest on a self-cited existence result that is internally inconsistent with the known non-existence of complete MOLS of order 6, and the proofs of Theorem 2 contain an unstated assumption about the entries of the C1 row. These issues materially affect the advertised scope.","major_comments":[{"comment":"The unconditional applicability of Theorems 1 and 2 is claimed on the basis of the self-cited Theorem 2.6 of [5], which is said to construct a complete set of LSESC of size n-1 for every n. This contradicts the facts stated earlier in the same section: the paper asserts N'(n)=N(n) and that the complete-MOLS conjecture is verified for orders at most 6, so a complete set of 5 LSESC of order 6 would conjugate to a complete set of 5 MOLS of order 6, contradicting N(6)=1. Consequently, Theorem 2 with n=14 requires a complete tensor LSESC set of order 6, which cannot exist under the paper's own assumptions, and Corollary 3 cannot hold for all r as stated. The authors should either supply a valid existence proof for the required LSESC sets, or explicitly restrict Theorems 1-2 and Corollary 3 to orders for which complete LSESC sets are known to exist (e.g., prime powers), and remove the conflicting claims about [5].","section":"Section 2 and Section 4, Corollary 3"},{"comment":"The proof of Theorem 2 uses 'xlxl = 1 for all l' to cancel the factors x_l^2 in the inner products of rows of Gamma_k. The row (x1 ... xn) is only required by condition C1 to be a row of H with the alternating-sign row also present; its entries are arbitrary m-th roots of unity and are not necessarily +-1. Unless the construction explicitly chooses (x1 ... xn) to be the all-ones row or the {1,-1} row from condition C2, the factorization x_l^2=1 is unjustified and the orthogonality argument for case 3 does not go through. Please state this choice, prove it, or restrict Theorem 2 to a subclass where it holds.","section":"Theorem 2, case 3, Eq. (4.10)"}],"minor_comments":[{"comment":"The abstract states that both constructions work when n and m are even, but Theorem 1 does not require m even; only Section 4's construction uses the evenness through Lemma 1. Please correct or qualify the abstract.","section":"Abstract"},{"comment":"The sentence 'j^2 = j' is a typo; for j = e^{2*pi*i/3}, j^2 = j^{-1}, not j. Please correct.","section":"Example 1"},{"comment":"The expression 'BH(2(2r + 1), 2r+1(2r + 1))' appears to have a missing superscript and does not match the output order n(n/2-1) = 4r(2r+1) when n=2(2r+1). Please correct the formula and clarify the notation.","section":"Corollary 3"},{"comment":"The count '|MOLS(n-1)| * |BH(m,n)|^2 * n' is stated without proof; since the constructions depend on complete tensor LSESC sets, the count should be justified in terms of the number of such sets and the choices of rows and columns.","section":"Section 3, counting paragraph"},{"comment":"There are numerous typos and typesetting errors, including 'dimenssions' in the Introduction, 'imputed' instead of 'input' in Section 4, and garbled matrix displays in Lemma 2. A thorough proofread is needed.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper's main theorems are conditional and appear to be correct in their algebraic core. The major problem is the reliance on a self-cited existence theorem from [5] that is almost certainly false in view of the MOLS order-6 obstruction; this internal inconsistency must be resolved before publication. The technical gap in Theorem 2 regarding x_l^2=1 can likely be fixed by restricting to the normalized case used in the examples, but as stated the proof is incomplete. The construction results may be salvageable as conditional statements for orders where complete LSESC sets exist."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nYou should know this paper has a real conditional construction hidden behind an overblown universal claim. Theorems 1 and 2 give explicit block matrices that are BH(m,(n-1)n) and BH(m,(n/2-1)n) whenever you have a complete set of LSESC of the appropriate order and, for Theorem 2, an input matrix in A(1,2)(m,n). The row-orthogonality proofs are straightforward and, as far as I can tell, correct. I checked the four cases in Theorem 1 and the analogous ones in Theorem 2; the use of the LSESC property in the cross-block case is exactly what you'd need. The two-input variants in Corollaries 1 and 4 are a genuine extension of [8], which only handles the single-input, prime-power case. The tensor formalism for LSESC is a little heavy but it works.\n\nThe soft spot is the existence problem. The paper claims the construction applies to all n because [5, Theorem 2.6] allegedly produces n-1 LSESC of every order n. But Section 2 itself says LSESC are conjugate to MOLS and that a complete set of MOLS of order 6 does not exist (the conjecture is verified through order 6). If Theorem 2.6 were true for n=6, it would give 5 LSESC, hence 5 MOLS of order 6, contradicting N(6)=1. That is an internal inconsistency in the manuscript, not just a gap. The author needs to either restrict the theorems to orders for which complete LSESC sets exist or provide a correct existence proof that is compatible with MOLS bounds. As a consequence, Corollary 3 is false for r=3 (n=14), where the construction would require a complete LSESC set of order 6. The order formula in Corollary 3 also appears wrong: for n=2(2r+1), the construction gives order 4r(2r+1), not the displayed value.\n\nThe paper deserves a serious referee because the conditional construction is novel and likely correct, and the flaws are fixable in revision. But it is not acceptable as-is. A referee should ask for the existence claim to be substantiated or weakened, and for the corollaries to be rechecked.\n\nIn short: worth engaging, but the advertised scope currently overreaches.","headline":"The conditional construction is real and the proofs hold, but the paper's universal claim rests on a self-cited existence theorem that contradicts known MOLS bounds.","tokens_in":15722,"tokens_out":4014,"would_cite":false,"duration_ms":32374,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["15B20","05C31"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper gives block constructions that turn a Butson matrix of order $n$ into Butson matrices of orders $n(n-1)$ and, under extra conditions, $n(n/2-1)$, using Latin-square arrangements called LSESC to control orthogonality.","keywords":["Butson matrices","complex Hadamard matrices","Scarpis construction","Latin squares","mutually orthogonal Latin squares","LSESC","Fourier matrices","Hadamard matrices"],"falsifier":"Run the claimed algorithm from [5] for order 6 and check whether it really produces a complete set of LSESC of order 6; because the paper equates LSESC sets with mutually orthogonal Latin squares, the known nonexistence of a complete set of MOLS of order 6 would already predict failure, and this would limit the constructions to orders where complete LSESC sets exist.","tokens_in":14740,"feed_emoji":"🧮","tokens_out":10569,"duration_ms":87212,"temperature":0.7,"pith_summary":"This paper proposes two families of constructions that produce larger Butson matrices from smaller ones. Theorem 1 turns any normalized Butson matrix of order $n$ into a Butson matrix of order $n(n-1)$, and Theorem 2 turns any Butson matrix in a specified subset $A(1,2)(m,n)$ into one of order $n(n/2-1)$, with $n$ and $m$ even in the second case. Both constructions come in one-input and two-input versions, so the procedures can produce many matrices of the same order. The output entries remain $m$-th roots of unity, which makes this a systematic way to enlarge the supply of complex Hadamard-style orthogonal matrices, including ordinary Hadamard matrices. The author also counts the outputs and shows that Fourier matrices can serve as inputs.","feed_headline":"Butson matrices grow to orders n(n-1) and n(n/2-1)","feed_subtitle":"Latin-square rearrangements of existing complex Hadamard-style matrices yield new orthogonal matrices.","key_machinery":"The load-bearing objects are LSESC, Latin squares eligible for Scarpis construction: two Latin squares $L,L'$ of order $q$ such that for every pair $(i,i')$ there is a unique $j$ with $L_{ij}=L'_{i'j}$. A complete set has $q-1$ pairwise eligible squares, and the paper notes that such sets are conjugate to mutually orthogonal Latin squares, so the maximal number $N'(q)$ equals $N(q)$. Each eligible square is encoded as a cubic tensor whose frontal slices are permutation matrices, and the tensor set is woven into the blocked matrix of (3.1) or (4.5). The eligibility condition is what makes cross-block row inner products collapse to zero, because it guarantees exactly one collision between the Latin-square row entries in each cross term.","core_discovery":"The central claim is that a Scarpis-style block matrix built from an input Butson matrix and a complete tensor set of LSESC is again a Butson matrix. Theorem 1 states that for $H \\in BH(m,n)$ with core $C$ and any complete tensor set $X$ of LSESC of order $n-1$, the matrix $\\Phi_X(H)$ defined in (3.1) lies in $BH(m,(n-1)n)$. Theorem 2 states that for $H \\in A(1,2)(m,n)$ with $n$ even and a complete tensor set of LSESC of order $n/2-1$, the matrix $\\Psi_X(H)$ defined in (4.5) lies in $BH(m,(n/2-1)n)$. Orthogonality is proved by checking row orthogonality in four cases, with column orthogonality following from the same structure. The paper also gives two-input variants $\\Phi_X(G,H)$ and $\\Psi_X(G,H)$, applies the machinery to Hadamard matrices, and identifies conditions under which Fourier matrices can serve as the input.","pith_inferences":["Editorial inference: because LSESC sets are identified with MOLS, the claim that complete LSESC sets exist for every order sits in tension with the classical fact that a complete set of MOLS of order 6 does not exist; if that tension is real, the constructions apply only where complete sets genuinely exist rather than for all $n$.","Editorial inference: the two-input variants separate the role of the first matrix (which supplies row-sign structure) from the second (which supplies the core), so inequivalent input pairs should often produce inequivalent outputs; the counting formulas then bound new classes only from below, and the paper leaves this classification open.","A testable extension would feed a fixed input matrix with all known complete LSESC sets of small order and compare normalized outputs against existing classifications of complex Hadamard matrices to count genuinely inequivalent outputs."],"forward_implications":["For every $n$ for which a complete set of LSESC of order $n-1$ exists, every $BH(m,n)$ produces a $BH(m,n(n-1))$, and the two-input version produces $|MOLS(n-1)|\\cdot|BH(m,n)|^2\\cdot n$ candidate matrices.","For even $n$ with a complete set of LSESC of order $n/2-1$, every input from $A(1,2)(m,n)$ produces a $BH(m,n(n/2-1))$, and the two-input version widens the eligible first matrix to all of $A(1)(m,n)$.","Applied to Hadamard matrices, the same block templates produce Hadamard matrices of orders $n(n-1)$ and $n(n/2-1)$, extending the author's earlier Scarpis-type Hadamard constructions.","Fourier matrices satisfy the needed conditions in the relevant cases, so the constructions give explicit new matrices, such as the $BH(6,12)$ example built from $F_6$."],"supporting_citations":[{"why":"Provides Theorem 2.6 asserting complete sets of LSESC of every order, which is the input both new constructions require.","marker":"[5]"},{"why":"Gives the earlier Scarpis-style construction of complex Hadamard matrices that the new block constructions generalize and compare against.","marker":"[8]"},{"why":"Supplies the MOLS background, the LSESC–MOLS conjugacy, the Galois-field complete sets, and the prime-power conjecture that frames the existential assumption.","marker":"[2]"},{"why":"Supplies the author's prior Hadamard construction from Paley matrices whose row-indexing idea leads to the LSESC formulation.","marker":"[4]"}],"fun_headline_variants":["Butson matrices: multiply orders with Latin-square blocks","Fourier inputs yield larger Butson matrices","Grow Butson matrices to n(n-1) and n(n/2-1)","New Butson orders via Latin square tensor sets","Butson matrix construction: from n to n(n-1) and more"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction needs a full set of specially arranged Latin squares of size $n-1$ (or $n/2-1$) for every relevant $n$, and the paper takes the existence of such sets from the author's earlier work; yet the same paper identifies these arrangements with mutually orthogonal Latin squares, and no complete set of those exists for order 6.","fun_headline_variants_meta":{"raw":{"variants":["Butson matrices: multiply orders with Latin-square blocks","Fourier inputs yield larger Butson matrices","Grow Butson matrices to n(n-1) and n(n/2-1)","New Butson orders via Latin square tensor sets","Butson matrix construction: from n to n(n-1) and more"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000821,"raw_usage":{"total_tokens":3575,"prompt_tokens":912,"completion_tokens":2663,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":528,"completion_tokens_details":{"reasoning_tokens":2576}},"tokens_in":528,"tokens_out":2663,"duration_ms":20177,"temperature":1.0,"reasoning_tokens":2576,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:14:16.727387+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the claimed algorithm from [5] for order 6 and check whether it really produces a complete set of LSESC of order 6; because the paper equates LSESC sets with mutually orthogonal Latin squares, the known nonexistence of a complete set of MOLS of order 6 would already predict failure, and this would limit the constructions to orders where complete LSESC sets exist.","supporting_citations":[{"cited_title":"Farouk, Q","cited_arxiv_id":null,"evidence_quote":"Provides Theorem 2.6 asserting complete sets of LSESC of every order, which is the input both new constructions require."},{"cited_title":"Sargent, K","cited_arxiv_id":null,"evidence_quote":"Gives the earlier Scarpis-style construction of complex Hadamard matrices that the new block constructions generalize and compare against."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the MOLS background, the LSESC–MOLS conjugacy, the Galois-field complete sets, and the prime-power conjecture that frames the existential assumption."},{"cited_title":"Farouk, Q","cited_arxiv_id":null,"evidence_quote":"Supplies the author's prior Hadamard construction from Paley matrices whose row-indexing idea leads to the LSESC formulation."}],"review_version":1}