{"id":"681652f3-cda6-4fa3-9132-f7446d74ca4b","arxiv_id":"2506.20778","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Using modular Hadamard matrices, the authors construct infinitely many SICs over finite fields, including finite-field SICs with trivial automorphism groups.","lead":"This paper constructs infinitely many new sets of equally spaced lines, called SICs, over finite fields, using modular Hadamard matrices, and exhibits the first known 'asymmetric' examples whose symmetry groups are trivial. The result gives mathematicians a new combinatorial route to configurations that are usually built from Weyl-Heisenberg symmetry, with possible implications for the long-open Zauner conjecture about such lines in complex space.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Example 23's total-asymmetry claim rests on two unreproduced computations with external matrices; if those computations are wrong, the paper's headline 'asymmetric SICs' is unsupported.","rationale":"The reader's weakest_assumption correctly identifies Example 23 as the crux of the asymmetry claim. I agree that this is the single most load-bearing assumption: the unconditional construction theorem is solid, but the title and abstract promise asymmetric SICs, and those are supported only by two unverified computations with external data. The paper is otherwise rigorous, with no fitted parameters, no circularity, and a clean Gram-matrix proof. My concern does not invalidate Theorem 2 or the infinite families; it only affects the novelty claim that some new SICs are totally asymmetric. Because the relevant scripts and matrices are not archived, the claim is not independently checkable from the manuscript. A conditional acceptance requiring the authors to provide a reproducible verification of Example 23 (and ideally the same verification for the control examples) would settle the matter. If the recomputation confirms trivial automorphism groups, the paper should be accepted as is; if not, the headline claim should be weakened to a conjecture. Thus I recommend CONDITIONAL rather than unconditional ACCEPT, but this is a verification condition, not a mathematical objection to the main construction.","tokens_in":13677,"tokens_out":36159,"duration_ms":378263,"concrete_test":"Download gs.36 from Spence's homepage, extract matrices #23 and #24, and verify they are ordinary Hadamard matrices of order 36 (H H^T = 36 I over Z) and reduce them mod 7 to valid modular Hadamard matrices. Reconstruct the 36^2 vectors over F_49 via Theorem 2 with z = -2(1+i). Implement Proposition 17 independently (e.g., build the four directed graphs Γ_ω on 4*36^2 vertices and compute simultaneous automorphisms respecting the C_4 fibers using nauty or GRAPE) to compute Aut_w. Run the same pipeline on Example 18 (known Aut = S_4) and Example 21 with q = 19 (known Aut = PΣL(2,19)) as controls. If #23/#24 yield a nontrivial Aut_w, the asymmetric SIC claim fails; if the controls mismatch, the computational method is unreliable.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 2 and Corollary 6 are unconditionally proven by an elementary Gram-matrix computation and standard Paley/Dirichlet arguments; I see no flaw there. The load-bearing concern is the paper's headline claim that some new SICs have trivial automorphism groups. That claim rests entirely on Example 23, where matrices #23 and #24 from Spence and Turyn's external catalog of Goethals-Seidel Hadamard matrices of order 36 are claimed to yield SICs over F_49 with trivial weak automorphism group, as computed with GAP/GRAPE via Proposition 17. This is load-bearing because if the catalog entries are misread, the matrices are not valid Hadamard matrices over F_7, or the GAP/GRAPE computation has any bug, then no totally asymmetric SIC is actually exhibited; only Conjecture 24 remains. The paper does not include the matrices #23/#24, the GAP scripts, or computation logs, so the asymmetric claim cannot be checked from the manuscript alone. Furthermore, the computation depends on the correctness of Proposition 17's graph criterion; an error in encoding the graphs or their simultaneous automorphisms could undercount automorphisms and yield a false 'trivial' result. This is a reproducibility and verification gap, not an internal inconsistency in the main theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper constructs SICs (symmetric, informationally complete sets of lines) over finite fields from modular Hadamard matrices. Theorem 2 states that if K has characteristic p, d ≡ 8 mod p, and H ∈ K^{d×d} satisfies H_ij^2 = 1 and H^T H = dI, then the vectors x_ij = h_j ∘ (1 + z e_i) with z = -2(1+i) form a (12,16,96)-SIC in L^d, where L = K(i). The proof is an explicit Gram-matrix and frame-operator computation. From this, Corollary 6 gives infinitely many SICs over F_{p^2} for each prime p ≡ 3 mod 4, using Paley's Hadamard constructions and Dirichlet's theorem, and Corollary 10 gives, conditionally on the p-modular Hadamard conjecture, SICs over some finite field for almost every dimension. Section 3 analyzes symmetries, establishing an automorphism-group sandwich (Theorem 15) and a graph criterion (Proposition 17) for computing SIC automorphism groups. Section 4 gives computational examples, including Example 23, which claims totally asymmetric SICs over F_49 from Spence–Turyn Hadamard matrices #23 and #24, and states Conjectures 22 and 24.","tokens_in":13895,"tokens_out":12489,"duration_ms":128223,"significance":"Assuming the computational claims are correct, the paper makes two notable contributions. Corollary 6 is an unconditional infinite-family result for finite-field SICs over each F_{p^2} with p ≡ 3 mod 4, extending previous results that were limited to p = 3. Example 23 provides the first claimed finite-field SICs with trivial automorphism group, directly addressing the question of whether SICs can exist without Weyl–Heisenberg symmetry. The central construction is elementary and self-contained: Theorem 2 is proven by a clean Gram-matrix computation with no fitted parameters, the infinite families follow from standard Paley/Dirichlet arguments, and the conditional statement in Corollary 10 is explicitly labelled as resting on Conjecture 9. The automorphism-group sandwich and the graph criterion are useful tools that reduce SIC symmetry computations to finite graph-isomorphism problems, and the paper is careful to distinguish theorems, computational observations, and conjectures.","major_comments":[{"comment":"The claim that matrices #23 and #24 from the Spence–Turyn catalog [27] produce totally asymmetric SICs is the only evidence for the abstract's assertion that 'some of our new SICs exhibit trivial automorphism groups.' The manuscript does not include the matrices, the GAP/GRAPE scripts, or the outputs, and the claim depends on a correct transcription of the external catalog and a correct encoding of the graphs Γ_ω in Proposition 17. If any of these steps is wrong, the exhibited examples of total asymmetry disappear and only Conjecture 24 remains. Please provide the matrices (or an explicit construction), the computational verification scripts, and the resulting automorphism-group output, or an independent certificate of the claimed triviality.","section":"§4, Example 23"}],"minor_comments":[{"comment":"The sentence 'We omit our proof for the sake of brevity' concerns a nontrivial containment PΓL(2,q) ≤ Aut_s(\\tilde H); since this is presented as progress on Conjecture 22, the proof should be supplied in an appendix or the statement should be explicitly labelled as a computational observation.","section":"§4, after Conjecture 22"},{"comment":"The phrase 'three have trivial weak automorphism group' is ambiguous, because matrix #3's SIC is then said to have weak automorphism group of order 2; clarify whether the first phrase refers to the weak automorphism group of the Hadamard matrix or of the SIC.","section":"§4, Example 23"},{"comment":"The p = 3 classification of modular Hadamard matrices is cited to [20], whose title concerns Hadamard matrices modulo 5; please verify the citation, which may need to be [19] or a different source.","section":"§2, Example 7"},{"comment":"The condition on the field automorphism γ is typeset incorrectly; it should state that γ commutes with conjugation, i.e., γ(ᾱ) = overline{γ(α)} for every α ∈ L.","section":"§3, Definition 13"},{"comment":"The congruence 'd ≡ 8 mod char K' is nonstandard when char K = 0; a parenthetical clarification that this means d = 8 in the characteristic-zero case would help the reader.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The central mathematical construction is sound, and the infinite-family results are well proven. My recommendation of major revision is driven entirely by the need to make Example 23 checkable; if the authors supply the matrices, scripts, and outputs for the total-asymmetry computation, I would support acceptance. I saw no evidence of a novelty or attribution problem, and the self-citations are appropriate background."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main result is the real thing. Theorem 2 takes any modular Hadamard matrix H over K with d ≡ 8 mod char K and produces a (12,16,96)-SIC by a one-line twist on each column. The proof is an elementary Gram-matrix computation, done carefully in Lemma 3, and it is unconditional. Corollary 6 then uses Paley's Hadamard constructions and Dirichlet's theorem to get infinitely many dimensions over every F_{p^2} with p ≡ 3 mod 4. That already extends what was known, where only p = 3 had such infinite families. The Hadamard-to-SIC dictionary is also useful: Theorem 15 gives a clean sandwich relating symmetries of the SIC to symmetries of H and its tensor lift, and Proposition 17 reduces automorphism computations to graph isomorphisms.\n\nThe genuinely new phenomenon is asymmetry. The authors exhibit, in Example 23, SICs over F_{49} whose weak automorphism group is trivial, which no one had before over any field. That is a strong claim, and it is the one place I want more evidence. The examples rest on matrices #23 and #24 from Spence and Turyn's online catalog of order-36 Hadamard matrices, plus GAP/GRAPE computations that are described but not shipped. No GAP scripts, no matrices in the paper, no computation log. This is a reproducibility gap, not an internal contradiction. The authors are transparent about the source and the method, and the graph criterion in Proposition 17 is well grounded, so I do not think the claim is likely wrong. But an editor should ask for a computational supplement before publication.\n\nTwo smaller points. After Conjecture 22 the authors say they can prove Aut_s(\\tilde H) contains PΓL(2,q) but omit the proof 'for brevity.' That is fine for a conjecture-supporting remark, but it should be marked as omitted rather than left dangling. And the density result in Corollary 10 is conditional on the p-modular Hadamard conjecture, which the paper states clearly. No circularity, no fitted parameters. The citations to the authors' own prior work are background and are appropriate.\n\nWho benefits: anyone working on equiangular lines over finite fields, SIC-POVMs, or Hadamard matrices. The main theorem is worth having on its own. I would send this to a serious referee, with the request that Example 23 either be made reproducible or softened to a conjecture with supporting computation.\n\nVerdict: accept after minor revision, with computational details for Example 23.","headline":"A clean construction of finite-field SICs from modular Hadamard matrices, with a solid main theorem and an asymmetric-symmetry claim that rests on reproducible computation the paper does not quite ship.","tokens_in":14443,"tokens_out":988,"would_cite":true,"duration_ms":12729,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05B20","11T30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves a Hadamard-matrix construction that yields infinitely many new SICs over finite fields and exhibits the first SICs with trivial automorphism groups.","keywords":["equiangular lines","SIC-POVM","finite fields","modular Hadamard matrices","automorphism groups","Weyl–Heisenberg symmetry","totally asymmetric SIC","Zauner's conjecture"],"falsifier":"Independently recompute the Gram matrix for the vectors produced from the two order-36 Hadamard matrices in Example 23 and search for any index permutation preserving all inner products up to the equivalence allowed in Definition 13; finding even one nontrivial permutation would destroy the totally asymmetric example. For the infinite-family claim, verifying that a Paley-constructed Hadamard matrix of order $d$ satisfies $H^T H = dI$ over the relevant prime and that $d \\equiv 8 \\pmod{p}$ would confirm or refute each individual dimension.","tokens_in":13454,"feed_emoji":"📐","tokens_out":8880,"duration_ms":92260,"temperature":0.7,"pith_summary":"The paper proves a general recipe for building symmetric, informationally complete sets of vectors, known as SICs, over finite fields. Starting from any modular Hadamard matrix of order $d$ with $d \\equiv 8$ modulo the ground-field characteristic, it forms $d^2$ vectors by perturbing each column with the same one-coordinate factor, and these vectors always form a SIC with the same fixed parameters. Because Hadamard matrices are abundant, the construction yields infinitely many new SICs over every field $\\mathbb{F}_{p^2}$ with $p \\equiv 3 \\pmod{4}$, and at least two of them, in dimension 36 over $\\mathbb{F}_{49}$, have no nontrivial automorphism at all. This matters because every previously known SIC, over any field, carried Weyl–Heisenberg symmetry; the paper's examples show that symmetry is not a necessary feature of SICs.","feed_headline":"Equiangular-line systems with no symmetry found over finite fields","feed_subtitle":"A Hadamard-matrix recipe produces infinitely many such systems, including two that are totally asymmetric.","key_machinery":"The load-bearing object is the modular Hadamard matrix together with the deformation $x_{ij} = h_j \\circ (1 + z e_i)$, in which the $j$-th column $h_j$ is pointwise multiplied by a vector that is $1+z$ in one coordinate and 1 everywhere else. This deformation converts the row and column orthogonality of $H$ into exact inner-product conditions: any two distinct constructed vectors have cross inner-product product 16, the diagonal inner product is 12, and the resolution identity is $96I$. A second object, the block matrix $\\widetilde{H}$ whose $(i,k)$ block is $r_k r_i^T$, mediates the symmetry analysis: symmetries of the SIC are trapped between symmetries of $H$ and symmetries of $\\widetilde{H}$.","core_discovery":"The central claim is Theorem 2: for a field $K$ with characteristic not 2, a dimension $d \\equiv 8 \\pmod{\\operatorname{char} K}$, and any modular Hadamard matrix $H \\in K^{d \\times d}$ (entries with square 1 and $H^T H = dI$), the vectors $x_{ij} = h_j \\circ (1 + z e_i)$ with $z = -2(1+i)$ form a $(12,16,96)$-SIC. The proof computes the Gram matrix entry by entry: diagonal inner products equal 12, off-diagonal products of mutually conjugate inner products equal 16, and the sum of the rank-one projectors is $96I$. Feeding in known families of Hadamard matrices gives infinitely many dimensions for each prime $p \\equiv 3 \\pmod{4}$, and feeding in two order-36 matrices from a published catalog gives SICs over $\\mathbb{F}_{49}$ with trivial strong and weak automorphism groups. The paper also proves a sandwich theorem showing that automorphisms of the SIC always sit between automorphisms of the Hadamard matrix and automorphisms of a related block matrix.","pith_inferences":["Beyond the paper, the same deformation could be tested with values of $z$ other than $-2(1+i)$; if any other choice works in positive characteristic, the fixed parameter triple $(12,16,96)$ is not special to the construction.","Beyond the paper, the sandwich theorem suggests a concrete search strategy for complex SICs without Weyl–Heisenberg symmetry: take a complex Hadamard matrix with a small automorphism group and try the same deformation, since the paper only shows why this fails below dimension 8.","Beyond the paper, if totally asymmetric SICs are truly abundant over finite fields, asymmetry itself is not an obstruction to informationally complete equiangular systems, which weakens the heuristic that complex SICs should be sought only among symmetric examples."],"forward_implications":["For every prime $p \\equiv 3 \\pmod{4}$, SICs exist in infinitely many dimensions over $\\mathbb{F}_{p^2}$, extending the previously known case $p = 3$.","Conditional on the modular Hadamard conjecture, SICs over some finite field exist for almost every dimension $d$.","For $p = 3$, the construction gives SICs in every dimension $d \\equiv 2 \\pmod{6}$, a much larger family than the earlier one with $d = 2^{2k+1}$.","The order-36 examples over $\\mathbb{F}_{49}$ are the first SICs with trivial automorphism groups, so finite-field SICs can be totally asymmetric.","The construction turns any modular Hadamard matrix of suitable order into a SIC, so future progress on modular Hadamard matrices automatically produces more SIC dimensions."],"supporting_citations":[{"why":"Supplies the catalog of order-36 Goethals–Seidel Hadamard matrices whose entries #23 and #24 produce the totally asymmetric SICs.","marker":"[27]"},{"why":"Provides the basic theory of frames and equiangular lines over finite fields plus the earlier Weyl–Heisenberg SIC family that this paper generalizes.","marker":"[6]"},{"why":"States the modular Hadamard conjecture and supplies the asymptotic existence results used in Corollary 10 and Example 8.","marker":"[19]"},{"why":"Provides the existence results for Hadamard matrices modulo 3 used to reach every dimension $d \\equiv 2 \\pmod{6}$.","marker":"[20]"},{"why":"Supplies Mertens' theorem for arithmetic progressions, used to estimate the density of dimensions covered in Corollary 10.","marker":"[29]"},{"why":"Gives the analogous construction with complex Hadamard matrices that motivated Theorem 2, though it only worked in a few complex dimensions.","marker":"[15]"},{"why":"Determines the automorphism groups of Paley I Hadamard matrices, used in Examples 20–21 and Conjecture 22.","marker":"[16]"}],"fun_headline_variants":["Infinitely many new SICs over finite fields, some totally asymmetric","Hadamard recipe yields asymmetric SICs over finite fields","Trivial automorphism SICs: new infinite families from Hadamard matrices","No Weyl-Heisenberg symmetry: new SICs over finite fields"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The strongest concrete claim of a totally asymmetric SIC rests on the computer-based automorphism-group computation for the two order-36 SICs in Example 23 being correct; if those computations or the underlying catalog entries are wrong, only Conjecture 24 remains, while the infinite-dimension claim separately relies on Paley's Hadamard constructions and Dirichlet's theorem.","fun_headline_variants_meta":{"raw":{"variants":["Infinitely many new SICs over finite fields, some totally asymmetric","Hadamard recipe yields asymmetric SICs over finite fields","Trivial automorphism SICs: new infinite families from Hadamard matrices","No Weyl-Heisenberg symmetry: new SICs over finite fields"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000839,"raw_usage":{"total_tokens":3614,"prompt_tokens":861,"completion_tokens":2753,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":477,"completion_tokens_details":{"reasoning_tokens":2678}},"tokens_in":477,"tokens_out":2753,"duration_ms":20624,"temperature":1.0,"reasoning_tokens":2678,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:43:14.510628+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Independently recompute the Gram matrix for the vectors produced from the two order-36 Hadamard matrices in Example 23 and search for any index permutation preserving all inner products up to the equivalence allowed in Definition 13; finding even one nontrivial permutation would destroy the totally asymmetric example. For the infinite-family claim, verifying that a Paley-constructed Hadamard matrix of order $d$ satisfies $H^T H = dI$ over the relevant prime and that $d \\equiv 8 \\pmod{p}$ would confirm or refute each individual dimension.","supporting_citations":[{"cited_title":"Spence, R","cited_arxiv_id":null,"evidence_quote":"Supplies the catalog of order-36 Goethals–Seidel Hadamard matrices whose entries #23 and #24 produce the totally asymmetric SICs."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the basic theory of frames and equiangular lines over finite fields plus the earlier Weyl–Heisenberg SIC family that this paper generalizes."},{"cited_title":"Kuperberg, Hadamard matrices modulo p and small modular Hadamard matrices, J","cited_arxiv_id":null,"evidence_quote":"States the modular Hadamard conjecture and supplies the asymptotic existence results used in Corollary 10 and Example 8."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the existence results for Hadamard matrices modulo 3 used to reach every dimension $d \\equiv 2 \\pmod{6}$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Mertens' theorem for arithmetic progressions, used to estimate the density of dimensions covered in Corollary 10."},{"cited_title":"Jedwab, A","cited_arxiv_id":null,"evidence_quote":"Gives the analogous construction with complex Hadamard matrices that motivated Theorem 2, though it only worked in a few complex dimensions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Determines the automorphism groups of Paley I Hadamard matrices, used in Examples 20–21 and Conjecture 22."}],"review_version":1}