{"id":"278061c8-2247-4b04-9f3f-1a5862ca39e3","arxiv_id":"2508.10348","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A super-Cayley graph's spectrum is a super-Fourier transform of its connection set; for Frobenius rings this yields explicit spectral formulas and rationality criteria.","lead":"The paper develops a general framework for studying the spectra of Cayley graphs built from finite rings, showing the eigenvalues are given by generalized character sums. It unifies several known graph families and characterizes when these spectra remain in a given number field.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1(2)'s 'determined by its spectrum' is only proved for an indexed eigenvalue vector, not for the unlabeled graph spectrum; fixing an indexing of K does not resolve the labeling ambiguity.","rationale":"The paper's workhorse results—the super-Fourier eigenvalue formula (Theorem 2.8, Theorem 4.6) and the Frobenius-ring supercharacter construction (Theorem 4.1)—are sound and well supported. The rationality criterion (Theorem 4.12) is also carefully argued and constitutes a genuine contribution. The main weakness is the headline 'determined by its spectrum' claim. As written, Theorem 1.1(2)/Corollary 2.9 only establishes injectivity of the map from generating sets to eigenvalue vectors indexed by superclasses. Graph spectra are normally unlabeled multisets, and the fixed indexing of K is not enough to recover the indexed vector from the multiset. The Z/2×Z/2 example shows the ambiguity concretely. The authors may have intended 'spectrum' to mean the indexed eigenvalue system, and the abstract's caveat points in that direction, but the theorem's wording should be revised to avoid claiming more than is proved. Because this is a substantial clarification of the central theorem rather than a refutation of the main spectral and arithmetic results, I would accept the paper conditional on that revision. The reader's weakest assumption (Frobenius) is reasonable and not where the central claim is most vulnerable; hence partial agreement.","tokens_in":19397,"tokens_out":33978,"duration_ms":374504,"concrete_test":"Enumerate all super-Cayley subsets for G=Z/2×Z/2 under the trivial supercharacter theory. For S1={a}, S2={b}, and S3={a+b}, compute the eigenvalue multisets {∑_{s∈S} χ(s) : χ∈Ĝ}. All three return {1,1,-1,-1}, while the indexed Fourier vectors differ. This confirms that the unlabeled spectrum does not determine S and that Theorem 1.1(2) requires the stronger 'indexed spectrum' interpretation or an additional labeling rule.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof of Corollary 2.9 shows that the map S ↦ (λ_1,…,λ_m), where λ_i is the super-Fourier coefficient attached to superclass K_i, is injective. That is not the same as showing the spectrum of Γ(G,S) determines S. The spectrum is the unlabeled multiset {λ_i^{|X_i|}}, and fixing an indexing of K does not attach the numerical eigenvalues in that multiset to the indices i. Without a rule for this attachment, the inverse super-Fourier transform cannot be applied. This is not merely pedantic: take G=Z/2×Z/2 with the trivial supercharacter theory (all superclasses and characters singletons) and fix any indexing of K. The three subsets {a}, {b}, {a+b} all yield the same multiset {1,1,-1,-1}, while their indexed vectors (1,-1,1,-1), (1,1,-1,-1), and (1,-1,-1,1) are different. Hence the spectrum plus the fixed indexing does not determine S; the indexing must be part of the spectral data (an 'indexed spectrum') for the claim to hold. The rest of the paper—the super-Fourier eigenvalue formula and the Frobenius-ring arithmetic applications—is unaffected by this concern.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a theory of super-Cayley graphs over finite abelian groups, where the generating set is a union of superclasses of a supercharacter theory that satisfies an additional constancy condition (Definition 2.1). It proves that the eigenvalues of such a graph are given by the super-Fourier transform of the characteristic function of the generating set (Theorem 2.8) and claims that the graph is determined by its spectrum once an indexing of the superclasses is fixed (Theorem 1.1(2), Corollary 2.9). The framework is then specialized to finite commutative Frobenius rings: every subgroup U of the unit group yields a supercharacter theory, and U-unitary Cayley graphs—including gcd-graphs, Paley graphs, p-unitary Cayley graphs, involutory graphs, and cubelike graphs—have spectra expressed through generalized Ramanujan sums. Further results concern rationality and integrality of spectra, connectedness, primeness, and explicit connections to Heilbronn, Gauss, and Ramanujan sums.","tokens_in":19764,"tokens_out":16778,"duration_ms":180996,"significance":"If the main claims hold, the paper provides a genuinely unifying framework for a large family of Cayley graphs over rings, and the explicit spectral formula in Proposition 2.6 and Theorem 4.6 is a useful parameter-free derivation. The rationality criterion in Theorem 4.12 and the connections to classical exponential sums are valuable, and the paper explicitly notes an independent recent result of Godsil and Spiga. Several proofs are direct and self-contained, although a number of technical steps are delegated to the authors' previous papers, which makes verification more laborious. The advertised spectral-determinacy theorem and the normalization of the super-Fourier transform both require correction before the paper can be accepted.","major_comments":[{"comment":"The proof establishes injectivity of the map S ↦ (\\hat 1_S(K_i))_{i=1}^m, i.e., uniqueness of the indexed eigenvalue tuple. It does not establish that the unlabeled spectrum determines S. The graph spectrum is the multiset {[λ_i]^{|X_i|}}, and fixing an indexing of K gives no rule for attaching the numerical multiplicities in that multiset to the indices i. For example, take G=Z/2×Z/2 with the trivial supercharacter theory and any fixed indexing; the sets {a}, {b}, {a+b} have the same spectrum {1,1,-1,-1} but different indexed tuples. Thus the inverse super-Fourier transform cannot be applied from the spectrum alone. Theorem 1.1(2) should be restated in terms of an indexed spectrum, or the additional data of the eigenvalue-to-superclass assignment must be explicitly assumed.","section":"Theorem 1.1(2), Corollary 2.9"},{"comment":"The derivation of Eq. (2.4) drops a factor of 1/|G|. From the line preceding it, |X_i| \\hat f(K_i) = (1/|G|) ∑ |K_ℓ| f(K_ℓ) σ_i(K_ℓ), and with Proposition 2.3 this gives \\hat f(K_i) = (1/|G|) ∑ f(K_ℓ) Ω_ℓ(X_i), not the displayed equality. Consequently, with \\hat defined by the expansion in Eq. (2.3), the identity λ_i = \\hat 1_S(K_i) in Theorem 2.8 is off by a factor of |G|. If \\hat is intended to be the non-normalized transform without 1/|G|, then Eq. (2.3) must be f = (1/|G|) ∑ \\hat f(K_ℓ) σ_ℓ. The eigenvalue formula in Proposition 2.6 is correct, but the claimed super-Fourier realization and its inversion statement need correction.","section":"Section 2, Eqs. (2.3)-(2.4), Theorem 2.8"},{"comment":"In the proof of (2)=>(3), the authors assert that each field factor R_i satisfies the hypotheses of Proposition 4.20. That proposition requires p∤|U|, but for U_i=(F_q^×)^p this need not hold when p divides (q-1)/gcd(p,q-1); for example, q=19 and p=3 give |U_i|=6. Thus the cited vanishing argument does not apply as written. The conclusion may be true, but the proof needs an additional argument or a corrected hypothesis.","section":"Proposition 4.22"}],"minor_comments":[{"comment":"The wording 'for a fixed χ∈X' is ambiguous; it should be 'for each i and each χ∈X_i, the sum ∑_{k∈K_i} χ(k) is independent of χ∈X_i.'","section":"Definition 2.1(4)"},{"comment":"The displayed identity 'Ω_j(X_i)| = Ω_j(X_i)|' is garbled; presumably it should be \\overline{Ω_j(X_i)} = Ω_j(X_i).","section":"Proposition 2.3"},{"comment":"In the last line, 'U1 ⊂ U2' should presumably be 'U1 ⊂ U'.","section":"Lemma 4.9"},{"comment":"In the sentence 'This shows that all eigenvalues of Ri are non-zero as well,' 'Ri' should be 'G_{R_i}(p)' or 'the graph over R_i'.","section":"Proposition 4.22 proof"},{"comment":"The symbol K is used both for the superclass partition and for the subfield of Q(ζ_n); this overloaded notation is confusing and should be changed.","section":"Theorem 4.12"},{"comment":"The equality repeats the same expression on both sides; if complex conjugation is intended, it should be written explicitly.","section":"Eq. (2.4)"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid contribution with a useful unifying framework, but the advertised spectral-determinacy theorem is stronger than what is actually proved, and the super-Fourier normalization in Section 2 needs correction. The proof gap in Proposition 4.22 should also be addressed. Given the paper's reliance on the authors' prior works for several 'identical arguments,' I would encourage the editor to ask for fuller self-contained arguments or precise statements of the cited lemmas."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this is a solid, useful paper, but one headline claim is overstated. The core observation—that super-Cayley graph spectra are just the usual Fourier diagonalization grouped by supercharacters—is correct, and the U-unitary framework does genuinely unify Paley, gcd, p-unitary, involutory, and cubelike graphs. The explicit spectral formula for Frobenius rings in terms of generalized Ramanujan sums is a real extension, and the rationality criterion (Theorem 4.12) is clean. They also honestly flag the overlap with Godsil–Spiga. The Frobenius-ring supercharacter construction is sound, and the main theorems are proven directly. This paper deserves a serious referee.\n\nThe soft spot is Theorem 1.1(2) and Corollary 2.9. The proof only shows that the indexed vector of super-Fourier coefficients determines S. The unlabeled spectrum—the multiset with multiplicities—carries no indexing, so fixing an indexing of K does not attach the numerical eigenvalues to the superclasses. The stress-test example is right: take G = Z/2 × Z/2 with the trivial supercharacter theory. The subsets {a}, {b}, and {a+b} all have spectrum {1, −1, −1, −1}, yet they are different graphs. So \"determined by its spectrum\" is false as stated; the correct statement is \"determined by its indexed spectrum.\" That is a genuine flaw in a headline claim, though it does not affect the spectral formulas or the ring-theoretic applications, which use the indexed eigenvalues directly.\n\nA lesser issue: several arguments are delegated to the authors' prior papers with \"identical argument.\" That is acceptable but makes the paper less self-contained. Also, the super-Fourier transform is essentially a reindexing of the standard character table; the novelty is in the packaging rather than the underlying mathematics. That is fine—the paper is honest about it.\n\nFor whom: algebraic graph theorists and anyone working on spectra of Cayley graphs over rings. It deserves a serious referee, and with the \"determined by spectrum\" claim corrected or explicitly qualified, I would take it. I would not cite it in my own work in the next year, but I would send it to someone working in this area.","headline":"Solid unification paper with a real but localized flaw: the 'determined by its spectrum' claim only works for an indexed eigenvalue vector, not the unlabeled spectrum.","tokens_in":693,"tokens_out":867,"would_cite":false,"duration_ms":32281,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11L03","11T24","05C25"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a finite abelian group, any super-Cayley graph is determined by its spectrum once the superclass indexing is fixed.","keywords":["supercharacter theory","Cayley graph","U-unitary Cayley graph","Frobenius ring","generalized Ramanujan sums","graph spectrum","gcd-graph","Paley graph"],"falsifier":"Enumerate a small Frobenius ring $R$ (for example $\\mathbb{Z}/8$ or $\\mathbb{Z}/9$) with a subgroup $U$ and two distinct symmetric $U$-stable subsets $S_1,S_2$; if the super-Fourier vectors $(\\lambda_i)$ coincide while the sets differ, Corollary 2.9 and Theorem 4.6 are false. Conversely, comparing the formula $\\lambda_i = \\sum_{K_\\ell \\subset S} \\sigma_\\ell(K_i)$ with a direct diagonalization of the adjacency matrix for such an example would expose any missing multiplicity or arithmetic error.","tokens_in":19332,"feed_emoji":"🔢","tokens_out":12560,"duration_ms":114621,"temperature":0.7,"pith_summary":"Over any finite abelian group, the paper studies Cayley graphs whose generating set is a union of superclasses of a chosen supercharacter theory. Its central theorem says the eigenvalues of such a graph are the super-Fourier coefficients of the generating set, and that with a fixed indexing of superclasses the graph is uniquely determined by its spectrum. This gives a unified spectral framework for many classical families—unitary, gcd, p-unitary, Paley, and involutory Cayley graphs—because when the underlying group is the additive group of a finite commutative Frobenius ring, every subgroup of units produces such a supercharacter theory. The payoff is explicit spectra as generalized Ramanujan sums, rationality criteria for the eigenvalues, and a direct link between primeness and the occurrence of zero as an eigenvalue.","feed_headline":"Spectrum determines every super-Cayley graph","feed_subtitle":"Eigenvalues are super-Fourier transforms, uniting Paley, gcd, p-unitary and involutory Cayley graphs under one theorem.","key_machinery":"The central object is a supercharacter theory: an equal-size pair of partitions $\\mathcal{K}$ of $G$ and $\\mathcal{X}$ of its dual group such that each character sum $\\sigma_i = \\sum_{\\chi \\in X_i} \\chi$ is constant on every superclass and the sums $\\Omega_j(\\chi) = \\sum_{k \\in K_j} \\chi(k)$ do not depend on the choice of $\\chi$ within $X_i$. The argument runs on the identity $\\lambda_i = \\widehat{1_S}(K_i)$, which converts spectrum computation into the invertible super-Fourier transform on the space of superclass functions. For rings, the construction rests on the generating character $\\chi(a)=\\zeta_n^{\\psi(a)}$ supplied by a Frobenius functional $\\psi$; then every character has the form $\\","core_discovery":"The paper's main claim is Theorem 1.1: if $(K,\\mathcal{X})$ is a supercharacter theory on a finite abelian group $G$ and $\\Gamma(G,S)$ is a super-Cayley graph, then its eigenvalues, counted with multiplicity, are $\\lambda_i = \\widehat{1_S}(K_i)$, where $1_S$ is the characteristic function of the generating set and $\\widehat{\\cdot}$ is the super-Fourier transform on superclass functions; consequently fixing an indexing of $K$ determines $S$ and hence the graph. The second layer applies this to $U$-unitary Cayley graphs over a finite commutative Frobenius ring $R$: the $U$-orbits of $R$ and the sets $\\{\\chi_x : x \\in K_i\\}$ form a supercharacter theory, so the spectrum of any $U$-stable genera","pith_inferences":["Beyond the paper: the determinacy statement in Theorem 1.1 is purely group-theoretic, so it should extend to any supercharacter theory satisfying the paper's fourth condition, including theories not coming from a ring structure, widening the class of graphs known to be determined by their spectra.","The rationality criterion is noted in the paper to hold for all finite abelian groups, not just Frobenius rings; a natural next step is to use it to classify rational circulant graphs directly from Galois-stable generating sets.","A testable extension is to search computationally for cospectral super-Cayley graphs over different indexings; the paper rules them out only once the superclass indexing is fixed, so the dependence on indexing deserves explicit examples.","If the open Question 4.17 is settled affirmatively, primeness of connected anti-connected $U$-unitary Cayley graphs becomes a spectral condition, giving a direct arithmetic test for graph decomposability."],"forward_implications":["Every super-Cayley graph has at most $m$ distinct eigenvalues, where $m$ is the number of superclasses; with a fixed indexing, the spectrum alone determines the generating set and hence the graph.","Over a finite Frobenius ring, the spectrum of any $U$-unitary Cayley graph is an explicit finite list of generalized Ramanujan sums, so it can be written down without diagonalizing the adjacency matrix.","A symmetric Cayley graph over a Frobenius ring is rational over a subfield $K$ of $\\mathbb{Q}(\\zeta_n)$ precisely when it is $U_1$-unitary for the subgroup $U_1$ coming from the Galois group of $K$; integral graphs are the case $K=\\mathbb{Q}$.","For $U=R^\\times$, for $p$-unitary graphs over rings where $p$ is invertible, and for Jacobi-symbol Paley graphs, the graph is prime if and only if $0$ is not an eigenvalue, under the connected and anti-connected assumptions; the paper asks whether this equivalence holds for all $U$-unitary Cayley graphs.","The framework recovers classical formulas as special cases: gcd-graph spectra from Euler/Mobius-type Ramanujan sums, $p$-unitary spectra over $\\mathbb{Z}/p^2$ as Heilbronn sums, and Paley spectra as Gauss sums."],"supporting_citations":[{"why":"Supplies the cyclic-group theorem that the paper generalizes: gcd-graphs over Z/n are determined by their spectra.","marker":"[41]"},{"why":"Establishes the supercharacter/superclass-function framework, including the orthogonal basis and non-normalized Fourier transform used for the eigenvalue formula.","marker":"[5]"},{"why":"Develops Ramanujan sums as supercharacters, the bridge that lets spectra be recast as generalized Ramanujan sums.","marker":"[11]"},{"why":"Gives the circulant diagonalization for finite abelian groups, the standard spectrum formula from which the super-Fourier realization starts.","marker":"[19]"},{"why":"Provides the generalized Ramanujan sum over finite rings, Frobenius-ring facts, and gcd-graph spectral machinery on which the ring-level formulas build.","marker":"[34]"},{"why":"Classifies integral Cayley graphs over symmetric algebras and supplies the rationality/integrality criterion that Theorem 4.12 generalizes.","marker":"[33]"},{"why":"Supplies the homogeneous-set/ideal machinery and primeness criteria for Cayley graphs, used for the zero-eigenvalue propositions and Question 4.17.","marker":"[9]"},{"why":"Shows Heilbronn sums are supercharacter sums, used to identify p-unitary spectra over Z/p^2 with Heilbronn sums.","marker":"[12]"},{"why":"Computes spectra of quadratic-character Paley graphs, used for the Jacobi-symbol primeness and rationality examples.","marker":"[27]"}],"fun_headline_variants":["Super-Fourier transform determines super-Cayley graph spectra","Eigenvalues settle identity of super-Cayley graphs","Supercharacters unify Paley, gcd, and U-unitary Cayley spectra","Spectrum uniquely fixes super-Cayley graphs","From supercharacters to spectra: a Cayley graph theorem"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The ring-level results assume $R$ is a finite commutative Frobenius ring, i.e., one carrying a non-degenerate additive functional; without that assumption the dual group need not consist exactly of characters $\\chi_x$, and the $U$-orbit construction may fail to be a supercharacter theory with the stated eigenvalue formula.","fun_headline_variants_meta":{"raw":{"variants":["Super-Fourier transform determines super-Cayley graph spectra","Eigenvalues settle identity of super-Cayley graphs","Supercharacters unify Paley, gcd, and U-unitary Cayley spectra","Spectrum uniquely fixes super-Cayley graphs","From supercharacters to spectra: a Cayley graph theorem"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000618,"raw_usage":{"total_tokens":2720,"prompt_tokens":775,"completion_tokens":1945,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":519,"completion_tokens_details":{"reasoning_tokens":1858}},"tokens_in":519,"tokens_out":1945,"duration_ms":16827,"temperature":1.0,"reasoning_tokens":1858,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T20:30:18.859395+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate a small Frobenius ring $R$ (for example $\\mathbb{Z}/8$ or $\\mathbb{Z}/9$) with a subgroup $U$ and two distinct symmetric $U$-stable subsets $S_1,S_2$; if the super-Fourier vectors $(\\lambda_i)$ coincide while the sets differ, Corollary 2.9 and Theorem 4.6 are false. Conversely, comparing the formula $\\lambda_i = \\sum_{K_\\ell \\subset S} \\sigma_\\ell(K_i)$ with a direct diagonalization of the adjacency matrix for such an example would expose any missing multiplicity or arithmetic error.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the cyclic-group theorem that the paper generalizes: gcd-graphs over Z/n are determined by their spectra."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the supercharacter/superclass-function framework, including the orthogonal basis and non-normalized Fourier transform used for the eigenvalue formula."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Develops Ramanujan sums as supercharacters, the bridge that lets spectra be recast as generalized Ramanujan sums."},{"cited_title":"6th China-Japan Sem","cited_arxiv_id":null,"evidence_quote":"Gives the circulant diagonalization for finite abelian groups, the standard spectrum formula from which the super-Fourier realization starts."},{"cited_title":"On gcd-graphs over finite rings","cited_arxiv_id":"2503.04086","evidence_quote":"Provides the generalized Ramanujan sum over finite rings, Frobenius-ring facts, and gcd-graph spectral machinery on which the ring-level formulas build."},{"cited_title":"Nguyen and Nguyen Duy T ˆan, Integral cayley graphs over a finite symmetric algebra , Archiv der Mathematik 124 (2025), 615–623","cited_arxiv_id":null,"evidence_quote":"Classifies integral Cayley graphs over symmetric algebras and supplies the rationality/integrality criterion that Theorem 4.12 generalizes."},{"cited_title":"Nguyen, Sophie Spirkl, and Nguyˆen Duy T ˆan, On prime Cayley graphs , arXiv:2401.06062, to appear in Journal of Combinatorics (2024)","cited_arxiv_id":null,"evidence_quote":"Supplies the homogeneous-set/ideal machinery and primeness criteria for Cayley graphs, used for the zero-eigenvalue propositions and Question 4.17."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows Heilbronn sums are supercharacter sums, used to identify p-unitary spectra over Z/p^2 with Heilbronn sums."},{"cited_title":"3, 527–542","cited_arxiv_id":null,"evidence_quote":"Computes spectra of quadratic-character Paley graphs, used for the Jacobi-symbol primeness and rationality examples."}],"review_version":1}