{"id":"2f5d0f39-b1be-497a-b4e3-3ada4faec763","arxiv_id":"2604.06513","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Generalized Paley graphs have real spectra exactly when undirected; directed versions with three eigenvalues are only oriented Paley graphs or their unions; weak Waring numbers reduce to classical Waring numbers over finite fields.","lead":"The paper characterizes when generalized Paley graphs over finite fields have real spectra and classifies directed versions with three eigenvalues, while constructing infinite families of integral ones via cyclotomic polynomials. It applies this framework to give an alternative reduction of weak Waring numbers to classical ones.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's identified assumption (standard character-sum formula with no extra restrictions on q) is not a point of vulnerability; the formula applies uniformly to all prime powers and the reality claim follows from normality rather than case-by-case sum evaluation. The three-eigenvalue classification aligns with the index of the multiplicative subgroup and the known behavior of generalized Gauss sums. No internal inconsistency appears in the stated results or the reduction to classical Waring numbers.","tokens_in":1826,"tokens_out":373,"duration_ms":167829,"concrete_test":"For q = 13 and k = 3 (index 3, directed), compute the set of values {∑_{s∈(F_{13}^*)^3} χ_a(s) | a ∈ F_{13}} explicitly via the standard additive-character formula and confirm it contains more than three distinct complex numbers.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claims hold by standard facts on abelian Cayley graphs. The adjacency matrix A is normal (diagonalized by the Fourier basis of additive characters), so its spectrum lies in ℝ if and only if A is Hermitian, which occurs precisely when the connection set S satisfies S = −S. This is equivalent to −1 lying in the subgroup of k-th powers, i.e., the graph being undirected. For directed GP-digraphs the spectrum is closed under complex conjugation and not entirely real, hence contains at least three distinct eigenvalues; the further claim that exactly three occur only for index-2 subgroups (oriented Paley graphs) or the degenerate S = {1} case with p = 3 (unions of 3-cycles) is consistent with the distinct values taken by the character sums over multiplicative subgroups of different indices.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The paper studies generalized Paley graphs Γ(k,q) = Cay(F_q, (F_q^*)^k) for q = p^m. It proves that the spectrum is real if and only if the graph is undirected (equivalently, -1 lies in the subgroup of k-th powers). It gives a cyclotomic-polynomial construction that produces infinite families of integral GP-graphs. For the directed case it shows that every GP-digraph has at least three distinct eigenvalues and that exactly three occur only for oriented Paley graphs or disjoint unions of them. It determines the periods of these digraphs and, as an application, reduces the computation of weak Waring numbers over finite fields to that of classical Waring numbers.","tokens_in":1985,"tokens_out":467,"duration_ms":29833,"significance":"The spectral classification rests on standard character-sum formulas for abelian Cayley graphs and is therefore immediately verifiable. The explicit construction of integral examples and the reproof of the Waring-number reduction via graph spectra are useful contributions that link spectral graph theory with additive combinatorics over finite fields. The fact that the central claims follow from well-known facts on normal adjacency matrices and multiplicative subgroups strengthens the manuscript.","major_comments":[],"minor_comments":[{"comment":"The definition of the period (or index of imprimitivity) of a digraph is used in §5 but is not recalled in the text; a one-sentence reminder with the standard definition would help readers.","section":null},{"comment":"The statement that GP-digraphs generically have period 1 would benefit from an explicit small example (e.g., q=13, k=3) showing the eigenvalue multiplicities and the resulting period.","section":null},{"comment":"The cyclotomic-polynomial construction in §3 is presented for prime-power q; it would be useful to state explicitly whether the same polynomials work when q is not prime.","section":null},{"comment":"A reference to the 2012 Cochrane-Cipra paper is given for the Waring-number result, but the manuscript does not indicate which parts of their proof are reproduced or simplified by the graph-theoretic approach.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and positive evaluation of the manuscript. The significance of the spectral classification, the construction of integral examples, and the reduction of weak Waring numbers via graph spectra is appreciated. We note the recommendation for minor revision and will prepare an updated version incorporating any necessary clarifications or corrections.","responses":[],"tokens_in":1395,"tokens_out":82,"duration_ms":35787,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The one or two things to know are that the spectrum of these generalized Paley graphs is real if and only if the graph is undirected, which follows directly from the connection set being a subgroup closed under inversion, and that among directed GP-digraphs only the oriented Paley graphs and their disjoint unions have exactly three eigenvalues. The cyclotomic construction for producing infinite families of integral GP-graphs is also new and concrete. The paper does well by relying on the standard Fourier analysis for eigenvalues of abelian Cayley graphs over finite fields, where the values come from character sums over the subgroup of k-th powers. This leads to straightforward characterizations without extra machinery, and the three-eigenvalue claim follows from counting the distinct sums that arise for multiplicative subgroups of varying indices. The stress-test note confirms these line up with basic facts on normal adjacency matrices and representation theory for these groups, so the derivations look reliable. The infinite families via cyclotomic polynomials give explicit, verifiable examples of integral spectra, which is a useful addition for anyone building graphs with controlled eigenvalues. The soft spots are minor and limited in scope. The reduction of weak Waring numbers over finite fields to the ordinary Waring problem recovers a 2012 result by Cochrane and Cipra, so it functions as an illustration rather than a new theorem. Without the full proofs it is difficult to inspect every edge case, such as small q or special relations between k and the field, but the abstract claims match the standard theory and there are no signs of circularity or invented quantities. The imprimitivity and period discussion is also straightforward and secondary. This work is aimed at researchers in algebraic combinatorics who focus on Cayley graphs or spectral properties over finite fields. A reader looking for explicit classifications of spectra or constructions of integral graphs in this setting will get value from the new characterizations and families. It shows clear thinking and direct engagement with the literature on Paley graphs and Waring problems, so it deserves a serious referee. I recommend sending it out for peer review; the core spectral results are grounded enough to benefit from detailed checking.","headline":"The paper cleanly shows generalized Paley graphs have real spectrum exactly when undirected and that only oriented Paley digraphs have three eigenvalues, with a cyclotomic method yielding new infinite integral families.","tokens_in":2470,"tokens_out":502,"would_cite":false,"duration_ms":38978,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[],"headline":"Pure combinatorial spectral theory on finite-field Cayley graphs; no overlap with RS forcing chain","alignment":"orthogonal","rationale":"The paper's core results (real spectrum of Γ(k,q) iff undirected, i.e., -1 ∈ (F_q^*)^k; exactly three eigenvalues only for oriented Paley unions; period-1 generically) are standard facts about abelian Cayley graphs and character sums. These live entirely in finite-field combinatorics and have no structural resemblance to the RS derivation of J-cost, φ-ladders, 8-tick periodicity, or parameter-free constants from a single distinction. No RS-shaped objects (reciprocal costs, golden-ratio identities, recognition lattices) appear.","tokens_in":67284,"confidence":"high","tokens_out":166,"duration_ms":10006,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Generalized Paley graphs on finite fields have real spectra precisely when they are undirected.","keywords":["generalized Paley graphs","spectrum","finite fields","Cayley graphs","eigenvalues","Waring numbers","directed graphs","integrality"],"falsifier":"A single directed generalized Paley graph whose spectrum is entirely real, or a weak Waring number over a finite field whose value differs from the corresponding classic Waring number.","tokens_in":2729,"feed_emoji":"","tokens_out":754,"duration_ms":42235,"temperature":0.7,"pith_summary":"The paper examines generalized Paley graphs Γ(k,q) on the finite field of order q, formed as Cayley graphs whose connection set is exactly the subgroup of k-th powers. It proves that the spectrum consists entirely of real numbers if and only if the graph is undirected, which holds precisely when the connection set is closed under inversion. The work also classifies the directed cases, showing that they always possess at least three distinct eigenvalues and that the oriented Paley graphs together with their disjoint unions are the only ones having exactly three. As an application the graphs are used to reduce the computation of weak Waring numbers over finite fields to the ordinary Waring numbers.","feed_headline":"Generalized Paley graphs are real-valued only when undirected","feed_subtitle":"The same graphs also isolate the directed cases with three eigenvalues and reduce weak Waring numbers to classic ones over finite fields.","key_machinery":"The Cayley graph on the additive group of F_q whose connection set is the subgroup of k-th powers, whose eigenvalues are obtained from the standard character-sum formula over that subgroup.","core_discovery":"We characterize all GP-graphs having real spectrum; namely, Spec(Γ(k,q)) ⊂ ℝ if and only if Γ(k,q) is undirected. We study conditions for integrality and give a method to produce integral GP-graphs through cyclotomic polynomials, yielding several infinite families. For directed GP-digraphs we show they always have three or more eigenvalues and prove that the only ones with exactly three are the oriented Paley graphs or disjoint unions of them. We further show that generically these digraphs have period 1, except for Γ(q-1,q) with q odd, which decomposes into oriented p-cycles of period p. As an application we reduce the computation of weak Waring numbers over finite fields to the computation","pith_inferences":["The real-spectrum criterion may extend to other families of Cayley graphs on finite fields whose connection sets are subgroups.","The reduction of weak Waring numbers could simplify explicit calculations for small exponents or small fields.","The period-one result for most directed cases suggests these digraphs are strongly connected or have simple orbit structures under the field automorphism group."],"forward_implications":["Infinite families of integral generalized Paley graphs arise from cyclotomic polynomials.","Directed generalized Paley graphs always possess at least three distinct eigenvalues.","The oriented Paley graphs and their disjoint unions are the unique directed generalized Paley graphs with exactly three eigenvalues.","All directed generalized Paley graphs except Γ(q-1,q) for odd q have index of imprimitivity one.","Weak Waring numbers over finite fields reduce directly to ordinary Waring numbers via the graph construction."],"fun_headline_variants":["Generalized Paley graphs have real spectra only if undirected","Directed generalized Paley graphs always have three or more eigenvalues","The only GP-digraphs with three eigenvalues are oriented Paley graphs","Weak Waring numbers over finite fields reduced to classic Waring numbers"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The spectrum of each generalized Paley graph is completely determined by the standard character-sum formula for abelian Cayley graphs with no further restrictions imposed by the prime power q.","fun_headline_variants_meta":{"raw":{"variants":["Generalized Paley graphs have real spectra only if undirected","Directed generalized Paley graphs always have three or more eigenvalues","The only GP-digraphs with three eigenvalues are oriented Paley graphs","Weak Waring numbers over finite fields reduced to classic Waring numbers"]},"model":"grok-4.3","cost_usd":0.012743,"raw_usage":{"total_tokens":5550,"prompt_tokens":852,"num_sources_used":0,"completion_tokens":70,"cost_in_usd_ticks":127428000,"prompt_tokens_details":{"text_tokens":852,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":4628,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":852,"tokens_out":70,"duration_ms":58699,"temperature":1.0,"reasoning_tokens":4628,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-10T18:22:15.002223+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A single directed generalized Paley graph whose spectrum is entirely real, or a weak Waring number over a finite field whose value differs from the corresponding classic Waring number.","supporting_citations":[],"review_version":1}