{"id":"a09d5121-b5bc-416b-aab1-9d4af2b48599","arxiv_id":"2607.20907","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Structured cosine sums on cyclic groups are classified through vanishing sums of roots of unity, yielding a small-weight Fourier rigidity theorem and eigenvalue multiplicity bounds for cyclic Cayley graphs.","lead":"This paper studies finite sums of cosines at rational multiples of 2π by encoding them as evaluations in integral group rings, and uses vanishing sums of roots of unity to decide when such sums vanish and how many inputs give the same value. The payoff is a set of multiplicity bounds and rigidity results for eigenvalues of cyclic Cayley graphs.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.2's induction leaves the self-conjugate minimal-summand case u in C_p unaddressed; Theorem B's vanishing criterion is not fully proven.","rationale":"The paper's central rigidity theorem, Theorem 4.1, survives my stress-test: the square-free induction in Proposition 4.4, the symmetric disjoint case in Proposition 4.10, Lemma 4.7, and the coset-lifting argument in Section 4.2 all appear internally consistent, with only minor implicit symmetry justifications. The load-bearing concern is therefore not in the rigidity proof but in Theorem 3.2, exactly the gap the reader identified: the self-conjugate minimal-summand case u in C_p is omitted, and the reduction to eσ(C_p)+fσ(C_q) is not justified by the written induction. This does affect the paper's advertised vanishing criteria (Theorem B, Corollary 3.3, and the zero-eigenvalue applications), so a CONDITIONAL verdict remains appropriate. I do not share the reader's broader worry about the completeness of the cited Lam–Leung and Poonen–Rubinstein classifications; those are published results and no specific misapplication was identified. Hence agreement_with_reader is partial, and the recommended verdict is UNCHANGED.","tokens_in":31819,"tokens_out":46203,"duration_ms":401759,"concrete_test":"Write a small exact program to enumerate all symmetric even elements X in N[Z/45Z] with ev_45(X) = 0 and weight at most 24, and check that each is a nonnegative integer combination of the blocks A(u;15), A(v;9), and D_{3,5}(45) listed in Theorem 3.2; repeat for N = 63 = 3^2·7. A counterexample would refute Theorem 3.2 as stated. If the search passes, the missing u in C_p case still needs to be supplied as an explicit induction step, but the theorem is likely repairable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Theorem 3.2 (pp. 10–11), after choosing a minimal vanishing submultiset Y of X, the proof treats only the case Y = σ(u+C_p) with u not in C_p, where Y and Y* are disjoint and A(u; N/p) = Y+Y* can be subtracted. The case u in C_p, i.e. Y = Y* = C_p, is never handled, yet the text jumps to 'we are reduced to X = eσ(C_p)+fσ(C_q)'. This is not a cosmetic omission: take X = C_p + C_q on N = p^a q^b. The multiset X is symmetric, of even weight, and vanishes; it equals the block D_{p,q}(N). But if the induction chooses the self-conjugate minimal summand C_p, subtracting Y leaves C_q, which has odd weight and no longer satisfies the symmetric-even induction hypothesis. Closing the gap requires proving that a self-conjugate minimal summand can be paired with a second summand (another C_p, giving A(0; N/p), or a C_q, giving D_{p,q}) before subtraction. Without that, Corollary 3.3 and the zero-eigenvalue criteria in Theorems 5.2 and 5.4 are not established. By contrast, the rigidity engine in Theorem 4.1, namely Propositions 4.4, 4.10, 4.11 and the lifting argument in Section 4.2, does not use Theorem 3.2; I checked the main steps of that proof and found no comparable gap, although several symmetry-based inequalities are left implicit. The cited Lam–Leung and Poonen–Rubinstein results are published, and I see no independent basis to doubt their completeness.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies structured cosine sums C_S(k)=Σ cos(2π k s_j/N) through the group-ring evaluation ev_N and vanishing sums of roots of unity. It gives a roots-of-unity proof of Włodarski's four-cosine classification, a sharp bound |R(c)|≤3 for two-cosine fibers, and block decompositions for symmetric vanishing multisets when N=p^a q^b (Theorem 3.2) and for weight 2p_1 (Theorem 3.5). The central rigidity theorem (Theorem 4.1) states that for odd N with smallest prime divisor p, equality of one nonzero primitive Fourier coefficient determines a multiset of weight ≤p uniquely, and in the symmetric case weight ≤2p uniquely. The final section applies these results to cyclic Cayley graphs: zero-eigenvalue criteria, multiplicity bounds mult(μ)≤|S| for |S|≤2p when S contains a unit, an irrationality corollary for μ_1, and, for square-free N with S a subgroup of units, a complete description of the spectrum as scaled Gaussian periods with a criterion for layer intersection.","tokens_in":32121,"tokens_out":13593,"duration_ms":118780,"significance":"If the proofs are completed, the paper's main contributions are solid and well motivated. The Fourier rigidity theorem is a clean, sharp result with explicit counterexamples showing the optimality of its hypotheses (Examples 4.5 and 4.13). The multiplicity bound mult(μ)≤|S| in the small-support case and its optimality construction are valuable, and the Gaussian-period description in Theorem 5.25 gives a complete spectral decomposition for square-free cyclic unit-subgroup Cayley graphs, including a precise condition for layer collisions. The paper is careful about external dependencies: it states the Lam–Leung and Poonen–Rubinstein results it uses, and the main rigidity proof in Section 4 does not rely on the problematic vanishing criterion in Theorem 3.2. I found no circularity and no fitting of free parameters. The remaining issue is a genuine but local gap in the proof of Theorem 3.2.","major_comments":[{"comment":"The induction in the proof of Theorem 3.2 (pp. 10–11) treats only the case where the chosen minimal vanishing summand is Y=σ(u+C_p) with u∉C_p. The self-conjugate case u∈C_p, and its analogue for C_q, is never handled. The sentence 'Iterating this process for p and q, we are reduced to the case X=eσ(C_p)+fσ(C_q)' is therefore not justified: if the selected minimal summand is C_p, one cannot form a conjugate pair to subtract, and subtracting the single block C_p changes the parity of the weight, so the induction hypothesis (symmetric, even weight) no longer applies. The gap is not cosmetic, since the theorem is true for X=C_p+C_q=D_{p,q}(N) but the proof fails if C_p is the minimal summand chosen. A repair is available: first delete all non-self-conjugate translates in conjugate pairs, then use e+f≡0 mod 2 to subtract one of A(0;N/p), A(0;N/q), or D_{p,q}(N). This argument must be written into the proof. Because Corollary 3.3 and the zero-eigenvalue criteria in Theorems 5.2 and 5.4 rely on Theorem 3.2, the stated if-and-only-if results are not fully proven as written.","section":"Section 3, proof of Theorem 3.2"},{"comment":"The block decomposition for N=2^a q^b is asserted in Remark 3.4 with only a reference to 'the same inductive argument', but no proof is supplied. The statement involves additional self-conjugate blocks, and handling those requires the same parity-pairing argument that is missing in the odd case of Theorem 3.2. Since Remark 5.3 converts this classification into a zero-eigenvalue criterion for even cyclic Cayley graphs, the criterion is currently conditional on an unproved assertion. Either a proof of Remark 3.4 should be included, or the statement should be explicitly marked as unproved.","section":"Section 3, Remark 3.4 and Section 5.2, Remark 5.3"}],"minor_comments":[{"comment":"The phrase 'This is z special case' contains a typo and should read 'This is a special case'.","section":"Section 1, after (1.3)"},{"comment":"The sentence 'Then Theorem 2.3(3) then ε(A)=ε(B) to p,2p' is garbled. More importantly, the claimed equivalence of Propositions 4.10 and 4.11 requires an argument that symmetric disjoint vanishing multisets of equal weight at most 2p cannot have weight q, the second smallest prime divisor of M; this fact is not stated and should be justified, or the remark should be softened.","section":"Section 4.1, Remark 4.12"},{"comment":"The assertion that at most one pair in R(c) has equal reduced denominators is justified only by 'inspection' of the Włodarski list; a short table of the relevant sporadic quadruples would make this step easier to verify.","section":"Section 2.2, Proposition 2.12"}],"recommendation":"major_revision","confidential_remarks":"The main rigidity theorem in Section 4 appears sound and is independent of the problematic vanishing criterion. The gap in Theorem 3.2 is local and fixable, so I do not recommend rejection; however, it affects the central if-and-only-if statements in Theorems B, 5.2, and 5.4, so the manuscript should be revised before publication. I saw no circularity or parameter-fitting concerns."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new part is the small-weight Fourier rigidity (Theorem C) and the two-cosine fiber bound (Theorem A). The vanishing-sum classification in Theorem B is not fully proven as written.\n\nThe group-ring translation is standard, but Theorem C is a real addition: for odd N with smallest prime p, equality of one nonzero primitive Fourier coefficient forces equality of multisets up to weight 2p (symmetric case). I followed the square-free proof and the coset lifting in Section 4.2; the argument is long but structurally coherent, and the examples showing sharpness of p and 2p are useful. The Lam–Leung and Poonen–Rubinstein inputs are cited and used correctly. Theorem A is a neat byproduct: the root-of-unity proof of Wlodarski plus a finite inspection gives |R(c)| ≤ 3, with the exceptional triples listed.\n\nThe soft spot is real. In Theorem 3.2, after choosing a minimal vanishing summand Y = σ(u + C_p), the proof only handles u ∉ C_p, where Y and Y* are disjoint and can be subtracted. The self-conjugate case u ∈ C_p, where Y = Y* = C_p, is never treated; the text jumps straight to the reduced form X = eσ(C_p) + fσ(C_q). That is exactly the situation X = C_p + C_q, which vanishes and is symmetric, but subtracting C_p leaves C_q of odd weight. You need an argument that pairs the self-conjugate summand with something before subtracting. Without that, Corollary 3.3 and the zero-eigenvalue criteria in Theorems 5.2 and 5.4 are not established. This is patchable, most likely, but it is a gap in a stated if-and-only-if theorem. The stress-test note is right that Section 4 (the rigidity engine) does not rely on Theorem 3.2; I checked the dependency chain, and the multiplicity bounds in Theorem D stand independently.\n\nMinor issues: Proposition 2.12's proof relies on a finite inspection of Wlodarski's sporadic quadruples that is summarized rather than shown in full; that is a routine check but should be spelled out. The optimality discussion in Section 5 is honest about what happens when S contains no unit.\n\nVerdict: this deserves a serious referee, but not acceptance in current form. The author needs to close the self-conjugate gap or restrict Theorem B accordingly. If that is done, the paper is a solid contribution for number theorists working on vanishing sums of roots of unity and for spectral graph theorists interested in cyclic Cayley graphs. I would not desk-reject it; I would send it out and ask for a revision focusing on Theorem 3.2.","headline":"New rigidity theorem and a sharp two-cosine bound are real, but the vanishing-sum classification has a self-conjugate proof gap that needs fixing before acceptance.","tokens_in":32676,"tokens_out":3149,"would_cite":true,"duration_ms":27827,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11L03","11R18","05C50","11T22","20C05"],"pacs":[],"model":"deepseek-v4-flash","headline":"For odd N, one nonzero Fourier coefficient can force two small multisets to be identical, and this rigidity bounds eigenvalue multiplicities of cyclic Cayley graphs.","keywords":["vanishing sums of roots of unity","cosine sums","Fourier rigidity","group ring","cyclic Cayley graphs","eigenvalue multiplicity","Gaussian periods","trigonometric Diophantine equations"],"falsifier":"Enumerate all multisets on $\\mathbb{Z}/15\\mathbb{Z}$ of weight at most 3 and compare their nonzero evaluations $ev_{15}$; two distinct multisets with the same nonzero value would refute Theorem 4.1. Enumerate symmetric multisets of weight 6 on $\\mathbb{Z}/15\\mathbb{Z}$: a distinct pair with the same nonzero evaluation would refute the symmetric form.","tokens_in":31570,"feed_emoji":"📐","tokens_out":12563,"duration_ms":103201,"temperature":0.7,"pith_summary":"The paper studies finite sums of cosines of rational angles attached to a multiset S in Z/NZ by rewriting them as evaluations $2C_S(k)=ev_N(\\operatorname{Sym}(kS))$ in the integral group ring $\\mathbb{Z}[\\mathbb{Z}/N\\mathbb{Z}]$. Its central result is a small-weight Fourier rigidity: for odd N with smallest prime divisor p, two multisets of weight at most p that share a nonzero Fourier coefficient are the same multiset, and the same holds for symmetric multisets of equal weight at most 2p. This rigidity yields concrete spectral consequences for cyclic Cayley graphs: nonzero eigenvalues of small-support symmetric generating sets have multiplicity at most |S|, and the first cosine sum is either 0 or irrational for odd composite N. The paper also gives block-decomposition criteria for vanishing cosine sums and, for square-free N, an exact description of the spectrum as scaled Gaussian periods when S is a subgroup of the unit group.","feed_headline":"One nonzero Fourier value fixes small multisets","feed_subtitle":"For odd Z/NZ, weights up to the smallest prime factor are rigid, bounding eigenvalue multiplicity.","key_machinery":"The load-bearing object is the evaluation map $ev_N:\\mathbb{Z}[\\mathbb{Z}/N\\mathbb{Z}]\\to\\mathbb{Z}[\\zeta_N]$, together with the identity $2C_S(k)=ev_N(\\operatorname{Sym}(kS))$, which converts cosine sums into vanishing-sum questions in the integral group ring. The argument for rigidity reduces N to its square-free radical M, decomposes $\\mathbb{Z}/M\\mathbb{Z}$ by the Chinese remainder theorem along the smallest prime factor p, and slices the group ring into fibers; the comparison lemma shows that equal evaluations force the fiber evaluations to differ by a constant, after which weights and the known classification of minimal vanishing sums force the fibers to agree. The vanishing criteria use the same classification: for two-prime moduli, minimal vanishing sums are exactly the prime subgroups up to translation, so symmetric vanishing multisets decompose into the three block types $A(u;N/p)$, $A(v;N/q)$, and $D_{p,q}(N)$. For the two-term fiber result, the proof uses the six-form list of conjugation-stable vanishing sums of weight 8 to recover the four-cosine classification and then sharpens it to the bound $|R(c)|\\leq 3$ with explicit equality cases.","core_discovery":"The central claim is Theorem 4.1. Let N be odd and let p be its smallest prime divisor. If X and Y are multisets on $\\mathbb{Z}/N\\mathbb{Z}$ with max weight at most p and $ev_N(X)=ev_N(Y)\\neq 0$, then $X=Y$; if X and Y are symmetric with equal weight at most 2p and the same nonzero evaluation, then $X=Y$. Because Galois automorphisms move $ev_N(1)$ to every primitive Fourier coefficient, equality at one nonzero primitive coefficient forces equality of all primitive coefficients, so the statement is genuinely a Fourier rigidity. The nonzero condition and the weight bounds are each necessary, as the paper shows by examples where zero evaluation or weight $2p+1$ breaks rigidity. A companion vanishing theorem says that for $N=p^a q^b$ with distinct odd primes, any symmetric vanishing multiset is a finite sum of the explicit blocks $A(u;N/p)$, $A(v;N/q)$, and $D_{p,q}(N)$; for weight equal to twice the smallest prime divisor, vanishing occurs exactly when the multiset is a single block $A(u;N/p_1)$.","pith_inferences":["The rigidity mechanism suggests a general principle beyond the paper's statement: on any cyclic group, once the weight falls below the smallest nonzero vanishing weight, one nonzero Fourier coefficient should determine the multiset; identifying that threshold for arbitrary N is a natural next step.","The zero-eigenvalue examples show that failure of rigidity is concentrated at evaluation zero, where translations of prime subgroups give identical zero spectra with different supports; this points to the kernel of the evaluation map, rather than the smallness of support, as the true obstruction to uniqueness.","The square-free layer criterion translates spectral collisions into the number-theoretic condition $\\mu(r)/\\varphi(r)=\\mu(t)/\\varphi(t)$, so the existence of eigenvalue coincidences across layers can be decided by factoring N; this makes the $N=1365$ phenomenon a member of a predictable family rather than an isolated example."],"forward_implications":["For odd composite N with smallest prime factor p, any symmetric generating set S with $|S|\\leq 2p$ that contains a unit has the property that every nonzero eigenvalue $\\mu$ satisfies $\\operatorname{mult}(\\mu)\\leq |S|$; the bound is attained when a p-th power congruence has nontrivial solutions modulo N.","For even $N>3$ with $S=\\{\\pm s_1,\\pm s_2\\}$ containing a unit, every nonzero eigenvalue has multiplicity at most 6, and this is sharp, as shown by $N=30$ with $S=\\{\\pm 1,\\pm 3\\}$, where the eigenvalue 1 has multiplicity 6.","For odd composite N, the eigenvalue $\\mu_1=\\sum_{s\\in S}2\\cos(2\\pi s/N)$ is either 0 or irrational; it is 0 exactly when $|S|=2p$ and $S=A(u;N/p)$ for some u.","For square-free N, when S is a subgroup of the unit group, the eigenvalue in the m-layer is a scaled Gaussian period $\\Lambda_{m,c}$; each layer contributes $\\varphi(m)/|S_m|$ distinct eigenvalues, each with layer multiplicity $|S_m|$ and degree $\\varphi(m)/|S_m|$ over $\\mathbb{Q}$.","Two layers m and n share an eigenvalue exactly when they have the same eigenvalue set, which happens exactly when the kernels of reduction to $\\gcd(m,n)$ lie in $S_m$ and $S_n$ and $\\mu(r)/\\varphi(r)=\\mu(t)/\\varphi(t)$ for $r=m/g$ and $t=n/g$."],"supporting_citations":[{"why":"Supplies the classification of minimal vanishing sums of roots of unity and the kernel/comparison theorems on which the block-decomposition and rigidity proofs rest.","marker":"[LL00]"},{"why":"Supplies the six-form list of conjugation-stable vanishing sums of weight 8 used to recover the four-cosine classification and prove Theorem A.","marker":"[PR98]"},{"why":"Provides the four-cosine vanishing classification that the paper reproves from roots of unity and then sharpens to the two-cosine fiber bound.","marker":"[W lo69]"},{"why":"Provides the coset-slicing and radical-reduction technique used in the rigidity proof and the normal-basis fact used in the square-free spectral analysis.","marker":"[Len79]"},{"why":"Gives the character formula identifying eigenvalues of abelian Cayley graphs with the cosine sums, connecting the algebraic results to spectral statements.","marker":"[Bab79]"},{"why":"Supplies the Möbius-function identity for sums of primitive roots of unity used in the normal-basis criterion and in the layer intersection analysis.","marker":"[HW08]"}],"fun_headline_variants":["Single Fourier value pins small multisets","Weight up to smallest prime factor: rigidity","Vanishing sums classify symmetric multisets","Cayley graph spectra via cosine sum rigidity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The arguments depend on the completeness of the previously published lists of minimal ways roots of unity can sum to zero, and on one case in the induction (where a vanishing piece equals its own reflection) being closable; if either assumption fails, the paper's vanishing and rigidity conclusions are not established.","fun_headline_variants_meta":{"raw":{"variants":["Single Fourier value pins small multisets","Weight up to smallest prime factor: rigidity","Vanishing sums classify symmetric multisets","Cayley graph spectra via cosine sum rigidity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000921,"raw_usage":{"total_tokens":3927,"prompt_tokens":897,"completion_tokens":3030,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":513,"completion_tokens_details":{"reasoning_tokens":2976}},"tokens_in":513,"tokens_out":3030,"duration_ms":21541,"temperature":1.0,"reasoning_tokens":2976,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:32:33.141435+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate all multisets on $\\mathbb{Z}/15\\mathbb{Z}$ of weight at most 3 and compare their nonzero evaluations $ev_{15}$; two distinct multisets with the same nonzero value would refute Theorem 4.1. Enumerate symmetric multisets of weight 6 on $\\mathbb{Z}/15\\mathbb{Z}$: a distinct pair with the same nonzero evaluation would refute the symmetric form.","supporting_citations":[],"review_version":1}