{"id":"4d691eef-39cc-4652-bb64-a7c18ee47837","arxiv_id":"2607.18334","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"All signings that make every quadrilateral of C_n(1,2) unbalanced have spectral radius exactly 2√2 or 2√(cos²(π/n)+cos²(2π/n)), the latter conjecturally minimal.","lead":"This paper finds exact 'vibration sizes' for a special family of 4-regular circular graphs whose edges carry plus and minus signs, and shows the best ones are smaller than a famous theoretical limit. It gives a concrete test case for a major open problem about improving graph spectra.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The reader's weakest assumption — the unproved quadrilateral classification — is indeed the most delicate point of the paper, and the proof of Proposition 1 does rely on it. However, a direct step-sequence analysis confirms the classification is true for n≥10. The concern therefore does not land as a correctness objection; it is a presentational gap (a proof sketch could be added). The central claims — the explicit sub-Kesten signing of Proposition 1 and the twisted-class formula of Proposition 2 — are independently checkable and were verified on inspection. The conjectural global-minimum statement is clearly labeled and supported by exhaustive enumeration, so it does not undermine the proven results. I find no load-bearing flaw and recommend leaving the ACCEPT verdict unchanged.","tokens_in":4375,"tokens_out":26604,"duration_ms":232152,"concrete_test":"Enumerate all 4-step closed walks in C_n(1,2) for a generic even n≥10: for each (a,b,c,d)∈{±1,±2}^4, check whether the partial sums are distinct mod n and return to 0, and compare the resulting edge set to {Q_i}. Since |a|+|b|+|c|+|d|≤8<n, the integer sum must be 0; solving a+b+c+d=0 with entries in {±1,±2} yields only the three multisets described above. This reduces the classification to a short check (or a script over n=8..18) and settles the omitted proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The one genuinely load-bearing point is the unproved classification at the end of §1 that, for n≥10, the quadrilaterals of C_n(1,2) are exactly Q_i=(i,i+1,i+3,i+2). This underpins the identification of 'every quadrilateral unbalanced' with system (1), and hence the rank n−1/four-class conclusion. On inspection the assertion is correct: a 4-cycle is a closed walk with four steps in {±1,±2}. Since the total displacement has absolute value at most 8<n, the step sequence must sum to 0 as an integer, not merely mod n. The only multisets of four steps from {±1,±2} summing to 0 are {+2,+1,−1,−2}, {+2,+2,−2,−2}, and {+1,+1,−1,−1}. The latter two force immediate vertex repetition, while the first, up to rotation/reversal, yields precisely the edge sets {i,i+1}, {i+1,i+3}, {i+3,i+2}, {i+2,i} — exactly Q_i. The extra step-2 4-cycles seen at n=8 require total displacement ±8, excluded by n≥10. Thus the flagged assumption holds. All spectral computations in Propositions 1 and 2 are consistent on rechecking; no internal inconsistency or circular step was found.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies signings of the 4-regular circulant graph C_n(1,2) subject to the F2 constraint that every quadrilateral is unbalanced. For even n≥10 it proves that this constraint system has exactly four switching classes, and it computes the spectrum of the canonical class (step-1 edges all +1, step-2 edges alternating signs) as {±2√(cos²θ_k+cos²2θ_k)}, giving spectral radius exactly 2√2, below the Kesten bound 2√3. It then shows the quadrilateral condition is equivalent to alternating triangle fluxes, parametrizes the four classes by (τ0, α), proves that the spectral radius depends only on α, and computes ρ_-(n)=2√(cos²(π/n)+cos²(2π/n))<2√2 for the two twisted classes. For odd n the system is inconsistent; the exceptional n=8 case is handled separately. The paper concludes with a conjecture, supported by exhaustive enumeration for n=8,...,18, that ρ_-(n) is the global minimum over all 2^{n+1} switching classes.","tokens_in":4656,"tokens_out":19516,"duration_ms":175953,"significance":"If the result stands, it supplies an explicit, parameter-free family of 4-regular signings with spectral radius strictly below the Kesten floor, and an exactly solvable instance of the companion's parity-family method. The spectral derivations are self-contained and do not rely on the companion manuscript for their main conclusions. The exhaustive enumeration is reproducible from the supplied repository and gives solid computational evidence for the conjecture. The conjecture itself, with its Lieb flux-phase analogy, is a meaningful open problem. The paper is honest in distinguishing theorem from conjecture.","major_comments":[],"minor_comments":[{"comment":"The assertion that for n≥10 the quadrilaterals of C_n(1,2) are exactly Q_i=(i,i+1,i+3,i+2) is load-bearing: it converts 'every quadrilateral unbalanced' into system (1) and underlies the rank n−1 incidence argument. It is stated without proof. I checked the claim and it is correct: a 4-cycle is a closed walk with four steps in {±1,±2}; for n≥10 the total displacement has absolute value at most 8<n, so the step multiset must sum to zero over Z, and the only simple such walks are the Q_i. Please insert a short proof of this classification and flag the n=8 exception explicitly at that point.","section":"§1 (quadrilateral classification)"},{"comment":"The diagonal-unitary conjugacy criterion is invoked without proof: two Hermitian matrices with the same connected support, all off-diagonal entries of modulus one, and equal cycle holonomy are conjugate by a diagonal unitary. This is standard, but since it is essential for the isospectrality of A(φ) with the real signed adjacency matrix, a one-sentence justification should be added. It would also be helpful to state explicitly that checking the triangle holonomies and the Hamilton-cycle holonomy suffices because those cycles form a basis of the cycle space.","section":"Proposition 2 proof"},{"comment":"The side claim that uniformly random signings concentrate above the Kesten bound 2√3 from n≈480 is stated without proof and relies on the companion manuscript [2]. Since [2] is not part of this submission, this assertion should either be proved, replaced by a precise bound, or explicitly marked as conditional on [2].","section":"Remark 5 / Conjecture 3"},{"comment":"The sentence 'the maximum over the shifted lattice is attained at π/n' is terse. The full argument is that g(π/n)>1 for all even n≥8, while all other shifted momenta in [π/2, π−π/n] have g(θ)≤1 by symmetry of g and the already-established monotonicity on [0,π/2]. Spelling this out would improve readability.","section":"Proposition 2 proof"}],"recommendation":"minor_revision","confidential_remarks":"The main theorems are independent of the companion manuscript [2], but the paper's framing and the random-signing concentration remark lean on it. The editor may wish to confirm that the journal is comfortable with a companion-note format. No circularity or fitted parameters were found; the only substantive gap is the missing proof of the quadrilateral classification, which is easily supplied."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nBottom line: this paper deserves a serious referee. It gives explicit 4-regular signings of the circulant C_n(1,2) with spectral radius 2√2, and a twisted family with radius 2√(cos²(π/n)+cos²(2π/n)), strictly below the Kesten bound 2√3. That is a concrete, verifiable instance of the Bilu-Linial phenomenon for a non-bipartite family, and the formulas are new as far as the cited literature goes.\n\nWhat is actually well done: the spectral derivation is clean and self-contained. The 2x2 Fourier block decomposition in Propositions 1 and 2 is standard but correctly executed, and the holonomy transfer from the Hermitian block matrix to a real signing is justified. The cycle-space basis argument (triangles plus Hamilton cycle) is neat. The conjectured optimality, supported by exhaustive enumeration over all switching classes for n=8,...,18, is a reasonable conjecture and honestly labeled. The paper also flags its own exceptional case n=8 and handles it separately.\n\nSoft spots, in proportion: the most load-bearing combinatorial assertion — that for n≥10 the quadrilaterals are exactly Q_i — is stated without proof. It is true (the stress-test argument is simple: four steps in {±1,±2} summing to zero force that shape), but it should be a one-paragraph lemma. The odd-n claim \"at most n-1 quadrilaterals can be unbalanced\" is asserted without proof; minor, since the main results are even-n. The companion manuscript [2] is cited for motivation and for the random-signing concentration remark, not for the spectral claims; that is fine, but the concentration claim is not established here. No commit hash for the verification suite is given, which is a small reproducibility nit. The conjecture itself is open, as the abstract says.\n\nOverall: no load-bearing flaw. The math checks out, the result is modest but real, and the paper is honest about what is proven and what is conjectural.","headline":"Explicit sub-Kesten signings of C_n(1,2); the spectral math is correct, the main conjecture is open, and the paper deserves refereeing.","tokens_in":5148,"tokens_out":1788,"would_cite":true,"duration_ms":16367,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C50","05C22"],"pacs":[],"model":"deepseek-v4-flash","headline":"A parity condition on quadrilaterals drives a signed degree-4 cyclic graph to a spectral radius below 2√2.","keywords":["signed graphs","circulant graphs","spectral radius","switching classes","quadrilateral parity","invariant subspaces","flux minimization","4-regular graphs"],"falsifier":"Exhaustively enumerate all 2^{n+1} switching classes for n=20 and compare the minimum spectral radius with ρ_-(20)=2√(cos²(π/20)+cos²(π/10)); a class with smaller radius would refute Conjecture 3. This is the first even size past the paper's verification range (n≤18).","tokens_in":4244,"feed_emoji":"📉","tokens_out":14423,"duration_ms":141015,"temperature":0.7,"pith_summary":"The paper studies signings of the degree-4 cyclic graph C_n(1,2), whose vertices are the integers modulo n and whose edges join vertices at distance 1 or 2. A signing marks each edge +1 or -1; switching changes signs around a vertex without changing the spectrum. The paper imposes the mod-2 rule that every four-cycle has sign product -1, and proves that for even n≥10 this rule leaves exactly four switching classes. On these classes the signed adjacency matrix splits into two-dimensional invariant subspaces, and the best spectral radius comes out to 2√(cos²(π/n)+cos²(2π/n)), which is below 2√2 and therefore below the degree-4 spectral floor 2√3. The exceptional case n=8 is treated separately, and exhaustive enumeration through n=18 supports the conjecture that no other switching class attains a smaller spectral radius.","feed_headline":"Parity rule pushes a 4-regular graph below 2√2","feed_subtitle":"Four switchings survive; the twisted one beats the 2√3 degree-four bound for every even n.","key_machinery":"The load-bearing object is the signed adjacency matrix A_σ = C_1 + D C_2, where C_1 and C_2 are the unsigned adjacency matrices of the step-1 and step-2 circulants and D is the diagonal matrix with entries (-1)^j. Its invariance on the two-dimensional subspaces spanned by the pair of exponential vectors indexed by k and k+n/2 turns the spectral problem into eigenvalues of a traceless 2×2 matrix, hence ±√(a²+b²) with a=2cosθ_k, b=2cos2θ_k. The combinatorial selector is the mod-2 system s·χ_{Q_i}=1; its incidence matrix has rank n-1, so the solution set is four switching classes, coordinatized by the sign of one triangle and the sign product along the step-1 spanning cycle. For the twisted cla","core_discovery":"The central discovery is that the parity rule 'every quadrilateral unbalanced' selects the near-optimal signings. For even n the solutions form four switching classes; the representative with +1 on step-1 edges and (-1)^i on step-2 edges has spectrum ±2√(cos²θ_k+cos²2θ_k), spectral radius exactly 2√2. The two classes with odd sign product around the step-1 spanning cycle have the smaller radius ρ_-(n)=2√(cos²(π/n)+cos²(2π/n))<2√2. The paper proves these identities by diagonalizing the signed adjacency matrix on the invariant subspaces spanned by paired exponential vectors, and conjectures, with exhaustive verification through n=18, that the smaller value is the minimum over all switching cla","pith_inferences":["The paper leaves implicit that the two-dimensional block structure is a recipe: for any two-step circulant with a shift-invariant quadrilateral incidence matrix, the same rank calculation would isolate a small family of candidate optimizer signings, making the method transferable.","The phase-twisted operator A(φ) has not been optimized over φ by the paper; a natural extension is to minimize the spectral radius over φ between 0 and π/n and see whether ρ_-(n) is the endpoint minimum.","The contrast between random signings, which concentrate above the floor, and the parity family, which sits below it, suggests a testable general mechanism: constraining all short even cycles to be unbalanced may be a deterministic way to beat random spectral concentration on other 4-regular graphs."],"forward_implications":["For every even n≥10 there is an explicit signing of C_n(1,2) with spectral radius 2√2, strictly below the degree-4 bound 2√3; for the twisted classes the radius is even smaller, 2√(cos²(π/n)+cos²(2π/n)).","The quadrilateral parity system has exactly four switching-class solutions; every signing in the family has one of two spectral radii depending only on the sign product around the step-1 spanning cycle.","No signing can unbalance all quadrilaterals when n is odd; at most n-1 quadrilaterals can be made unbalanced.","If Conjecture 3 holds, the twisted classes are global optimizers among all 2^{n+1} switching classes, a flux-minimization statement for this family; this is verified exhaustively through n=18.","At n=8, requiring all ten quadrilaterals to be unbalanced selects exactly the twisted classes and excludes the 2√2 class."],"fun_headline_variants":["Parity rule on quadrilaterals yields minimal spectral radius <2√2","Four switching classes, two with radius below 2√2","Signed circulants beat the Ramanujan bound with a quadrilateral rule","Conjecture: signings from quadrilateral parity minimize radius, verified to n=18","Quadrilateral parity rule: only four signings survive, minimal radius <2√2"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof assumes without proof that for n≥10 every quadrilateral of C_n(1,2) is one of the n cycles Q_i=(i,i+1,i+3,i+2); if another four-cycle existed for some n, the mod-2 system (1) would no longer encode the intended condition, and the four-class conclusion would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Parity rule on quadrilaterals yields minimal spectral radius <2√2","Four switching classes, two with radius below 2√2","Signed circulants beat the Ramanujan bound with a quadrilateral rule","Conjecture: signings from quadrilateral parity minimize radius, verified to n=18","Quadrilateral parity rule: only four signings survive, minimal radius <2√2"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00237,"raw_usage":{"total_tokens":8998,"prompt_tokens":819,"completion_tokens":8179,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":563,"completion_tokens_details":{"reasoning_tokens":8079}},"tokens_in":563,"tokens_out":8179,"duration_ms":61067,"temperature":1.0,"reasoning_tokens":8079,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T18:14:46.775049+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhaustively enumerate all 2^{n+1} switching classes for n=20 and compare the minimum spectral radius with ρ_-(20)=2√(cos²(π/20)+cos²(π/10)); a class with smaller radius would refute Conjecture 3. This is the first even size past the paper's verification range (n≤18).","supporting_citations":[],"review_version":1}