{"id":"0a4f53fa-40ca-4f78-b493-0700068c72dd","arxiv_id":"1907.10881","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":4.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Explores quadratic cyclic sequences and their links to cyclotomic polynomials and fixed-angle plane walks, identifying non-symmetric phenomena for n≥12 from algebraic numbers of modulus one outside the roots of unity.","lead":"The paper explores connections between sequences defined by a quadratic recurrence, cyclotomic polynomials, and closed walks in the plane that turn left or right by a fixed angle 2π/n. It notes non-symmetric behavior for n at least 12 arising from certain algebraic numbers on the unit circle that are not roots of unity.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"Non-root-of-unity unit-modulus algebraic numbers cannot generate periodic (cyclic) sequences satisfying the quadratic recurrence unless coefficients vanish.","rationale":"The reader's weakest_assumption already isolates the algebraic realization step; the periodicity obstruction is the precise point at which that assumption fails for the non-cyclotomic case. The rest of the paper (Eulerian digraphs, geometric walks) may still be valid for the cyclotomic part, so the verdict moves only to CONDITIONAL pending verification that the non-root examples are not actually used for the cyclic sequences themselves.","tokens_in":1565,"tokens_out":331,"duration_ms":39397,"concrete_test":"Locate the explicit algebraic number α and the associated sequence (or recurrence) given for any n≥12; generate the first 200 terms numerically and test whether the sequence is periodic with period dividing a small multiple of n; if the values fail to repeat, the cyclic claim does not hold for that example.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The quadratic difference relation is a linear homogeneous recurrence whose characteristic roots are algebraic numbers of modulus one. A non-trivial linear combination A α^k + B β^k is periodic if and only if every root lies on the unit circle and is a root of unity (otherwise the orbit is dense on the circle by equidistribution). The central claim asserts that examples for n≥12 arise precisely from such non-root-of-unity numbers. This directly contradicts the requirement that the sequences be cyclic and that the corresponding walks close after finitely many steps with fixed turn angle 2π/n.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript explores relations between cyclic sequences determined by a quadratic difference relation, cyclotomic polynomials, Eulerian digraphs and walks in the plane. These walks correspond to closed paths with a fixed turn angle of 2π/n at each step. The central claim is that non-symmetric phenomena occur for n≥12, with examples arising from algebraic numbers of modulus one which are not nth roots of unity.","tokens_in":1677,"tokens_out":277,"duration_ms":15070,"significance":"If the claimed examples could be realized, the work would connect quadratic recurrences, algebraic units on the unit circle, and periodic planar walks, potentially yielding new combinatorial and number-theoretic constructions.","major_comments":[{"comment":"Abstract: the assertion that non-symmetric phenomena for n≥12 arise from algebraic numbers of modulus one that are not nth roots of unity is load-bearing for the paper's main observation. This directly contradicts the fact that a non-trivial solution to a linear homogeneous recurrence with characteristic roots on the unit circle is periodic if and only if those roots are roots of unity (otherwise the powers are dense on the circle by Weyl equidistribution). No derivation or example in the manuscript can circumvent this property of the recurrence.","section":"Abstract"}],"minor_comments":[],"recommendation":"reject","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading of the manuscript and for identifying the central issue in the abstract. We respond to the major comment below.","responses":[{"response":"We agree with the referee that the claim in the abstract is inconsistent with the standard theory of linear homogeneous recurrences. The assertion that non-symmetric phenomena for n≥12 arise from algebraic numbers of modulus one that are not nth roots of unity cannot be maintained, as no derivation or example can circumvent the periodicity requirement. We will revise the abstract and related sections of the manuscript to remove this claim and instead describe the non-symmetric phenomena strictly in terms of the Eulerian digraphs and fixed-angle planar walks, without invoking such algebraic numbers as characteristic roots of a recurrence. This revision will be made in the next version.","revision_made":"yes","referee_comment":"[Abstract] Abstract: the assertion that non-symmetric phenomena for n≥12 arise from algebraic numbers of modulus one that are not nth roots of unity is load-bearing for the paper's main observation. This directly contradicts the fact that a non-trivial solution to a linear homogeneous recurrence with characteristic roots on the unit circle is periodic if and only if those roots are roots of unity (otherwise the powers are dense on the circle by Weyl equidistribution). No derivation or example in the manuscript can circumvent this property of the recurrence."}],"tokens_in":1128,"tokens_out":298,"duration_ms":21786,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main point is that this paper asserts non-symmetric phenomena for n≥12 arising from algebraic numbers of modulus one that are not nth roots of unity, yet that setup cannot yield periodic sequences under a quadratic difference relation. A linear homogeneous recurrence has solutions Aα^k + Bβ^k. When both roots lie on the unit circle but are not roots of unity, equidistribution makes the sequence dense rather than periodic, so the corresponding fixed-angle walk cannot close after finitely many steps. The abstract presents these numbers as the source of the claimed examples, which suggests either the sequences are not truly cyclic or the coefficients reduce to the trivial case. Either way the central observation does not hold up on standard recurrence theory. The paper does connect the quadratic relation to fixed-turn walks in the plane and to Eulerian digraphs, and it correctly flags the role of cyclotomic polynomials for the root-of-unity cases. That geometric framing is a reasonable way to visualize the problem and could be useful for readers already working on recurrence sequences or combinatorial walks. The soft spot sits at the core: the non-symmetric claim for larger n relies on precisely the numbers that prevent periodicity. No indication appears that the paper modifies the recurrence, chooses special initial conditions, or restricts to cases where one coefficient vanishes. Without explicit sequences or a proof that bypasses the equidistribution obstruction, the observation stays unsupported. This work would interest a narrow group of people studying algebraic recurrences or angle-constrained paths, but only after the periodicity issue is resolved. A reader looking for new examples or theorems on unit-modulus roots would find little to take away. It does not merit sending to referees.","headline":"The paper's claim that non-root-of-unity unit-modulus algebraic numbers produce cyclic quadratic sequences for n≥12 runs into a direct contradiction with the periodicity conditions for linear recurrences.","tokens_in":2164,"tokens_out":413,"would_cite":false,"duration_ms":21213,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":{"model":"grok-4.3","evidence":[],"headline":"Quadratic cyclic sequences and fixed-angle walks in combinatorial geometry","alignment":"orthogonal","rationale":"The paper's core machinery is a quadratic difference relation on cyclic sequences (eq. 1-2) that generates polynomials, Eulerian digraphs, and planar walks with constant turning angle 2π/n. This is discrete/combinatorial geometry (math.CO) with no overlap to RS forcing from a single distinction. No J-cost functional equations, φ-ladder, 8-tick periodicity, or parameter-free constant derivations appear; the structures are unrelated to AbsoluteFloorClosure, Cost, or AlexanderDuality modules.","tokens_in":66020,"confidence":"high","tokens_out":149,"duration_ms":10052,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"For turn angles of 2π/n with n at least 12, quadratic cyclic sequences include non-symmetric cases generated by algebraic numbers of modulus one that are not roots of unity.","keywords":["quadratic cyclic sequences","cyclotomic polynomials","Eulerian digraphs","fixed turn angle","plane walks","algebraic numbers of unit modulus","roots of unity","non-symmetric phenomena"],"falsifier":"A calculation for n=12 showing that every quadratic cyclic sequence remains symmetric and arises only from nth roots of unity.","tokens_in":2463,"feed_emoji":"","tokens_out":447,"duration_ms":15287,"temperature":0.7,"pith_summary":"The paper connects cyclic sequences defined by a quadratic difference relation to closed walks in the plane that turn left or right by a fixed angle at each step. These walks correspond to Eulerian digraphs and are analyzed using cyclotomic polynomials. The key finding is that when the fixed angle is 2π/n, symmetry breaks for n greater than or equal to 12. Examples come from algebraic numbers on the unit circle that are not nth roots of unity. This reveals new phenomena in how such sequences behave beyond the cases tied to roots of unity.","feed_headline":"Non-symmetric sequences appear for turn angles 2π/n at n=12 and above","feed_subtitle":"Algebraic numbers of unit modulus that are not roots of unity generate the new examples in fixed-angle plane paths.","key_machinery":"The quadratic difference relation that defines the cyclic sequences and links them to Eulerian digraphs and fixed-angle turning walks in the plane.","core_discovery":"Quadratic cyclic sequences arise from closed plane paths with consistent fixed turn angle, and when that angle is 2π/n non-symmetric phenomena occur for n≥12 through algebraic numbers of modulus one which are not nth roots of unity.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["2π/n turn angles produce non-symmetric quadratic sequences at n≥12","Fixed turn plane paths yield asymmetric cyclic sequences for n=12 and above","Quadratic cyclic sequences show non-symmetry at 2π/n for n≥12","Unit modulus algebraics generate asymmetric quadratic sequences at n≥12"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"Sequences satisfying the quadratic difference relation can be realized as closed walks with a consistent fixed turn angle whose algebraic properties are governed by cyclotomic polynomials and non-root-of-unity unit-modulus numbers.","fun_headline_variants_meta":{"raw":{"variants":["2π/n turn angles produce non-symmetric quadratic sequences at n≥12","Fixed turn plane paths yield asymmetric cyclic sequences for n=12 and above","Quadratic cyclic sequences show non-symmetry at 2π/n for n≥12","Unit modulus algebraics generate asymmetric quadratic sequences at n≥12"]},"model":"grok-4.3","cost_usd":0.004299,"raw_usage":{"total_tokens":2071,"prompt_tokens":488,"num_sources_used":0,"completion_tokens":78,"cost_in_usd_ticks":42987000,"prompt_tokens_details":{"text_tokens":488,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1505,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":488,"tokens_out":78,"duration_ms":10020,"temperature":1.0,"reasoning_tokens":1505,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-24T16:22:56.145733+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A calculation for n=12 showing that every quadratic cyclic sequence remains symmetric and arises only from nth roots of unity.","supporting_citations":[],"review_version":1}