Quadratic Cyclic Sequences
Pith reviewed 2026-05-24 16:22 UTC · model grok-4.3
The pith
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.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
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.
What carries the argument
The quadratic difference relation that defines the cyclic sequences and links them to Eulerian digraphs and fixed-angle turning walks in the plane.
Load-bearing premise
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.
What would settle it
A calculation for n=12 showing that every quadratic cyclic sequence remains symmetric and arises only from nth roots of unity.
read the original abstract
We explore 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 for which at each step one must turn either left or right through a fixed angle. In the case when this angle is $2 \pi /n$, then non-symmetric phenomena occurs for $n\geq 12$. Examples arise from algebraic numbers of modulus one which are not $n$'th roots of unity.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (1)
- [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.
Simulated Author's Rebuttal
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.
read point-by-point responses
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Referee: [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.
Authors: 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: yes
Circularity Check
No circularity; derivation self-contained against external algebraic inputs
full rationale
The paper defines quadratic difference relations on sequences, links them to closed walks with fixed turn angle 2π/n, and invokes cyclotomic polynomials plus unit-modulus algebraic numbers as external sources for examples when n≥12. No step equates a derived quantity to a fitted parameter by construction, renames a known result, or reduces the central claim to a self-citation chain. The algebraic numbers are treated as independent inputs whose properties (modulus one, non-root-of-unity) are imported from standard field theory rather than defined via the sequences themselves. The derivation therefore remains non-circular even if the resulting periodicity claims later prove incorrect on equidistribution grounds.
discussion (0)
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