Short-arc flat root relations and what they force

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An explicit degree-49 flat relation

Bzdęga's jump-one theorem for ternary cyclotomic coefficients, Theorem 1.5 of Bounds on ternary cyclotomic coefficients (printed page 6, proof in Section 5, pages 14--15, checked in the held copy), implies that $(1-x)\Phi_{pqr}(x)$, and hence $(1+x)\Phi_{2pqr}(x)$, are flat for distinct odd primes $p,q,r$.

The particular polynomial $F=(1+x)\Phi_{210}$ has positive exponents

$$\{0,3,5,7,8,10,12,37,39,41,42,44,46,49\}$$

and negative exponents

$$\{18,20,21,22,23,24,25,26,27,28,29,31\}.$$

Its elementary exact certificate is

$$F(x)(1+x^{15})(1+x^{21})(1+x^{35}) =(1+x^{105})(1+x^3)(1+x^5)(1+x^7).$$

Evaluation at a primitive 210th root is legitimate because the three factors on the left other than $F$ are nonzero there, and $1+x^{105}=0$. This gives a 26-point signed relation in an arc of angular length $2\pi\cdot49/210=7\pi/15$. Polynomial multiplication checks the identity exactly; it is short exposed arithmetic and needs no separate checker.

Degree 49 is minimal for a nonzero flat multiple of $\Phi_{210}$: $\deg\Phi_{210}=48$, its height is two, and a degree-48 multiple would be a scalar multiple with scalar $\pm1$, by its constant coefficient. The same degree argument applies to $\Phi_{105}$. This is a finite fact, not an $o(n)$-degree family. Within the displayed ternary-derived even-conductor family, the degree/conductor ratio is minimized at $(p,q,r)=(3,5,7)$, giving $49/210$: $(\varphi(pqr)+1)/(2pqr)$ increases in each prime separately.

What a short-arc relation actually forces

For a proper coloring $c:\mathbb Z/n\to[q]$, put $\rho_c(t)=n^{-1}|\{x:c(x+t)\ne c(x)\}|$. If $S$ supports a signed relation and $T$ is any spanning tree on $S$, then

$$\sum_{\{a,b\}\in T}\rho_c(b-a)\geq1.$$

Every translated support is nonmonochromatic, so at least one of its tree edges is bichromatic; averaging proves the inequality. If $S\subset[0,d]$ and $B=n\rho_c(1)$, ordering its exponents and using $\rho_c(t)\leq tB/n$ gives $B\geq n/d$. Thus hypothetical arbitrarily short flat relations force unbounded angular color variation, not impossibility of arbitrary finite coloring.

Neither a finite palette nor averaging implies angular continuity. Write a root as $e^{2\pi ix}$, let $x_2\in[0,1)$ be its 2-primary component, and color by $\lfloor2x_2\rfloor$. For every odd prime $p$, $e^{-\pi i/p}\to1$ has 2-primary component $1/2$, so multiplication by it flips every color. This example is not asserted proper; it refutes only the proposed automatic continuity inference. Bare measurability also adds nothing on this countable group: every coloring has a Borel-measurable extension to the circle, assigning all nonroots one color.

No $o(n)$-degree flat family or unrestricted-coloring contradiction was proved on this route.

Exact ranges of the exploratory searches

All runs below were bounded finite searches and supplied no new negative-solution ingredient. No script is retained.

• For $n=105,1155,15015$, exhaustively tested $\Phi_n(x)\prod_{p\in U}(1-x^p)(1-x)^\epsilon$, with $U\subseteq\{p:p\mid n\}$, $\epsilon\in\{0,1\}$. Minimum heights were respectively 1, 3, 14. The next conductor 255255 was interrupted during polynomial construction after the completed cases ceased to discriminate, and has no search conclusion.
• At $n=1155$, one integer-feasibility experiment used multiplier degree at most 48, constant coefficient 1, all remaining multiplier coefficients bounded in magnitude by 1000, product coefficients constrained to $[-1,1]$, and a 20-second solver limit. It timed out without a feasible point. This is not infeasibility.
• At $n=1155$, tested all 465 squarefree products of proper cyclotomic factors of total multiplier degree at most 58. Eligible divisors were $1,3,5,7,11,15,21,33,35,55,105$. None gave a flat product. The minimum height was two, attained by $\Phi_1\Phi_3\Phi_5$, degree seven.
• Consecutive-power differences $(1-x^s)\Phi_m(x)$ for $m\in\{105,165,195,1155\}$ and $s\in\{1,2,3,5,7\}$, a fixed total of 24 polynomials, were formed exactly to test whether a consecutive-power recurrence could yield a useful bounded-height mandatory root configuration. Height means the maximum absolute coefficient.

The low-height multiples did not supply a general recurrence construction; the prime staircase supplied the needed circuits directly. These numbers are recorded observations, not an asymptotic assertion.