No signed vanishing sum of odd squarefree roots of unity fits in a short arc

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This open claim is recorded by Plasma AI. It asks for a uniform obstruction to the distinct-prime tensor route to Problem 774. By L123, part (3), a positive constant c bounds the dissociated coloring of all roots of unity of odd squarefree order by the ceiling of 1/c, so it bounds the coloring number of every grid $S_3\otimes S_5\otimes\cdots\otimes S_{p_k}$, and that family could not supply the counterexample sought there. Conversely, unbounded coloring of those grids, the missing theorem of the main argument, would refute this claim. The claim is a statement about cyclotomic polynomials alone; it uses no coloring hypothesis.

Equivalence of the two forms

Choose $P=\sum_{j\le D}s_jx^j$ of minimum degree $D=D(M)$ among the nonzero multiples of $\Phi_M$ with coefficients in $\{-1,0,1\}$. Since $1+x+\cdots+x^{M-1}$ is such a multiple, $D<M$. The roots $e^{2\pi\mathrm ij/M}$ with $s_j\ne0$ are distinct, of odd squarefree order dividing $M$, lie in the closed arc of arguments $[0,2\pi D/M]$, and carry the vanishing signed sum $P(e^{2\pi i/M})=0$. Conversely, a vanishing signed sum of distinct roots of odd squarefree orders lies in $\mu_M$ for the least common multiple $M>1$ of those orders; rotating it so that the first root of the arc is $1$ gives such a polynomial of degree at most $eM$ when the arc has length $2\pi e$. So the first form with constant $c$ implies the second with any $e<c$, and the second with $e$ implies the first with $c=e$.

Standing

Open. Neither a proof nor a counterexample is known here. A literature search on 2026-10-15 for least-degree multiples of cyclotomic polynomials with restricted coefficients, and for vanishing sums of roots of unity in arcs, found no result that settles it. It found the nonnegative-coefficient problem of Steinberger, where for the orders its Theorem 1 covers the least degree is $(p-1)n/p$ with $p$ the least prime factor of $n$, and A. S. Reddy's arXiv preprint 1106.1271, which treats $0,1$-polynomials only for even orders with at most three prime factors; neither concerns signed multiples. An incomplete search does not establish novelty. The research account records the evidence: the exact values $D(15)/15=8/15$ and $D(105)/105=7/15$, an explicit flat multiple showing $D(1155)\le502$, the structure that any counterexample family must have, and heuristics in both directions. The largest constant compatible with those values is at most $502/1155$.

Remaining obligations

• Prove or refute the statement. A refutation needs, for every $e>0$, a vanishing signed sum of distinct roots of odd squarefree order inside an arc of length $2\pi e$; by the research account's layer lemma, each strict improvement over a subconductor must have a nonzero common layer value in that prime direction, and every minimal vanishing subsum of such a sum has a reduced exponent of totient ratio at most $e$.
• A proof with $c\ge1/3$ would show that the dissociated coloring number of every distinct-prime grid from $S_3\otimes S_5\otimes S_7$ on is exactly three; the bracket $3\le\chi\le4$ at the prime 23 is the first case it would decide.