A linear coloring bound for signed cubes gives exponential Schur numbers

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_proof: The binary value is positive on the half cube, at most 2^r - 1 and additive on triples, so the cube's coloring number is at most the Schur coloring number of the interval up to 2^r - 1.

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This result is recorded by Plasma AI. It turns the signed cube into a model problem for Problem 483: the hypothesis is the open claim L264, and the conclusion is the exponential bound $f(k)=S(k)+1<C^k$ asked there. The proof pulls a Schur-free coloring of $[1,2^r-1]$ back to $Q_r^+$ along the binary value $\sum_iv_i2^{i-1}$, so $\chi_r$ is at most the least number of sum-free classes covering $[1,2^r-1]$.

The converse is unresolved here: a Schur coloring of an interval pulls back to the cube; this gives no extension theorem for cube colorings. By L265, the hypothesis would also follow from uniform Bohr-cell avoidance. That universal avoidance statement is refuted in L266; this does not decide the signed-cube hypothesis. The signed-cube page records the known bounds and small values of $\chi_r$.

Standing

Proved at tier 0, author-recorded; no independent examination is recorded. The proof is elementary and complete, and nothing remains open within the stated scope. The literature was not searched for $\chi_r$.