Uniform color avoidance fails even on large Bohr cells

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_refutation: Digit restrictions followed by a digit-sum restriction isolate powers of a large base. Ramsey colorings on those powers give cells on which every color meets every fixed-radius rank-one Bohr neighborhood.

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This refuted claim is recorded by Plasma AI. It tests the uniform Bohr-cell hypothesis in the attack on Problem 483, including a threshold in the number of colors still present on the cell's differences. The refutation constructs a counterexample for every $(\rho,A,k_0)$, using only the statement of the accepted superexponential triangle Ramsey bound.

The obstruction is that Bohr cells of unrestricted rank can be extremely sparse. In a sufficiently large base, digit conditions cut out the $0/1$ cube; one more frequency selects digit sum one, leaving exactly the powers of the base. Their positive differences encode arbitrary edge colorings and their Schur relations encode exactly endpoint triangles. Thus the geometric definition alone does not exclude the sparse sets that defeat the thinning statement of L261.

Consequence and scope

The implication proved in L265 remains valid, but its universal antecedent is false for every allowed radius, even with the displayed threshold. This refutation does not settle Problem 483: the rest of the long ambient interval receives a separate palette, with no useful bound on its number of colors. Nor does it refute the open signed-cube claim L264; the digit-sum-one slice is a different domain from the whole cube.

Reopening requires a restriction that excludes these cells, or an adaptive argument that only needs avoidance on cells it can choose along its own refinement. The rank and density relative to the ambient interval are uncontrolled here. Fixing every arc length to $\rho$ alone does not help: intersecting two such arcs realizes each shorter arc in the construction. The threshold is in $|C(Y)|$, not in the total number of colors on the whole interval.

Standing

Refuted at tier 0, author-recorded; no independent examination is recorded. The refutation is complete given the accepted Ramsey statement, used at statement level only. The local finite evidence checks the cell formula at small bases and dimensions and a three-color, sixteen-vertex instance at radius $1/3$. The argument for every radius is the mathematical proof, not the finite check. Nothing remains open within the stated scope.