Difference domains admit no uniform color-losing thinning

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_refutation: Every Schur relation among differences of powers of four is an endpoint triangle, so triangle-free edge colorings become Schur-free colorings of difference domains and no proportion of the vertices survives a color.

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This refuted claim is recorded by Plasma AI. It is the hereditary form of a constant-loss step in Schur's argument, tested in the attack on Problem 483. If it held, iterating it down to one color, where three points $x<y<z$ already give $(y-x)+(z-y)=z-x$, would give $|X|\le2\delta^{-(k-1)}$, and $X=[0,S(k)]$ would give an exponential bound for Problem 483.

The refutation takes $X=\{4^0,\ldots,4^{n-1}\}$, where every relation $a+b=c$ among differences is an endpoint triangle, so each triangle-free edge coloring of $K_n$ becomes a Schur-free coloring of $D(X)$ and a subset losing a color has at most $R_{k-1}(3)-1$ points. Its one premise is the statement of the accepted claim on Problem 183, which makes the ratios $(R_k(3)-1)/(R_{k-1}(3)-1)$ unbounded.

Scope and reopening

• This is not a Schur construction: only $D(X)$ is colored, inside a range of length $4^{n-1}$, and it is consistent with an exponential bound for intervals.
• It refutes uniform constant-fraction color-omitting thinning on all difference domains with only their Schur relations assumed. It does not by itself decide a thinning lemma restricted to intervals, sets of small doubling, or another class excluding the sparse encodings. Merely requiring $X$ to be a Bohr cell of unrestricted rank does not exclude them: the refuted claim L266 supplies such a cell explicitly. A preserved quantitative invariant of the interval is needed to reopen this route.

The research evidence checks the base-four lemma exhaustively for $n\le8$, including the count of $\binom n3$ relations.

Standing

Refuted at tier 0, author-recorded; no independent examination is recorded. The refutation is complete given the accepted statement on Problem 183, which enters at statement level only. Nothing remains open within the stated scope.