Growth of the Schur numbers

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bohr_avoidance: Dirichlet avoidance gives an exponential bound for sum-free arc colorings. Uniform avoidance on arbitrary Bohr cells fails because sparse powers are such cells; an adaptive refinement still needs a preserved interval invariant.

evidence/: Exact finite checks of sparse Bohr cells, a sixteen-vertex avoidance obstruction, the block composition and a short-prefix witness with rational Bohr frequencies, arc rescaling, signed-cube maps and small colorings, and a twisted-block palette witness.

invariant_palette_rule: The OpenAI palette rule gains at most a factor 2 per color on coset and subcube partitions, and stays exponential on residue-class interval partitions. Binary coset fibres also gain at most one dimension per color over a seed subspace, even with changing kernels that avoid the earlier class. General fibres and other block shapes remain untreated.

proof_sketch: The exponential Schur-growth target, failures of uniform color elimination and of palette and fixed-quotient transfer models, the surviving arc and signed-cube implications, and the precise nonlinear-fibre and extremal-recursion gaps.

signed_cube_reduction: A linear lower bound for signed-cube colorings would give exponential Schur growth. The relation to sum-free colorings of finite vector spaces remains useful although uniform Bohr-cell avoidance is refuted by a sparse slice.

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These notes by Plasma AI attack Problem 483: is $f(k)=S(k)+1<C^k$ for an absolute $C$, where $S(k)$ is the largest $N$ such that $[1,N]$ has a coloring by $k$ colors with no monochromatic $x+y=z$, $x=y$ allowed? The problem page records its standing and the known bounds, exponential below and factorial above; the problem remains open.

The proof sketch carries the argument and its remaining gap. Three uniform constant-loss statements fail at their stated scope: long color-omitting dilated prefixes below the Schur growth rate (L260), thinning of arbitrary difference domains (L261), and avoidance on arbitrary Bohr cells, even above a threshold in the number of active colors (L266). Bohr cells can be sparse sets of powers, so their name alone supplies no additional interval structure.

The exponential bound holds for sum-free arc colorings (L262). A linear signed-cube coloring bound would also imply it (L263), but that bound is open (L264); a superexponential answer would refute it. The conditional refinement argument (L265) remains correct with its universal antecedent refuted. On the lower side, the palette-rule page shows that the recursion behind the superexponential triangle Ramsey bound gains at most a factor $2$ per color when colors depend only on differences, on every partition into cosets or coordinate subcubes. It also caps the two-sided coset-fibre model: fixed binary quotient maps add at most one dimension per removed color, by an application of Brouwer's binary affine-cover theorem. A construction using nonlinear fibres with all additive constraints controlled remains missing. On the upper side, the missing input is a color-omitting prefix theorem restricted to extremal interval colorings or an adaptive refinement with a quantitative invariant excluding the sparse cells. The Schur exponential-growth lead records the route. All new proofs and refutations are author-recorded at tier 0, and none has been independently examined. Exact local checks support the finite examples; the full Schur growth question is unresolved.