Lemma 10. Individual-edge arc consistency implies a block-preserving winning strategy with r+1 pairs on the valuation graphs.

For the set U of blocks containing pebbled vertices, maintain offsets (a_B)_ B∈U ∈ H_U . A homogeneous vertex z in block B is matched to the affine vertex z+a_B . On discarding pebbles, discard offsets only for blocks no longer occupied; the invariant is retained by Lemma 9 ([Exact projection]). Before placing a new pair there are at most r occupied blocks. For each unoccupied block B , independently extend the retained offsets to a member of H_ U∪B , and use its offset a_B . For occupied blocks keep the old offsets. On every block, translation by a_B is a bijection from its full homogeneous solution space onto its full affine solution coset. The union is therefore a bijection of the vertex sets. Only one new block is selected, so its chosen extension preserves the invariant. The family H_ B need not contain all affine valuations: any one legal offset already supplies the required block bijection. Offsets on shared readouts agree. Adding the same offset to the readouts of two pebbled valuations preserves both their agreement and their disagreement, including when several readouts are shared. Thus adjacency and nonadjacency between distinct blocks are preserved. Within one block, injectivity of translation preserves equality and there are no edges. Multiple pebbles in a block cause no difficulty. These bijections therefore maintain a partial isomorphism forever.