Stopped common-span test for the seven-layer prime coloring

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Purpose and outcome

The prime reset and plateau page exhibits an explicit seven-layer three-coloring of $S_3\otimes S_5\otimes S_7$ and proves its root-omission properties by an algebraic cyclic signed-sum lemma. Before that lemma was found, one fixed computation tried to decide the common signed-span intersection of the three classes directly. It was designed to form the $96\times105$ matrix whose kernel consists of the coefficient triples with equal signed sums in all three classes, and to enumerate at most $3^{12}$ ternary values of the free original coefficients only if the nullity was at most 12. The nullity is 22, so no enumeration ran and the test decides nothing about the intersection. The algebraic lemma supersedes this route; no zero-intersection conclusion is drawn from the stopped computation.

The coloring itself was discovered in an exact search of at most 120 column permutations of its specified Latin coloring; the search stopped at permutation 2. Its coordinate proof on the route page supersedes discovery.

What is checked

Run, from the repository root,

uv run --no-sync python wiki/research/erdos_774/evidence/prime_reset_span/main.py

The checker rebuilds the seven-layer coloring, forms the equality matrix, and reduces it by exact rational elimination, using the standard library and the shared root tools checker; it performs no search and finishes in a few seconds. Its named checks are the class sizes $(35,35,35)$, the matrix shape $96\times105$, rank 83, nullity 22, and that the nullity exceeds the declared gate, so that the ternary enumeration is not run. A failed clause exits nonzero. These are exact facts about the matrix; they certify nothing about dissociated coloring.