Exact boundary certificate for the 3-by-4-by-3 tensor

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Purpose and conclusion

This certificate supplies the boundary case of the root-coverage recurrence: for the base $S_3\otimes S_4$ with one further factor $S_3$, the universal signed-span set of three colors is $U_3=\{0\}$, and the same holds after every exact ternary-relation-preserving linear encoding. The mathematical argument that turns the checked facts into that conclusion, a disjointness-connected hyperplane argument, is on the route page; the same argument is used by the Grow boundary certificate.

What is checked

Run, from the repository root,

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

The checker uses the standard library and the shared root tools checker, performs no search, and finishes in under a second. Its named checks are:

• the fixed three-layer coloring of the twelve base columns of $S_3\otimes S_4$ has, for each of its three colors, four columns per $S_3$ layer with 81 distinct signed sums, and successive signed-span intersection sizes $81,7,1$, $81,7,1$ and $81,17,1$, so each color is dissociated;
• the three tensor color classes have twelve columns each, and the third lies in the rational hyperplane with normal $(0,0,0,0,0,0,0,0,1,-1,-1,1)$;
• for each of twelve automorphic colorings, given by a permutation of each factor and a color relabeling, the image of one specified color class is disjoint from a specified class of the original coloring, and the image of the third class lies in the transported hyperplane;
• the twelve transported hyperplane normals have rational rank twelve.

Nine base layers with $3^4=81$ signed sums each are enumerated; no enumeration of $3^{36}$ tensor coefficients is used. A failed clause exits nonzero.

Discovery of the certificate

The certificate was found by bounded exact enumeration, which is not part of the normal run. For $S_3\otimes S_4$, all $6^4=1296$ column-permutation colorings were enumerated, retaining 216 whose colorwise row counts permute $(2,1,1)$; their 36 four-point classes were checked by 81 ternary sums each. With seed 3774 and a cap of 50,000 triples, the first triple, indices $[96,70,44]$, gave the proper boundary coloring, with exact ranks $12,12,11$. A contemplated $3^{12}$ enumeration was not performed. All 864 coordinate automorphisms were checked for disjoint-class connectivity and normal rank; a refinement to neighbors of the original coloring stopped after 332 permutations, including 43 qualifying neighbors, and produced the twelve-coloring certificate. A weighted-span diagnostic with the pattern $(123),(231),(312),(123)$, all 144 automorphisms, three classes, and 81 sums per class (34,992 occurrences) found, among 1,134 nonzero vectors, a largest membership count of 144, attained by the 24 signed roots, with all other counts at most 84; this was not used to claim a general weighted separation theorem.