Inconclusive two-coloring searches for small simplicial boundaries

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Question

The simplicial route has uniform extraction at $n/k\sim C\log n$ but no unbounded actual coloring theorem. The columns of $E(n,3)$ are the oriented triangle boundaries $\partial[i,j,k]=[j,k]-[i,k]+[i,j]$ for $0\le i<j<k<n$. The searches below asked whether the smallest interesting instances admit two fully dissociated colors. There is no mathematical conclusion from these tests.

Exact enumeration at n = 6

For $n=6$, all $3^{10}=59049$ assignments on triangles avoiding zero were enumerated, the other coefficients were solved exactly, and all $2^{20}=1048576$ subsets were checked. The enumeration reported 1,011 signed supports, 450 minimal supports, maximum dissociated size 11 (with 120 maximizers), and 13,884 ordered two-colorings. No randomness or solver was used, and no script is retained.

Inconclusive solver searches at n = 7

For the 35-column $n=7$ input, mixed-integer searches used default solver settings with no specified seed:

• initial 35 tetrahedron constraints, a cap of 150 rounds, each with at most one 10-second coloring call and two 5-second relation calls; the round cap was reached without deciding two-colorability;
• 2,870 augmented constraints from six-vertex restrictions, a cap of 300 rounds with the same call limits, invoked twice; the retained output records a timeout without incumbent in the first coloring call;
• one relaxed maximum-size call capped at 20 seconds, followed by at most one 5-second relation call; its 17-column incumbent (reported upper bound 21) had a verified ten-column relation and was not a dissociated witness.

The retained alternating search is the $(n,k,q)=(7,3,2)$ instance. It constructs the complete triangle-to-edge boundary matrix, starts with the elementary tetrahedral boundary supports, and alternates a binary coloring feasibility problem with two ternary relation-separation problems, adding each monochromatic relation found in a color class as a new support. Every returned relation is checked by exact integer matrix multiplication before it is added. The caps are 100 rounds and 10 seconds per solver call, with at most three calls per round. In the original run the round cap was reached in under a second with 156 support constraints, and neither a proper-color certificate nor an infeasibility certificate was produced. Solver-dependent choices need not reproduce the same intermediate constraints; the bounds and the question are exact.

Scripts

The script needs NumPy and SciPy, which the repository does not declare, and writes its final state to the ignored output/ folder beside it. There is no checker: an inconclusive search is not evidence, and this folder contributes nothing to the aggregate evidence entry point.