Exact checks of the eight- and twenty-point prime-tensor circuits

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Purpose

The twenty-point circuit page exhibits a dissociated rational circuit in $S_3\otimes S_5\otimes S_7$ of rank nineteen with a saturated column lattice and a primitive relation having one coefficient of magnitude two. Its algebraic argument is the durable justification; this probe checks the displayed matrices exactly.

What is checked

Run, from the repository root,

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

The checker constructs only two prescribed matrices and three prescribed minors, using exact Bareiss elimination, the standard library and the shared root tools checker; it performs no search and finishes in under a second. Its named checks are:

• the eight-point seed in $S_3\otimes S_3\otimes S_3$ gives an $8\times8$ matrix, its coefficient vector $(-1,1,-1,1,-1,1,-1,-2)$ is a zero relation, and its prescribed $7\times7$ minor equals $-1$;
• the twenty-point lift has twenty distinct labels, gives a $48\times20$ matrix, its displayed coefficient vector is a zero relation, and its two prescribed $19\times19$ minors equal $2$ and $-1$;
• both relations have full support, a unit coefficient, and a coefficient $-2$.

A nonzero $(n-1)$-minor with a full-support relation proves a rational circuit; a unit coefficient makes the displayed integer relation primitive, and its coefficient two excludes every nonzero ternary relation. A failed clause exits nonzero. The lemma page states the exact scope: the seed check inspected at most 64 rank-seven minors, and neither computation is a search for unbounded coloring.

Retained exploratory probes

Two bounded enumerations preceded the graph-cut circuits of the prime staircase, which made them unnecessary. They are recorded observations, not inputs to any theorem; the checker above does not run them, and their arithmetic is short enough to repeat with a few lines of code.

Powers of $1+\sqrt2$ through ten columns. The question was whether algebraic integrality and being a unit, without the root-of-unity conjugate constraints, could exclude mandatory coordinates. For each $n=3,\ldots,10$, encode $\beta^j$, $0\le j<n$, for $\beta=1+\sqrt2$ by the integer pair $(a_j,b_j)$ with

enumerate all $\epsilon\in\{-1,0,1\}^n$, group them by the exact integer pair $\sum_j\epsilon_j(a_j,b_j)$, and intersect the support bitmasks within each group. For every $n$ from $3$ to $10$, only the zero coefficient vector maps to zero, and every nonzero value group has nonempty support intersection; the enumeration covered $\sum_{n=3}^{10}3^n=88{,}560$ vectors. A separate pass at $n=6$ over its $729$ vectors found $336$ distinct nonzero value classes. No infinite conclusion is drawn: these powers are not roots of unity, and the staircase's mandatory-coordinate proof does not use them.

Twenty-seven seed layer choices. The question was whether duplicating one old layer in each seed coordinate could keep the circuit size linear in three growing side lengths. For the eight-point seed $\{000,012,021,102,110,211,222,001\}$ of the twenty-point circuit page, enumerate all $a\in\{0,1,2\}^3$ and test whether each seed point agrees with $a$ in at most one coordinate. None of the $27$ choices passes, after at most $27\cdot8=216$ point-choice comparisons. This rules out only that particular no-cross-duplication condition, not linear-size circuits, which the staircase's graph-cut construction then supplied directly, with no further search.