A saturated coefficient-two circuit in the prime grid

---

Purpose and standing

The live prime-root coloring problem cannot be replaced by rational independence or by saturation of a column lattice. The following explicit example establishes both failures inside the actual prime grid, not only for an unrelated scalar configuration. The layer-duplication lemma below also gives a portable way to preserve a full integral relation lattice.

There is a twenty-point set $C\subset S_3\otimes S_5\otimes S_7$ of rational rank nineteen whose column image is a saturated sublattice, but which is dissociated. This is a diagnostic, not an unbounded-coloring theorem. Precisely, it refutes the implication from rational dependence and a saturated column image to the existence of a nonzero signed relation. It does not exclude refined lattice or topological methods that use additional hypotheses.

An eight-point seed

Use zero-based labels $s_0=(1,0),s_1=(0,1),s_2=(-1,-1)$ for $S_3$. In $S_3^{\otimes3}$, take, in this order,

$$000,\ 012,\ 021,\ 102,\ 110,\ 211,\ 222,\ 001.$$

Their columns in lexicographic tensor coordinates form

$$M=\begin{pmatrix} 1&0&0&0&0&0&-1&0\\ 0&0&-1&0&0&0&-1&1\\ 0&-1&0&0&0&0&-1&0\\ 0&-1&-1&0&0&-1&-1&0\\ 0&0&0&-1&0&0&-1&0\\ 0&0&0&-1&0&0&-1&0\\ 0&0&0&0&1&0&-1&0\\ 0&0&0&0&0&-1&-1&0 \end{pmatrix}.$$

Solving the rows gives

$$\ker_{\mathbb Z}M =\mathbb Z(-1,1,-1,1,-1,1,-1,-2). \tag{1}$$

Every coefficient is nonzero, so the support is a rational circuit. Every nonzero integral relation has a coefficient of absolute value at least two, so the set is dissociated. The minor on zero-based rows $(0,1,2,3,4,6,7)$ and columns $(0,1,2,3,4,5,7)$ has determinant $-1$, independently certifying rank seven and saturation of its image.

Layer duplication preserves the integral relation lattice

Suppose a rational circuit in $B\otimes S_m$ meets at least two layers, and replace one layer by several copies of exactly its support in a larger simplex. Each original layer is rationally independent, since it is a proper subset of a circuit.

A relation in a simplex tensor has the same vector sum in every layer. The coefficients in a copied layer are uniquely determined over $\mathbb Q$ by that sum, because its support is independent. Thus all copied coefficient vectors agree. Restriction and duplication are inverse isomorphisms between the old and new integral relation lattices. The new support remains a circuit, with the old primitive coefficients repeated in the copied layers. If a coefficient of absolute value two persists, the new circuit is still dissociated.

Explicit lift to the distinct-prime grid

Write $J=\{0,3,4,5,6\}$, $K=\{2,3,4\}$. The following groups, in table order and increasing index order within each group, specify twenty columns and their primitive relation coefficients.

These duplicate the seed's second-coordinate layer 2 three times and its third-coordinate layer 0 five times. Consequently their rank is nineteen and their integral kernel is generated by the displayed coefficient vector.

For an explicit check, let $a+b+c=0$ be the first simplex; let $d=u_0,e=u_1$ be second-coordinate vectors and $f=v_1,g=v_2$ be third-coordinate vectors. Set $U=\sum_{y\in K}u_y=-d-e$, $V=\sum_{z\in J}v_z=-f-g$. The proposed relation is

$$a\otimes(-d\otimes V+e\otimes g-U\otimes f-2d\otimes f) +b\otimes(d\otimes g-e\otimes V) +c\otimes(e\otimes f-U\otimes g).$$

Each parenthesis equals $d\otimes g+e\otimes f+e\otimes g$, so it vanishes. The duplication argument proves uniqueness.

In the standard $48$-coordinate column matrix, with coordinates $(i,j,\ell)\in[2]\times[4]\times[6]$ ordered lexicographically, take zero-based rows

$$0,1,2,3,4,5,7,8,13,14,19,20,26,30,31,32,33,34,35$$

and columns $0,1,\ldots,17,19$. The determinant is $-1$. Thus these nineteen columns generate a saturated rank-nineteen sublattice: projection onto the chosen rows identifies it with $\mathbb Z^{19}$, and any integer vector in its rational span has integral coordinates in this basis. The omitted column is an integral combination of them, as also follows from its coefficient $-1$ in the primitive relation. Hence the full column image is saturated. This does not turn (1) into a signed relation.

Exact verification and discovery bounds

The exact certificate check constructs only the displayed $8\times8$ and $48\times20$ matrices. It checks the two zero relations and the specified unit minors by exact integer elimination. No relation-support search is needed to verify the proof.

The bounded discovery search used the fixed $3^3$ grid, RNG seed 774, and a cap of 2,000 sampled eight-column bases; it stopped at zero-based trial 217. At most nineteen outside columns were inspected per trial. Floating determinants were only a screening device; accepted candidates were checked exactly. The seed audit inspected at most 64 rank-seven minors. The lift then checked one fixed $48\times20$ matrix and the two fixed $19\times19$ minors with columns $0,\ldots,18$ (determinant 2) and $0,\ldots,17,19$ (determinant $-1$). These finite computations served the stated coefficient/saturation check, not a search for unbounded coloring. The algebraic argument and the fixed certificate are the durable justification. The certificate check is author-recorded; the discovery search is not part of it.