verified certificate script asserted: CERTIFIED chi(T3^3) >= 7, by an argument declared exact throughout (no floating-point margin)
2 runs; 0.15 CPU-hours; 0.04 h wall-clock; on Apple silicon laptop, 8 workers: 99 s wall and 270 s CPU in the worktree, 135 s wall and 486 s CPU on the rebased branch; about three minutes on the 4-core CI runner.
Pairs with the verified rank-8 witness to put this cell at 7 ≤ χ ≤ 8. The Galois descent is made explicit: |T3⟩⊗3 = (ψ0 + ω9 ψ1 + ω9^2 ψ2)/√27 with ψr supported on the coset x1+x2+x3 = r mod 3 and entries ω3^q (q the carry), so a stabilizer span contains the state iff it contains V3 = span{ψ0, ψ1, ψ2}, and the problem is over Q(ω3). Six states whose span contains V3 have images mod V3 spanning at most 3 dimensions; the scan runs mod ell = 65521 with exact hashing (reduction and linear projection can only add candidates), the first pivot over the 45 orbits of the 2916 monomial (anti)unitary symmetries of V3, the second pivot minimal in its orbit under the pivot stabilizer (326,368 pivot pairs). All members of a candidate class lie in one 6-dim space containing V3, so one rank pair per class (class set, class set with the ψr) decides every subset; the 259,423 classes are all rejected. Ranks are computed mod 2^31 - 1 and are exact by a Hadamard bound (minors of size ≤ 9 of root-of-unity matrices have norm below the prime), asserted per class, so the rejection is exact arithmetic, not least squares, and the candidate harvest (exact hashing mod 65521) is exact too; the certificate declares exact: true and takes the verified tier. Control inside the script: at m = 2 exactly three states lie in V2 and the five-state scan finds class sets containing it. Outside it, the same kernel recovers the Lean rank-4 decomposition of |S⟩⊗3 as a class set, and shows the 45 rank-5 groups of the earlier certificate are one configuration under symmetry: five states spanning only three dimensions. Rank 7 was not excluded: the three-pivot scan costs about N^4/(6|G|) = 5e13 inner steps, about 70 CPU-hours in this kernel; simulated annealing at k = 7 stalls near residual 0.08 and an exact swap-neighbourhood search around the eight-term witness (three or more shared states) finds no seven-state span containing V3.
this repository · stabrank PR #41 · Galois descent to Q(ω3), then exhaustive coplanarity scan of six-state configurations in exact arithmetic mod two primes
{
"schema_version": "0.1",
"orbit": "T3",
"m": 3,
"direction": "lower",
"rank": 7,
"certificate": {
"script": "verify_challenge/cert_t3m3_rank7.py",
"expect": "CERTIFIED chi(T3^3) >= 7",
"exact": true
},
"provenance": {
"author": "this repository",
"reference": "stabrank PR #41",
"method": "Galois descent to Q(w3), then exhaustive coplanarity scan of six-state configurations in exact arithmetic mod two primes",
"date": "2026-09-16",
"github": [
"vprusso"
],
"compute": {
"cpu_hours": 0.15,
"wall_clock_hours": 0.04,
"runs": 2,
"hardware": "Apple silicon laptop, 8 workers: 99 s wall and 270 s CPU in the worktree, 135 s wall and 486 s CPU on the rebased branch; about three minutes on the 4-core CI runner"
}
},
"notes": "Pairs with the verified rank-8 witness to put this cell at 7 <= chi <= 8. The Galois descent is made explicit: |T3>^3 = (psi_0 + w9 psi_1 + w9^2 psi_2)/sqrt(27) with psi_r supported on the coset x1+x2+x3 = r mod 3 and entries w3^q (q the carry), so a stabilizer span contains the state iff it contains V_3 = span{psi_0, psi_1, psi_2}, and the problem is over Q(w3). Six states whose span contains V_3 have images mod V_3 spanning at most 3 dimensions; the scan runs mod ell = 65521 with exact hashing (reduction and linear projection can only add candidates), the first pivot over the 45 orbits of the 2916 monomial (anti)unitary symmetries of V_3, the second pivot minimal in its orbit under the pivot stabilizer (326,368 pivot pairs). All members of a candidate class lie in one 6-dim space containing V_3, so one rank pair per class (class set, class set with the psi_r) decides every subset; the 259,423 classes are all rejected. Ranks are computed mod 2^31 - 1 and are exact by a Hadamard bound (minors of size <= 9 of root-of-unity matrices have norm below the prime), asserted per class, so the rejection is exact arithmetic, not least squares, and the candidate harvest (exact hashing mod 65521) is exact too; the certificate declares exact: true and takes the verified tier. Control inside the script: at m = 2 exactly three states lie in V_2 and the five-state scan finds class sets containing it. Outside it, the same kernel recovers the Lean rank-4 decomposition of |S>^3 as a class set, and shows the 45 rank-5 groups of the earlier certificate are one configuration under symmetry: five states spanning only three dimensions. Rank 7 was not excluded: the three-pivot scan costs about N^4/(6|G|) = 5e13 inner steps, about 70 CPU-hours in this kernel; simulated annealing at k = 7 stalls near residual 0.08 and an exact swap-neighbourhood search around the eight-term witness (three or more shared states) finds no seven-state span containing V_3."
}