reproduced certificate script asserted: CERTIFIED chi(T5^2) >= 5
2 runs; 0.04 CPU-hours; 0.04 h wall-clock; on Apple silicon laptop, one core: 82 s for the first rank-4 run, 51 s for the certificate.
With the rank-5 witness of T5-m2-upper-5.json (the five Z(x)Z eigensectors of |T5⟩⊗2, found 2026-09-24 by the census of research/t5q_m2_rank5) this settles χ(|T5⟩⊗2) = 5, against the product bound 9. Rank 3 (and rank 2 in the same pass) is excluded over the 3,900 two-ququint stabilizer states from one pivot per orbit of the order-100 symmetry group of |T5⟩⊗2 (the order-5 Clifford stabilizer of |T5⟩ on each copy, the copy swap and one antiunitary symmetry; 66 orbits), with no collinear candidate at all and a closest approach to parallel of 0.9337 against the threshold 1 - 1e-6. Rank 4 is excluded by the pivot-pair search of slice_lift.decompositions_with_pivot from the 98 representatives of the unitary symmetry group of order 50, with partners reduced by each pivot's stabilizer; since rank 3 is excluded first, every rank-4 decomposition would be minimal and the listing is complete. Both exclusions are numerical and rest on margins, so the bound is reproduced rather than verified. Controls: the same pivot search finds the five rank-3 decompositions of |T5⟩, and the rank-4 search from the pivot |00⟩ finds the product decomposition of |N5⟩⊗2 = (|+⟩ - |0⟩)^2, whose rank is 4.
this repository · stabrank docs/notes/new_orbits_design.md · rank-3 exclusion by pivot-and-quotient from 66 symmetry representatives, then rank-4 exclusion by the pivot-pair search of slice_lift.py from 98 representatives, over the 3,900 two-ququint stabilizer states
{
"schema_version": "0.1",
"orbit": "T5",
"m": 2,
"direction": "lower",
"rank": 5,
"certificate": {
"script": "verify_challenge/cert_t5_m2_rank4.py",
"expect": "CERTIFIED chi(T5^2) >= 5"
},
"provenance": {
"author": "this repository",
"reference": "stabrank docs/notes/new_orbits_design.md",
"method": "rank-3 exclusion by pivot-and-quotient from 66 symmetry representatives, then rank-4 exclusion by the pivot-pair search of slice_lift.py from 98 representatives, over the 3,900 two-ququint stabilizer states",
"date": "2026-09-21",
"github": [
"vprusso"
],
"compute": {
"cpu_hours": 0.04,
"wall_clock_hours": 0.04,
"runs": 2,
"hardware": "Apple silicon laptop, one core: 82 s for the first rank-4 run, 51 s for the certificate"
}
},
"notes": "With the rank-5 witness of T5-m2-upper-5.json (the five Z(x)Z eigensectors of |T5>^2, found 2026-09-24 by the census of research/t5q_m2_rank5) this settles chi(|T5>^2) = 5, against the product bound 9. Rank 3 (and rank 2 in the same pass) is excluded over the 3,900 two-ququint stabilizer states from one pivot per orbit of the order-100 symmetry group of |T5>^2 (the order-5 Clifford stabilizer of |T5> on each copy, the copy swap and one antiunitary symmetry; 66 orbits), with no collinear candidate at all and a closest approach to parallel of 0.9337 against the threshold 1 - 1e-6. Rank 4 is excluded by the pivot-pair search of slice_lift.decompositions_with_pivot from the 98 representatives of the unitary symmetry group of order 50, with partners reduced by each pivot's stabilizer; since rank 3 is excluded first, every rank-4 decomposition would be minimal and the listing is complete. Both exclusions are numerical and rest on margins, so the bound is reproduced rather than verified. Controls: the same pivot search finds the five rank-3 decompositions of |T5>, and the rank-4 search from the pivot |00> finds the product decomposition of |N5>^2 = (|+> - |0>)^2, whose rank is 4."
}