attested · 2/859 re-run certificate script asserted: CERTIFIED chi(qubit_sqrtT^6) >= 7; re-ran 2 of 859 stored batches from scratch chosen from seed 20261023 and only checked the SHA-256 of the other 857 against research/sqrtt_m5_rank6/batch_manifest.json; the full enumeration (46.3 CPU-h on Perlmutter CPU node (two AMD EPYC 7763, 128 worker processes), NERSC job 60159879, 2026-10-23 16:04 to 16:41 UTC; the representatives stage single-threaded (846 s), the pivot census 46.1 CPU-h over 128 workers (about 23 min wall)) was not re-run
1 run; 47 CPU-hours; 0.7 h wall-clock; on Perlmutter CPU node, 128 workers (job 60159879); controls and the collection and lift stages on an Apple silicon laptop; claude-fable-5-1, designed the plane census and the lift, wrote the runner and the certificate.
Every stabilizer decomposition of |sqrtT⟩⊗m has a span containing the four-dimensional Galois-forced subspace V4 (cert_sqrtt_galois.py), and χ(|sqrtT⟩⊗5) = 6 exactly (qubit_sqrtT-m5-lower-6.json). Six terms at m = 5 are therefore independent with residues modulo V4 spanning a plane, and pivoting on one term reduces the census to residue classes modulo V4 + span(pivot), computed exactly modulo two primes with every candidate class re-decided over Q(zeta_16). Over all 8,585 orbit representatives under the qubit permutations and the diagonal Clifford S^(x)5 (which maps the set of decompositions to itself through the Galois automorphism zeta → zeta^13), the only candidate set is the 18 permutation-symmetric five-qubit stabilizer states, so every six-term decomposition of |sqrtT⟩⊗5 lies inside Sym^5: exactly 3,375 of them, 852 orbits. Both computational slices of |sqrtT⟩⊗6 are |sqrtT⟩⊗5 up to the scalar zeta, so a six-term decomposition of |sqrtT⟩⊗6 would slice to two six-term decompositions of |sqrtT⟩⊗5 related term by term by a Pauli and a fourth root of unity; the lift searches every pair in the complete list and finds none. Hence χ(|sqrtT⟩⊗6) ≥ 7, and the cell is 7 ≤ χ ≤ 12 (qubit_sqrtT-m6-upper-12.json). The enumeration was not re-run under the budget: it is attested by the stored block outputs and the committed runner; the certificate re-runs two blocks from a printed seed, checks the orbit coverage of the stored pivots, and runs the exact collection and the lift in full. χ ≤ 7 at this cell would beat the published exponent 1/2.
Vincent Russo · this repository (stabrank-research, round 33) · slice-and-lift from the complete rank-6 census of |sqrtT⟩⊗5 through the Galois-forced subspace: no six-term decomposition of |sqrtT⟩⊗5 lifts to six copies
{
"schema_version": "0.1",
"orbit": "qubit_sqrtT",
"m": 6,
"direction": "lower",
"rank": 7,
"certificate": {
"script": "verify_challenge/cert_sqrtt_m6_rank6_attested.py",
"expect": "CERTIFIED chi(qubit_sqrtT^6) >= 7",
"budget_s": 3600,
"attested": {
"batches": "research/sqrtt_m5_rank6/batch_manifest.json",
"recomputed": 2,
"compute_hours": 46.3,
"hardware": "Perlmutter CPU node (two AMD EPYC 7763, 128 worker processes), NERSC job 60159879, 2026-10-23 16:04 to 16:41 UTC; the representatives stage single-threaded (846 s), the pivot census 46.1 CPU-h over 128 workers (about 23 min wall)",
"note": "The complete list of six-term stabilizer decompositions of |sqrtT>^5 lives in the stored batch outputs under research/sqrtt_m5_rank6/results/: for each of the 8,585 orbit representatives of the five-qubit stabilizer states under S_5 x <S^(x)5>, the residue classes modulo V4 + span(pivot) under two primes and every class that could carry a decomposition (one candidate set in all, the 18 symmetric states). The certificate re-enumerates the dictionary and checks its hash, verifies that the stored pivots meet every orbit exactly once, re-runs two blocks from scratch (seed printed) and compares them byte for byte, re-decides the candidate set exactly over Q(zeta_16) (3,375 decompositions, their exact coefficients), closes the list under the group, and runs the lift to six copies in full; completeness rests on the stored outputs and on research/sqrtt_m5_rank6/census.py, which regenerates any block (census.py run 5 OUT --pivots ...)."
}
},
"provenance": {
"author": "Vincent Russo",
"reference": "this repository (stabrank-research, round 33)",
"method": "slice-and-lift from the complete rank-6 census of |sqrtT>^5 through the Galois-forced subspace: no six-term decomposition of |sqrtT>^5 lifts to six copies",
"date": "2026-10-23",
"github": [
"vprusso"
],
"compute": {
"cpu_hours": 47.0,
"wall_clock_hours": 0.7,
"runs": 1,
"hardware": "Perlmutter CPU node, 128 workers (job 60159879); controls and the collection and lift stages on an Apple silicon laptop",
"llm": [
{
"model": "claude-fable-5-1",
"role": "designed the plane census and the lift, wrote the runner and the certificate"
}
]
}
},
"notes": "Every stabilizer decomposition of |sqrtT>^m has a span containing the four-dimensional Galois-forced subspace V4 (cert_sqrtt_galois.py), and chi(|sqrtT>^5) = 6 exactly (qubit_sqrtT-m5-lower-6.json). Six terms at m = 5 are therefore independent with residues modulo V4 spanning a plane, and pivoting on one term reduces the census to residue classes modulo V4 + span(pivot), computed exactly modulo two primes with every candidate class re-decided over Q(zeta_16). Over all 8,585 orbit representatives under the qubit permutations and the diagonal Clifford S^(x)5 (which maps the set of decompositions to itself through the Galois automorphism zeta -> zeta^13), the only candidate set is the 18 permutation-symmetric five-qubit stabilizer states, so every six-term decomposition of |sqrtT>^5 lies inside Sym^5: exactly 3,375 of them, 852 orbits. Both computational slices of |sqrtT>^6 are |sqrtT>^5 up to the scalar zeta, so a six-term decomposition of |sqrtT>^6 would slice to two six-term decompositions of |sqrtT>^5 related term by term by a Pauli and a fourth root of unity; the lift searches every pair in the complete list and finds none. Hence chi(|sqrtT>^6) >= 7, and the cell is 7 <= chi <= 12 (qubit_sqrtT-m6-upper-12.json). The enumeration was not re-run under the budget: it is attested by the stored block outputs and the committed runner; the certificate re-runs two blocks from a printed seed, checks the orbit coverage of the stored pivots, and runs the exact collection and the lift in full. chi <= 7 at this cell would beat the published exponent 1/2."
}