attested · 2/459 re-run certificate script asserted: CERTIFIED chi(T3^3) >= 8; re-ran 2 of 459 stored batches from scratch chosen from seed 20260919 and only checked the SHA-256 of the other 457 against research/t3_rank7/batch_manifest.json; the full enumeration (80.1 CPU-h on Apple silicon laptop (18 cores), four batches at a time at nice 19 under a shared load; 2026-09-19 13:33 to 2026-09-20 11:33 UTC) was not re-run
4 runs; 468.7 CPU-hours; 115.8 h wall-clock; on Apple silicon laptop, four loop.py instances at nice 19 over the 459-batch partition, 21 ns per inner step averaged (80.1 CPU-h, 22 h wall); independent exclusion of research/t3_rank7/independent: RunPod Xeon Gold 6342 pod, 6 workers at nice 10 under a shared 7.65-core quota, 37 ns per step per worker (378.5 CPU-h over 92.9 h wall in two sessions), plus the laptop supplement of the skipped third pivots, 14 workers (10.1 CPU-h, 0.75 h wall).
With the verified rank-8 witness this settles χ(T3^3) = 8. A rank-7 decomposition would be seven independent stabilizer states whose images modulo V3 span four dimensions; the three-pivot scan in orbit-block order with Stab(i, j) masks (docs/notes/t3_rank7_exclusion.md) lists every candidate class set and decides exactly, mod 2^31 - 1 under the Hadamard bound, whether V3 lies in its span. 1,213,458,815 class sets over 459 batches, none containing V3, none spurious or undecided. The rare-type pivot lemma that would have cut the scan 28x is false (docs/notes/t3_rank7_rare_pivot.md). Cost 80 CPU-hours at 21 ns per step; batch outputs, partition and manifests are committed under research/t3_rank7/, and aggregate.py re-derives any batch from scratch. Attested rather than reproduced because no certificate budget re-runs the enumeration; a certificate that re-runs the enumeration under the budget would lift the tier. Independently excluded a second time by the scan of research/t3_rank7/independent (certificate pivot order, 5.52e13 inner steps, own kernel and case split, RunPod pod plus a laptop supplement, 2026-09-22 to 26): 1,935,849,726 class sets, none containing V3; the tier is unchanged, since it is fixed by what the certificate re-runs.
this repository · stabrank research/t3_rank7 · Galois descent to Q(ω3), then exhaustive three-pivot scan of seven-state configurations in exact arithmetic, batched and attested
{
"schema_version": "0.1",
"orbit": "T3",
"m": 3,
"direction": "lower",
"rank": 8,
"certificate": {
"script": "verify_challenge/cert_t3m3_rank8_attested.py",
"expect": "CERTIFIED chi(T3^3) >= 8",
"budget_s": 3600,
"attested": {
"batches": "research/t3_rank7/batch_manifest.json",
"recomputed": 2,
"compute_hours": 80.1,
"hardware": "Apple silicon laptop (18 cores), four batches at a time at nice 19 under a shared load; 2026-09-19 13:33 to 2026-09-20 11:33 UTC",
"note": "The enumeration of every rank-7 candidate class set (1.31e13 inner steps, 1.21e9 class sets, all decided exactly) lives in the 459 stored batch outputs; the certificate verifies their hashes, geometry and totals and re-runs two of them from scratch, so completeness rests on the stored outputs and on the committed runner that regenerates any batch."
}
},
"provenance": {
"author": "this repository",
"reference": "stabrank research/t3_rank7",
"method": "Galois descent to Q(w3), then exhaustive three-pivot scan of seven-state configurations in exact arithmetic, batched and attested",
"date": "2026-09-20",
"github": [
"vprusso"
],
"compute": {
"cpu_hours": 468.7,
"wall_clock_hours": 115.8,
"runs": 4,
"hardware": "Apple silicon laptop, four loop.py instances at nice 19 over the 459-batch partition, 21 ns per inner step averaged (80.1 CPU-h, 22 h wall); independent exclusion of research/t3_rank7/independent: RunPod Xeon Gold 6342 pod, 6 workers at nice 10 under a shared 7.65-core quota, 37 ns per step per worker (378.5 CPU-h over 92.9 h wall in two sessions), plus the laptop supplement of the skipped third pivots, 14 workers (10.1 CPU-h, 0.75 h wall)"
}
},
"notes": "With the verified rank-8 witness this settles chi(T3^3) = 8. A rank-7 decomposition would be seven independent stabilizer states whose images modulo V_3 span four dimensions; the three-pivot scan in orbit-block order with Stab(i, j) masks (docs/notes/t3_rank7_exclusion.md) lists every candidate class set and decides exactly, mod 2^31 - 1 under the Hadamard bound, whether V_3 lies in its span. 1,213,458,815 class sets over 459 batches, none containing V_3, none spurious or undecided. The rare-type pivot lemma that would have cut the scan 28x is false (docs/notes/t3_rank7_rare_pivot.md). Cost 80 CPU-hours at 21 ns per step; batch outputs, partition and manifests are committed under research/t3_rank7/, and aggregate.py re-derives any batch from scratch. Attested rather than reproduced because no certificate budget re-runs the enumeration; a certificate that re-runs the enumeration under the budget would lift the tier. Independently excluded a second time by the scan of research/t3_rank7/independent (certificate pivot order, 5.52e13 inner steps, own kernel and case split, RunPod pod plus a laptop supplement, 2026-09-22 to 26): 1,935,849,726 class sets, none containing V_3; the tier is unchanged, since it is fixed by what the certificate re-runs."
}