attested · 2/171 re-run certificate script asserted: CERTIFIED chi(N^4) >= 6; re-ran 2 of 171 stored batches from scratch chosen from seed 20260922 and only checked the SHA-256 of the other 169 against research/qutrit_m4_rank5/batch_manifest_N.json; the full enumeration (38.1 CPU-h on RunPod CPU pod, 16 vCPU AMD EPYC 4564P on a shared host (load 20 to 30), fifteen batches in parallel at nice 19, 2026-09-23 14:25 to 18:08 UTC (three sample batches at 14:25, the 141 shared batches 15:31 to 17:34, the 27 single-cover stage C batches 17:34 to 18:08); the aggregate's two seeded re-runs (batches 121 and 169, 720 s and 583 s) on the same pod, 2026-09-23 18:50 to 19:12 UTC) was not re-run
1 run; 38.1 CPU-hours; 4.1 h wall-clock; on RunPod CPU pod, 16 vCPU AMD EPYC 4564P on a shared host (load 20 to 30), fifteen batches in parallel at nice 19, 2026-09-23 14:25 to 18:08 UTC (three sample batches at 14:25, the 141 shared batches 15:31 to 17:34, the 27 single-cover stage C batches 17:34 to 18:08); the aggregate's two seeded re-runs (batches 121 and 169, 720 s and 583 s) on the same pod, 2026-09-23 18:50 to 19:12 UTC.
With the Lean rank-7 witness bounds/N-m4-upper-7.json this moves the cell to 6 ≤ χ(N^4) ≤ 7. The argument (docs/notes/qutrit_m4_rank5_exclusion.md, sections 2 and 10) depends on: (1) Fact A: every one-qutrit slice of a rank-5 decomposition of |N⟩⊗4 has all five terms nonzero, from χ(N^3) = 4, the unique rank-4 decomposition of |N⟩⊗3, and the 0 lifts of relaxed_lift.py at R = 5 (declared, not re-run). (2) Fact B of PR #86 (two_qutrit_slice.py, not re-run): no two-qutrit slice point of a rank-5 decomposition has exactly three nonzero terms. (3) The base-point argument (note, section 2): every term is a nine-slice or diagonal-line term along qutrits 1, 2, at most one is a line term, and every diagonal line meets the monomial-symmetry orbit of (0, 0), so one base point per cover suffices. (4) The two-qutrit slice structure lemma of PR #86; the matcher allows every flat through x0 and does not use Facts A and B. (5) Exactness: amplitudes lie in Q(omega_3); 65521 and 2013265921 are 1 mod 3, so the modular kernels list supersets, and every candidate and hit is re-decided mod 2013265921 and numerically. (6) Stage C reconstructs cancelling copies, refuses unpinned or ambiguous block reconstructions loudly (recorded as undecided), lists the cancel-at-base multisets, and carries a single block's coefficient family as an unknown; the 27 covers with two blocks in one Pauli orbit were decided without a refusal. (7) Controls (note, sections 5 and 7; results/N/control_*.json and results/controls/): cover counts against the stored lists and slice_lift, planted and product decompositions (270 of 270), the rank-4 decomposition of |N⟩⊗3 by 2 + 1 slicing, and the rank-7 witness recovered exactly once from base 4. All Python, no compiled kernel. Attested because no certificate budget re-runs the 38 CPU-hours.
this repository · stabrank research/qutrit_m4_rank5 · two-qutrit all-visible base slice: every rank-5 decomposition of |N⟩⊗4 is carried by a monomial symmetry of |N⟩ on qutrits 1, 2 to one whose slice at x0 = (0, 0) has all five terms nonzero (Facts A and B of the note), so that slice is a full 5-cover of |N⟩⊗2; every such cover is enumerated up to the unitary symmetry of |N⟩⊗2 and completed slice by slice under the two-qutrit slice structure lemma, modulo 65521 with exact re-decision
{
"schema_version": "0.1",
"orbit": "N",
"m": 4,
"direction": "lower",
"rank": 6,
"certificate": {
"script": "verify_challenge/cert_n_m4_rank5_attested.py",
"expect": "CERTIFIED chi(N^4) >= 6",
"budget_s": 3600,
"attested": {
"batches": "research/qutrit_m4_rank5/batch_manifest_N.json",
"recomputed": 2,
"compute_hours": 38.1,
"hardware": "RunPod CPU pod, 16 vCPU AMD EPYC 4564P on a shared host (load 20 to 30), fifteen batches in parallel at nice 19, 2026-09-23 14:25 to 18:08 UTC (three sample batches at 14:25, the 141 shared batches 15:31 to 17:34, the 27 single-cover stage C batches 17:34 to 18:08); the aggregate's two seeded re-runs (batches 121 and 169, 720 s and 583 s) on the same pod, 2026-09-23 18:50 to 19:12 UTC",
"note": "The exclusion of every rank-5 decomposition of |N>^4 lives in the 171 stored batch outputs under research/qutrit_m4_rank5/results/N/ (17 stage A batches over the 197,440 full 5-covers of |N>^2 with distinct independent base states, 98 stage B batches over the 12,175 dependent covers, 56 stage C batches over the 16,181 covers with a repeated state, including the cancel-at-base multisets T + (b, b) and the 27 covers with two blocks in one Pauli orbit, each alone in its batch), 225,796 covers matched at the base point x_0 = (0, 0) along qutrits 1, 2, every candidate re-decided mod 2013265921 and numerically, 0 hits, 0 refused, 0 undecided. The certificate checks the partition against the enumerator, re-enumerates the degenerate covers, verifies the hashes and geometry of every output, re-decides every stored hit, fails on any refusal or undecided run, and re-runs two batches from scratch; completeness rests on the stored outputs and on research/qutrit_m4_rank5/batch.py, which regenerates any batch (batch.py N K)."
}
},
"provenance": {
"author": "this repository",
"reference": "stabrank research/qutrit_m4_rank5",
"method": "two-qutrit all-visible base slice: every rank-5 decomposition of |N>^4 is carried by a monomial symmetry of |N> on qutrits 1, 2 to one whose slice at x_0 = (0, 0) has all five terms nonzero (Facts A and B of the note), so that slice is a full 5-cover of |N>^2; every such cover is enumerated up to the unitary symmetry of |N>^2 and completed slice by slice under the two-qutrit slice structure lemma, modulo 65521 with exact re-decision",
"date": "2026-09-23",
"github": [
"vprusso"
],
"compute": {
"cpu_hours": 38.1,
"wall_clock_hours": 4.1,
"runs": 1,
"hardware": "RunPod CPU pod, 16 vCPU AMD EPYC 4564P on a shared host (load 20 to 30), fifteen batches in parallel at nice 19, 2026-09-23 14:25 to 18:08 UTC (three sample batches at 14:25, the 141 shared batches 15:31 to 17:34, the 27 single-cover stage C batches 17:34 to 18:08); the aggregate's two seeded re-runs (batches 121 and 169, 720 s and 583 s) on the same pod, 2026-09-23 18:50 to 19:12 UTC"
}
},
"notes": "With the Lean rank-7 witness bounds/N-m4-upper-7.json this moves the cell to 6 <= chi(N^4) <= 7. The argument (docs/notes/qutrit_m4_rank5_exclusion.md, sections 2 and 10) depends on: (1) Fact A: every one-qutrit slice of a rank-5 decomposition of |N>^4 has all five terms nonzero, from chi(N^3) = 4, the unique rank-4 decomposition of |N>^3, and the 0 lifts of relaxed_lift.py at R = 5 (declared, not re-run). (2) Fact B of PR #86 (two_qutrit_slice.py, not re-run): no two-qutrit slice point of a rank-5 decomposition has exactly three nonzero terms. (3) The base-point argument (note, section 2): every term is a nine-slice or diagonal-line term along qutrits 1, 2, at most one is a line term, and every diagonal line meets the monomial-symmetry orbit of (0, 0), so one base point per cover suffices. (4) The two-qutrit slice structure lemma of PR #86; the matcher allows every flat through x_0 and does not use Facts A and B. (5) Exactness: amplitudes lie in Q(omega_3); 65521 and 2013265921 are 1 mod 3, so the modular kernels list supersets, and every candidate and hit is re-decided mod 2013265921 and numerically. (6) Stage C reconstructs cancelling copies, refuses unpinned or ambiguous block reconstructions loudly (recorded as undecided), lists the cancel-at-base multisets, and carries a single block's coefficient family as an unknown; the 27 covers with two blocks in one Pauli orbit were decided without a refusal. (7) Controls (note, sections 5 and 7; results/N/control_*.json and results/controls/): cover counts against the stored lists and slice_lift, planted and product decompositions (270 of 270), the rank-4 decomposition of |N>^3 by 2 + 1 slicing, and the rank-7 witness recovered exactly once from base 4. All Python, no compiled kernel. Attested because no certificate budget re-runs the 38 CPU-hours."
}