attested · 2/1021 re-run certificate script asserted: CERTIFIED chi(H3^4) >= 7; re-ran 2 of 1021 stored batches from scratch chosen from seed 20261002 and only checked the SHA-256 of the other 1019 against research/h3_m4_rank6/batch_manifest.json; the full enumeration (81.5 CPU-h on RunPod CPU pod ee4072210b09 (32 vCPU, Ubuntu 20.04, gcc-10 kernels, Python 3.13.5, numpy 2.4.4), 28 batches in parallel at nice 19 alongside another job's 30 workers, 2026-10-01 18:14 to 2026-10-02 00:58 UTC for the 1,021-batch loop (81.1 CPU-hours of batch records: A6 14.6, B6 60.9, C6 3.3, beta' 2.3, gamma 0.03); the four heaviest stage A6 batches (0, 3, 4, 5, the pivot-117 pairs with 0.7 to 2.1 million covers each) were starved of CPU by an external SIGSTOP in the first pass (40 to 146 s of CPU in 4,740 s of wall) and recorded their covers as undecided at the 3,600 s wall-clock deadline, so they were re-run without a deadline 00:32 to 00:58 UTC (9 to 26 minutes each, 0 hits) and their records replace the starved ones; the aggregate (1,331 s, the 6-cover census re-run over every pivot pair and the lists re-enumerated, every count equal) and its two seeded re-runs (batches 142 and 1009, seed 20261002, 183 s and 157 s, deterministic hashes equal) on the same pod 01:03 to 01:25 UTC) was not re-run
1 run; 81.5 CPU-hours; 7.2 h wall-clock; on RunPod CPU pod ee4072210b09 (32 vCPU, Ubuntu 20.04, gcc-10 kernels, Python 3.13.5, numpy 2.4.4), 28 batches in parallel at nice 19 alongside another job's 30 workers, 2026-10-01 18:14 to 2026-10-02 00:58 UTC for the 1,021-batch loop (81.1 CPU-hours of batch records: A6 14.6, B6 60.9, C6 3.3, beta' 2.3, gamma 0.03); the four heaviest stage A6 batches (0, 3, 4, 5, the pivot-117 pairs with 0.7 to 2.1 million covers each) were starved of CPU by an external SIGSTOP in the first pass (40 to 146 s of CPU in 4,740 s of wall) and recorded their covers as undecided at the 3,600 s wall-clock deadline, so they were re-run without a deadline 00:32 to 00:58 UTC (9 to 26 minutes each, 0 hits) and their records replace the starved ones; the aggregate (1,331 s, the 6-cover census re-run over every pivot pair and the lists re-enumerated, every count equal) and its two seeded re-runs (batches 142 and 1009, seed 20261002, 183 s and 157 s, deterministic hashes equal) on the same pod 01:03 to 01:25 UTC.
With the Lean rank-8 witness bounds/H3-m4-upper-8.json this moves the cell to 7 ≤ χ(H3^4) ≤ 8. Every rank-6 decomposition of |H3⟩⊗4 has four, five, or six terms visible at the base point (0, 0) along qutrits 1, 2 (Fact B of PR #86: no two-qutrit slice of a rank-5 or rank-6 decomposition of |H3⟩⊗4 has exactly three nonzero terms, and χ(H3^2) = 3 leaves at least three), so the base is a full 4-, 5-, or 6-multiset of |H3⟩⊗2, one per orbit of the unitary symmetry group of |H3⟩⊗2 (order 32), and the invisible terms lie on the 16 flats of F3^2 missing (0, 0): stages A6 (the 36,368,678 full 6-covers of distinct independent states), B6 (the dependent 6-sets), C6 (the repeated 6-multisets), β' (the full 5-multisets with one invisible term on each of the 16 flats), and γ (the full 4-multisets with each of the 136 flat multisets, the pairs cancelling at a common first point carried explicitly). The lists rest on the attested rank-5 lists of bounds/H3-m4-lower-6.json and on the 6-cover census; the facts of PR #86 are declared, not re-run. Exactness: amplitudes in Q(omega_3, √ 3), both primes 1 mod 12, every modular kernel a superset, every candidate and hit re-decided mod 2013265921 and numerically. Controls (research/h3_m4_rank6/results/control_*.json): planted decompositions of every kind for every stage, flat, and flat multiset, the orbit lemma, the rank-4 decomposition of |H3⟩⊗3, the lists and census by hash, the rank-8 witness recorded. The design, case split, and cost table are docs/notes/h3_m4_rank6_design.md. Attested because no certificate budget re-runs the pod hours.
this repository · stabrank research/h3_m4_rank6 · two-qutrit base slice at (0, 0): every rank-6 decomposition of |H3⟩⊗4 has four, five or six terms visible at that point (Fact B of PR #86), so its base is a full 4-, 5- or 6-multiset of |H3⟩⊗2 and the invisible terms lie on the 16 flats missing (0, 0); every base is enumerated up to the unitary symmetry of |H3⟩⊗2 and completed slice by slice under the two-qutrit slice structure lemma, modulo 65521 with exact re-decision
{
"schema_version": "0.1",
"orbit": "H3",
"m": 4,
"direction": "lower",
"rank": 7,
"certificate": {
"script": "verify_challenge/cert_h3_m4_rank6_attested.py",
"expect": "CERTIFIED chi(H3^4) >= 7",
"budget_s": 3600,
"attested": {
"batches": "research/h3_m4_rank6/batch_manifest.json",
"recomputed": 2,
"compute_hours": 81.5,
"hardware": "RunPod CPU pod ee4072210b09 (32 vCPU, Ubuntu 20.04, gcc-10 kernels, Python 3.13.5, numpy 2.4.4), 28 batches in parallel at nice 19 alongside another job's 30 workers, 2026-10-01 18:14 to 2026-10-02 00:58 UTC for the 1,021-batch loop (81.1 CPU-hours of batch records: A6 14.6, B6 60.9, C6 3.3, beta' 2.3, gamma 0.03); the four heaviest stage A6 batches (0, 3, 4, 5, the pivot-117 pairs with 0.7 to 2.1 million covers each) were starved of CPU by an external SIGSTOP in the first pass (40 to 146 s of CPU in 4,740 s of wall) and recorded their covers as undecided at the 3,600 s wall-clock deadline, so they were re-run without a deadline 00:32 to 00:58 UTC (9 to 26 minutes each, 0 hits) and their records replace the starved ones; the aggregate (1,331 s, the 6-cover census re-run over every pivot pair and the lists re-enumerated, every count equal) and its two seeded re-runs (batches 142 and 1009, seed 20261002, 183 s and 157 s, deterministic hashes equal) on the same pod 01:03 to 01:25 UTC",
"note": "The exclusion of every rank-6 decomposition of |H3>^4 lives in the stored batch outputs under research/h3_m4_rank6/results/ (88 stage A6 batches over the 36,368,678 full 6-covers of |H3>^2 with distinct independent base states, 802 stage B6 batches over the 378,843 G_2 orbit representatives of the dependent 6-sets, 92 stage C6 batches over the 354,012 representatives of the repeated 6-multisets, 36 stage beta' batches over the 61,413 representatives of the full 5-multisets with the invisible term on each of the 16 flats missing (0, 0), 3 stage gamma batches over the 565 representatives of the full 4-multisets with each of the 136 flat multisets), every base matched at 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-runs the 6-cover kernel over every pivot pair, re-enumerates the orbit representatives, 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/h3_m4_rank6/batch.py, which regenerates any batch (batch.py K)."
}
},
"provenance": {
"author": "this repository",
"reference": "stabrank research/h3_m4_rank6",
"method": "two-qutrit base slice at (0, 0): every rank-6 decomposition of |H3>^4 has four, five or six terms visible at that point (Fact B of PR #86), so its base is a full 4-, 5- or 6-multiset of |H3>^2 and the invisible terms lie on the 16 flats missing (0, 0); every base is enumerated up to the unitary symmetry of |H3>^2 and completed slice by slice under the two-qutrit slice structure lemma, modulo 65521 with exact re-decision",
"date": "2026-10-02",
"github": [
"vprusso"
],
"compute": {
"cpu_hours": 81.5,
"wall_clock_hours": 7.2,
"runs": 1,
"hardware": "RunPod CPU pod ee4072210b09 (32 vCPU, Ubuntu 20.04, gcc-10 kernels, Python 3.13.5, numpy 2.4.4), 28 batches in parallel at nice 19 alongside another job's 30 workers, 2026-10-01 18:14 to 2026-10-02 00:58 UTC for the 1,021-batch loop (81.1 CPU-hours of batch records: A6 14.6, B6 60.9, C6 3.3, beta' 2.3, gamma 0.03); the four heaviest stage A6 batches (0, 3, 4, 5, the pivot-117 pairs with 0.7 to 2.1 million covers each) were starved of CPU by an external SIGSTOP in the first pass (40 to 146 s of CPU in 4,740 s of wall) and recorded their covers as undecided at the 3,600 s wall-clock deadline, so they were re-run without a deadline 00:32 to 00:58 UTC (9 to 26 minutes each, 0 hits) and their records replace the starved ones; the aggregate (1,331 s, the 6-cover census re-run over every pivot pair and the lists re-enumerated, every count equal) and its two seeded re-runs (batches 142 and 1009, seed 20261002, 183 s and 157 s, deterministic hashes equal) on the same pod 01:03 to 01:25 UTC"
}
},
"notes": "With the Lean rank-8 witness bounds/H3-m4-upper-8.json this moves the cell to 7 <= chi(H3^4) <= 8. Every rank-6 decomposition of |H3>^4 has four, five, or six terms visible at the base point (0, 0) along qutrits 1, 2 (Fact B of PR #86: no two-qutrit slice of a rank-5 or rank-6 decomposition of |H3>^4 has exactly three nonzero terms, and chi(H3^2) = 3 leaves at least three), so the base is a full 4-, 5-, or 6-multiset of |H3>^2, one per orbit of the unitary symmetry group of |H3>^2 (order 32), and the invisible terms lie on the 16 flats of F_3^2 missing (0, 0): stages A6 (the 36,368,678 full 6-covers of distinct independent states), B6 (the dependent 6-sets), C6 (the repeated 6-multisets), beta' (the full 5-multisets with one invisible term on each of the 16 flats), and gamma (the full 4-multisets with each of the 136 flat multisets, the pairs cancelling at a common first point carried explicitly). The lists rest on the attested rank-5 lists of bounds/H3-m4-lower-6.json and on the 6-cover census; the facts of PR #86 are declared, not re-run. Exactness: amplitudes in Q(omega_3, sqrt 3), both primes 1 mod 12, every modular kernel a superset, every candidate and hit re-decided mod 2013265921 and numerically. Controls (research/h3_m4_rank6/results/control_*.json): planted decompositions of every kind for every stage, flat, and flat multiset, the orbit lemma, the rank-4 decomposition of |H3>^3, the lists and census by hash, the rank-8 witness recorded. The design, case split, and cost table are docs/notes/h3_m4_rank6_design.md. Attested because no certificate budget re-runs the pod hours."
}