attested · 3/4259 re-run certificate script asserted: CERTIFIED chi(S^5) >= 6; re-ran 3 of 4259 stored batches from scratch chosen from seed 20261023 and only checked the SHA-256 of the other 4256 against research/s5_rank5/batch_manifest.json; the full enumeration (1700 CPU-h on Perlmutter CPU nodes (AMD EPYC 7763, 128 workers per node): the level-3 census in four interactive allocations on 2026-10-23 (jobs 60160192, 60160573, 60170484, 60171488; about 12 node-hours for the kernel-2 pass, 4,096 chunks, 442,623,432 prefixes, 54,157,309,763 candidates decided modulo two primes), the level-5 kappa = 0 lifts (128 chunks, minutes) and the kappa = 1 family brute force (35 bases x 243 ranges, about 2 node-hours); the other stages on an Apple silicon laptop) was not re-run
2 runs; 1800 CPU-hours; 10 h wall-clock; on Perlmutter CPU nodes (AMD EPYC 7763) for the census (two passes: kernel 1 abandoned after 1,491 chunks when its one-prime dependency screen was replaced by the two-prime decision, kernel 2 complete), the level-5 lifts and the family brute force; Apple silicon laptop (loaded, load average 70 to 100) for the controls and the other stages; claude-fable-5-1, designed, wrote and ran the exclusion, its controls and this certificate.
Strict improvement of bounds/S-m5-lower-5.json. With bounds/S-m5-upper-8.json the cell is 6 ≤ χ(S^5) ≤ 8. The exclusion is a complete recursive slice-and-lift through non-minimal covers (research/s5_rank5/README.md): every five-term decomposition of |S⟩⊗5 has all terms full on every qutrit (the research record's Theorem F.3, cards S0540 to S0543), so it is a lift of its ⟨1| slice, a five-term decomposition of |S⟩⊗4 with all coefficients nonzero; those are lifts of level-3 bases (5-multisets of three-qutrit stabilizer states with |S⟩⊗3 in the span, generic coefficients nonzero: 2,013,120 independent full 5-covers in 154 orbits from the stored census, plus the four-term decompositions with a repeated letter; no letter lies in the span of a four-term decomposition beyond its own) or completions of a four-term decomposition of |S⟩⊗3 with a product term (756). The 2,609 level-4 bases (2,574 with unique coefficients, 35 one-parameter families) have no lift to |S⟩⊗5: meet in the middle over the 81 Pauli cosets and three phases per term for unique coefficients, a brute force over all 243^5 option tuples with a determinant screen for the families. Planted controls at every stage and the reproduction of the 2,160 and 9,936 four-term decompositions of |S⟩⊗3 and |S⟩⊗4 (S0541) are in the research record (attacks/round33-strange-rank5).
Vincent Russo · stabrank-research, round 33 (attacks/round33-strange-rank5) · recursive slice-and-lift over non-minimal covers: every five-term decomposition of |S⟩⊗5 is all full, hence a lift of a five-term decomposition of |S⟩⊗4 with nonzero coefficients; these are lifts of five-covers of |S⟩⊗3 (exact census) or completions of its four-term decompositions; no level-4 base lifts
{
"schema_version": "0.1",
"orbit": "S",
"m": 5,
"direction": "lower",
"rank": 6,
"certificate": {
"script": "verify_challenge/cert_s_m5_rank5_attested.py",
"budget_s": 3600,
"attested": {
"batches": "research/s5_rank5/batch_manifest.json",
"recomputed": 3,
"compute_hours": 1700.0,
"hardware": "Perlmutter CPU nodes (AMD EPYC 7763, 128 workers per node): the level-3 census in four interactive allocations on 2026-10-23 (jobs 60160192, 60160573, 60170484, 60171488; about 12 node-hours for the kernel-2 pass, 4,096 chunks, 442,623,432 prefixes, 54,157,309,763 candidates decided modulo two primes), the level-5 kappa = 0 lifts (128 chunks, minutes) and the kappa = 1 family brute force (35 bases x 243 ranges, about 2 node-hours); the other stages on an Apple silicon laptop",
"note": "Stored outputs: the census (4,096 chunks of 3 or 4 pivot pairs: pairs, covers, prefixes skipped because the third letter lies in span(t, u, a) mod P1, counters), the level-5 kappa = 0 lifts (128 chunks over the 2,574 bases of results/b4_reps_v2.json), and the kappa = 1 family brute force (35 files, 243 first-option ranges each). The certificate verifies every stored output by hash and recomputes in full: letters, symmetry group, pivot pairs, the rank-4 census and its closure (the 2,160 four-term decompositions of |S>^3), the premises chi(S^3) >= 4 and chi(S^5) >= 5 exactly (a decomposition with four or fewer terms fails the run), Theorem F.3 exactly over the three representatives of |S>^4 and the complete four-qutrit letter table, the closure of the stored covers (2,013,120 in 154 orbits, representatives confirmed over Z[omega]), the P2 recheck of every one-prime skip, every level 3 to 4 lift, the 9,936 four-term decompositions of |S>^4, the 756 product-pattern completions, the span scan of all 7,439,040 four-qutrit letters, and the assembly of the 2,609 level-4 bases, then checks that the stored level-5 results cover every base with nothing found. Whole work units re-run from the seed (recomputed = 3): one census chunk with stored covers, re-enumerated with fresh functionals and compared cover for cover and skip for skip, and two kappa = 0 lift chunks; on top, three 100-prefix slices, 120 further bases, and 4 of the 8,505 family ranges. Every modular decision is exact by Hadamard's bound (minors of norm at most 6^6 against P of order 2e9). Completeness rests on the stored outputs and the committed code in research/s5_rank5, which regenerates any unit; its README has the full account."
},
"expect": "CERTIFIED chi(S^5) >= 6"
},
"provenance": {
"author": "Vincent Russo",
"reference": "stabrank-research, round 33 (attacks/round33-strange-rank5)",
"date": "2026-10-23",
"github": [
"vprusso"
],
"method": "recursive slice-and-lift over non-minimal covers: every five-term decomposition of |S>^5 is all full, hence a lift of a five-term decomposition of |S>^4 with nonzero coefficients; these are lifts of five-covers of |S>^3 (exact census) or completions of its four-term decompositions; no level-4 base lifts",
"compute": {
"cpu_hours": 1800.0,
"wall_clock_hours": 10.0,
"runs": 2,
"hardware": "Perlmutter CPU nodes (AMD EPYC 7763) for the census (two passes: kernel 1 abandoned after 1,491 chunks when its one-prime dependency screen was replaced by the two-prime decision, kernel 2 complete), the level-5 lifts and the family brute force; Apple silicon laptop (loaded, load average 70 to 100) for the controls and the other stages",
"llm": [
{
"model": "claude-fable-5-1",
"role": "designed, wrote and ran the exclusion, its controls and this certificate"
}
]
}
},
"notes": "Strict improvement of bounds/S-m5-lower-5.json. With bounds/S-m5-upper-8.json the cell is 6 <= chi(S^5) <= 8. The exclusion is a complete recursive slice-and-lift through non-minimal covers (research/s5_rank5/README.md): every five-term decomposition of |S>^5 has all terms full on every qutrit (the research record's Theorem F.3, cards S0540 to S0543), so it is a lift of its <1| slice, a five-term decomposition of |S>^4 with all coefficients nonzero; those are lifts of level-3 bases (5-multisets of three-qutrit stabilizer states with |S>^3 in the span, generic coefficients nonzero: 2,013,120 independent full 5-covers in 154 orbits from the stored census, plus the four-term decompositions with a repeated letter; no letter lies in the span of a four-term decomposition beyond its own) or completions of a four-term decomposition of |S>^3 with a product term (756). The 2,609 level-4 bases (2,574 with unique coefficients, 35 one-parameter families) have no lift to |S>^5: meet in the middle over the 81 Pauli cosets and three phases per term for unique coefficients, a brute force over all 243^5 option tuples with a determinant screen for the families. Planted controls at every stage and the reproduction of the 2,160 and 9,936 four-term decompositions of |S>^3 and |S>^4 (S0541) are in the research record (attacks/round33-strange-rank5)."
}