attested · 2/323 re-run certificate script asserted: CERTIFIED chi(qubit_H^5) >= 6; re-ran 2 of 323 stored batches from scratch chosen from seed 20260923 and only checked the SHA-256 of the other 321 against research/h5_rank5/batch_manifest.json; the full enumeration (37.2 CPU-h on RunPod CPU pod, 16 vCPU AMD EPYC 4564P on a shared host (load 25 to 40), fifteen batches in parallel at nice 19, 2026-09-23 21:56 to 2026-09-24 00:28 UTC (stage A probe batch 0 at 21:56, the 323-batch run 22:02 to 00:28); the aggregate's two seeded re-runs (batches 19 and 319) on the same pod at 00:28 UTC) was not re-run
1 run; 37.2 CPU-hours; 2.5 h wall-clock; on RunPod CPU pod, 16 vCPU AMD EPYC 4564P on a shared host (load 25 to 40), fifteen batches in parallel at nice 19, 2026-09-23 21:56 to 2026-09-24 00:28 UTC (stage A probe batch 0 at 21:56, the 323-batch run 22:02 to 00:28); the aggregate's two seeded re-runs (batches 19 and 319) on the same pod at 00:28 UTC.
With the rank-6 witness bounds/qubit_H-m5-upper-6.json (Lean tier) this settles χ(H^5) = 6. The argument (docs/notes/h5_rank5_exclusion.md) depends on: (1) Fact 1, every term full along every qubit: the residual tables of research/constructions/two_qubit_slice.py for the 30 rank-4 decompositions of |H⟩⊗4 have no exact and no stabilizer combination at the ratio tan(π/8)^{+-1} (recomputed, results/tables.json; the completeness of the 30-element list up to symmetry is assumed as in the H^6 bound). (2) Fact 2, at most two absent terms at any two-qubit point: χ(H^3) = 3, bounds/qubit_H-m3-lower-3.json. (3) The flat lemma and the base-point case split of the note: stage α (all visible at 00 or 01) or stage β (four visible, one line term on the other diagonal); the configuration with two line terms on each diagonal is excluded by the same tables for the 6 rank-3 decompositions of |H⟩⊗3 at the ratio tan(π/8)^{+-2} (0 exact combinations, results/tables.json). (4) The slice structure lemma, which makes every option set exhaustive; stage β takes the invisible term present at both coordinate points and absent on the diagonal through the base point (Fact 1) and assumes nothing about the visible terms' flats. (5) The H^6 census (research/h6_rank5/results/kernel_census.json) and degenerate list (degenerate_covers_v2.json, cancel-at-base multisets included) reused by hash; the stage β list beta_covers.json re-enumerated by the certificate. (6) Exactness: amplitudes in Q(zeta_16), 65521 and 2013265921 are 1 mod 16, modular kernels list supersets, every candidate and hit re-decided mod 2013265921 and in floating point; a block or residual the matcher cannot decide raises and fails the batch (none did). (7) Controls (results/control_*.json): the rank-6 witness from its 24 all-visible pair bases, planted stage β instances of five kinds, the rank-4 decompositions of |H⟩⊗4 along a qubit pair.
this repository · stabrank research/h5_rank5 · two-qubit base slice: every rank-5 decomposition of |H⟩⊗5 has, along qubits 1, 2 at a base point in {00, 01}, either all five terms visible (a full 5-cover of |H⟩⊗3, the H^6 census) or four visible and one line term on the diagonal missing the base point (a full 4-cover of |H⟩⊗3); every base is completed slice by slice under the slice structure lemma, modulo 65521 with exact re-decision
{
"schema_version": "0.1",
"orbit": "qubit_H",
"m": 5,
"direction": "lower",
"rank": 6,
"certificate": {
"script": "verify_challenge/cert_qubit_h_m5_rank5_attested.py",
"expect": "CERTIFIED chi(qubit_H^5) >= 6",
"budget_s": 3600,
"attested": {
"batches": "research/h5_rank5/batch_manifest.json",
"recomputed": 2,
"compute_hours": 37.2,
"hardware": "RunPod CPU pod, 16 vCPU AMD EPYC 4564P on a shared host (load 25 to 40), fifteen batches in parallel at nice 19, 2026-09-23 21:56 to 2026-09-24 00:28 UTC (stage A probe batch 0 at 21:56, the 323-batch run 22:02 to 00:28); the aggregate's two seeded re-runs (batches 19 and 319) on the same pod at 00:28 UTC",
"note": "The exclusion of every rank-5 decomposition of |H>^5 lives in the stored batch outputs under research/h5_rank5/results/ (partition.json: 16 stage A batches over the 5,939,465 full 5-covers of |H>^3 with distinct independent base states, 293 stage B batches over the 12,390 dependent covers, 9 stage C batches over the 16,006 covers with a repeated state, including the cancel-at-base multisets, and 5 stage beta batches over the 3,466 full 4-covers, 6 of them with a repeated state), each cover matched at the base points 00 and 01 along qubits 1, 2 and every candidate re-decided mod 2013265921 and numerically: 0 hits, 0 refused, 0 undecided. The certificate checks the partition against the enumerator and the census hash, re-enumerates the degenerate covers and the stage beta bases, 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/h5_rank5/batch.py, which regenerates any batch (batch.py K)."
}
},
"provenance": {
"author": "this repository",
"reference": "stabrank research/h5_rank5",
"method": "two-qubit base slice: every rank-5 decomposition of |H>^5 has, along qubits 1, 2 at a base point in {00, 01}, either all five terms visible (a full 5-cover of |H>^3, the H^6 census) or four visible and one line term on the diagonal missing the base point (a full 4-cover of |H>^3); every base is completed slice by slice under the slice structure lemma, modulo 65521 with exact re-decision",
"date": "2026-09-24",
"github": [
"vprusso"
],
"compute": {
"cpu_hours": 37.2,
"wall_clock_hours": 2.5,
"runs": 1,
"hardware": "RunPod CPU pod, 16 vCPU AMD EPYC 4564P on a shared host (load 25 to 40), fifteen batches in parallel at nice 19, 2026-09-23 21:56 to 2026-09-24 00:28 UTC (stage A probe batch 0 at 21:56, the 323-batch run 22:02 to 00:28); the aggregate's two seeded re-runs (batches 19 and 319) on the same pod at 00:28 UTC"
}
},
"notes": "With the rank-6 witness bounds/qubit_H-m5-upper-6.json (Lean tier) this settles chi(H^5) = 6. The argument (docs/notes/h5_rank5_exclusion.md) depends on: (1) Fact 1, every term full along every qubit: the residual tables of research/constructions/two_qubit_slice.py for the 30 rank-4 decompositions of |H>^4 have no exact and no stabilizer combination at the ratio tan(pi/8)^{+-1} (recomputed, results/tables.json; the completeness of the 30-element list up to symmetry is assumed as in the H^6 bound). (2) Fact 2, at most two absent terms at any two-qubit point: chi(H^3) = 3, bounds/qubit_H-m3-lower-3.json. (3) The flat lemma and the base-point case split of the note: stage alpha (all visible at 00 or 01) or stage beta (four visible, one line term on the other diagonal); the configuration with two line terms on each diagonal is excluded by the same tables for the 6 rank-3 decompositions of |H>^3 at the ratio tan(pi/8)^{+-2} (0 exact combinations, results/tables.json). (4) The slice structure lemma, which makes every option set exhaustive; stage beta takes the invisible term present at both coordinate points and absent on the diagonal through the base point (Fact 1) and assumes nothing about the visible terms' flats. (5) The H^6 census (research/h6_rank5/results/kernel_census.json) and degenerate list (degenerate_covers_v2.json, cancel-at-base multisets included) reused by hash; the stage beta list beta_covers.json re-enumerated by the certificate. (6) Exactness: amplitudes in Q(zeta_16), 65521 and 2013265921 are 1 mod 16, modular kernels list supersets, every candidate and hit re-decided mod 2013265921 and in floating point; a block or residual the matcher cannot decide raises and fails the batch (none did). (7) Controls (results/control_*.json): the rank-6 witness from its 24 all-visible pair bases, planted stage beta instances of five kinds, the rank-4 decompositions of |H>^4 along a qubit pair."
}