χ_R(∣qubit_H⟩^⊗6) ≥ 6

{
  "schema_version": "0.1",
  "orbit": "qubit_H",
  "m": 6,
  "direction": "lower",
  "rank": 6,
  "certificate": {
    "script": "verify_challenge/cert_qubit_h_m6_rank5_attested.py",
    "expect": "CERTIFIED chi(qubit_H^6) >= 6",
    "budget_s": 3600,
    "attested": {
      "batches": "research/h6_rank5/batch_manifest.json",
      "recomputed": 2,
      "compute_hours": 74.7,
      "hardware": "187 batches on a RunPod CPU pod (16 vCPU AMD EPYC 4564P, shared host at load 30 to 45, ten batches in parallel, nice 19), 2026-09-22 01:37 to 08:31 UTC; batches 14, 15, and 173 on an Apple silicon laptop (one core, nice 19) during the pipeline test on 2026-09-21; the aggregate's two seeded re-runs (batches 11 and 148) on the pod; the stage C repair batches 190 to 196 on the same pod (seven in parallel, nice 19), 2026-09-23 09:17 to 09:32 UTC, and the aggregate's two seeded re-runs (batches 10 and 140) on the pod afterwards",
      "note": "The exclusion of every rank-5 decomposition of |H>^6 lives in 180 stored batch outputs under research/h6_rank5/results/: 15 stage A batches over the 5,939,465 full 5-covers of |H>^3 with distinct independent base states, 158 stage B batches over the 12,390 dependent covers (partition.json, batches 0 to 172), and the 7 stage C repair batches 190 to 196 (partition_stage_c_v2.json) over the 16,006 covers with a repeated state of the regenerated list degenerate_covers_v2.json, which supersede the 17 original stage C batches 173 to 189 over 13,852 covers (docs/notes/h6_rank5_stagec_repair.md: the original list missed the 2,154 multisets T + (b, b) with b outside span(T), and the block reconstruction missed cancelling copies). Each batch matches its covers at the four base points and re-decides every candidate mod 2013265921 and numerically. The certificate checks both partitions against the enumerator, re-enumerates the degenerate covers into the regenerated list, verifies the hashes and geometry of every output, re-decides every stored hit, and re-runs two batches from scratch; completeness rests on the stored outputs and on research/h6_rank5/batch.py, which regenerates any batch (`batch.py K`, with --partition research/h6_rank5/partition_stage_c_v2.json for 190 to 196): 74.7 CPU-hours on the pod, stage A 2.8, stage B 69.6, stage C 1.3."
    }
  },
  "provenance": {
    "author": "this repository",
    "reference": "stabrank research/h6_rank5",
    "method": "all-visible base slice: every rank-5 decomposition of |H>^6 restricts, along some three qubits at some base point, to a full 5-cover of |H>^3 with all five terms nonzero; every such cover is enumerated up to symmetry and completed slice by slice under the slice structure lemma, modulo 65521 with exact re-decision",
    "date": "2026-09-23",
    "github": [
      "vprusso"
    ],
    "compute": {
      "cpu_hours": 74.7,
      "wall_clock_hours": 7.2,
      "runs": 2,
      "hardware": "187 batches on a RunPod CPU pod (16 vCPU AMD EPYC 4564P, shared host at load 30 to 45, ten batches in parallel, nice 19), 2026-09-22 01:37 to 08:31 UTC; batches 14, 15, and 173 on an Apple silicon laptop (one core, nice 19) during the pipeline test on 2026-09-21; the aggregate's two seeded re-runs (batches 11 and 148) on the pod; the stage C repair batches 190 to 196 on the same pod (seven in parallel, nice 19), 2026-09-23 09:17 to 09:32 UTC, and the aggregate's two seeded re-runs (batches 10 and 140) on the pod afterwards"
    }
  },
  "notes": "With the QPG cat witness bounds/qubit_H-m6-upper-6.json this settles chi(H^6) = 6. The argument (docs/notes/h6_rank5_exclusion.md) depends on: (1) property P of PR #87, computed by research/constructions/two_qubit_slice.py (501 s of tables, declared here and not re-run): every four-qubit slice of a rank-5 decomposition of |H>^6 along a 2 + 4 bipartition has all five terms nonzero. (2) The all-visible slice lemma (note, section 2): some triple has at most one plane term and a base point where all five terms are nonzero. (3) The slice structure lemma of PR #87, which makes the matcher's option sets exhaustive; two copies of one base state may cancel at a slice or at the base point, which property P does not exclude, so the stage C list holds the multisets T + (b, b) with b outside span(T) and the block reconstruction allows cancelling copies (docs/notes/h6_rank5_stagec_repair.md, the 2026-09-23 repair). (4) Exactness: all amplitudes lie in Q(zeta_16), and 65521 and 2013265921 are 1 mod 16, so the modular kernels list supersets of the true covers and matches, and every candidate cover and every hit is re-decided mod 2013265921 and in floating point (research/h6_rank5/common.decide_terms), stored hits again by the aggregate; a block state the reconstruction cannot decide raises and fails the batch (none did). (5) The two positive controls (note, section 5): all 23 classes of rank-4 decompositions of |H>^4 and the rank-6 witness from its four all-visible bases (results/control_*.json; the repeated bases not re-run under the repaired matcher). Stage A ran through the compiled kernels cpp/src/cover5.cpp and cpp/src/slice_match.cpp, checked against the Python reference in tests/test_slice_cover.py and cpp/tests/; batch 14 was also re-run with STABRANK_NO_NATIVE=1 (results/nonative_14.json, note section 8) and agrees on every exact quantity. Attested rather than reproduced because no certificate budget re-runs the 75 CPU-hours."
}