χR(∣N⟩⊗4) ≥ 5

Norrell orbit, qutrit.

Participate

Run it yourself, or point a coding agent at it.

git clone https://github.com/unitaryfoundation/stabrank
cd stabrank
# describe your decomposition in bounds/mybound.json
make fit BOUND=bounds/mybound.json ARGS=--write
make verify BOUND=bounds/mybound.json

fit solves for the exact coefficients, or tells you no exact combination of your terms reproduces the target. verify rebuilds the identity in exact arithmetic. If it passes, open a pull request adding the file and the leaderboard picks it up on the next build.

A bound the pipeline cannot check is recorded as cited and never takes a record.

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Verification

reproduced · 3600 s   certificate script asserted: CERTIFIED chi(N^4) >= 5 within a declared 3600s budget

Compute

3 runs; 7.6 CPU-hours; 7.6 h wall-clock; on Apple silicon laptop at low priority for the 6.4-hour first enumeration; the certificate ran in 43 minutes on the 4-core CI runner with the numpy search, and re-runs here with the compiled kernel.

Notes

χ(N^3) = 4 exactly (Lean witness and rank-3 exclusion). A rank-4 decomposition of |N⟩⊗4 sliced along one qutrit gives a rank-4 decomposition of |N⟩⊗3 whose terms are related slice to slice by a Pauli and cube roots of unity. |N⟩⊗3 has one rank-4 decomposition up to its unitary symmetries, the Lean witness of M3StabRank, and no Pauli assignment satisfies the other two slices: every one misses by at least 1e-4. The pivot-pair search searches the two remaining members above the partner, which is the least non-pivot member, runs in the compiled kernel, and cuts each pivot's stabilizer transversal by the stabilizer's known order, of cpp/src/pivot_pair.cpp when stabrank_core provides it, with the numpy loop as the checked reference. Supersedes the rank-3 exclusion at m=4 (χ ≥ 4, 63 CPU-minutes), which never fit the certificate budget. Puts the cell at 5 ≤ χ ≤ 7.

Attribution

this repository · stabrank slice_lift.py · slice-and-lift: the single rank-4 decomposition of |N⟩⊗3 up to symmetry does not extend to |N⟩⊗4

Submission

JSON
{
  "schema_version": "0.1",
  "orbit": "N",
  "m": 4,
  "direction": "lower",
  "rank": 5,
  "certificate": {
    "script": "verify_challenge/cert_n_m4_lift.py",
    "expect": "CERTIFIED chi(N^4) >= 5",
    "budget_s": 3600
  },
  "provenance": {
    "author": "this repository",
    "reference": "stabrank slice_lift.py",
    "method": "slice-and-lift: the single rank-4 decomposition of |N>^3 up to symmetry does not extend to |N>^4",
    "date": "2026-09-18",
    "github": [
      "vprusso"
    ],
    "compute": {
      "cpu_hours": 7.6,
      "wall_clock_hours": 7.6,
      "runs": 3,
      "hardware": "Apple silicon laptop at low priority for the 6.4-hour first enumeration; the certificate ran in 43 minutes on the 4-core CI runner with the numpy search, and re-runs here with the compiled kernel"
    }
  },
  "notes": "chi(N^3) = 4 exactly (Lean witness and rank-3 exclusion). A rank-4 decomposition of |N>^4 sliced along one qutrit gives a rank-4 decomposition of |N>^3 whose terms are related slice to slice by a Pauli and cube roots of unity. |N>^3 has one rank-4 decomposition up to its unitary symmetries, the Lean witness of M3StabRank, and no Pauli assignment satisfies the other two slices: every one misses by at least 1e-4. The pivot-pair search searches the two remaining members above the partner, which is the least non-pivot member, runs in the compiled kernel, and cuts each pivot's stabilizer transversal by the stabilizer's known order, of cpp/src/pivot_pair.cpp when stabrank_core provides it, with the numpy loop as the checked reference. Supersedes the rank-3 exclusion at m=4 (chi >= 4, 63 CPU-minutes), which never fit the certificate budget. Puts the cell at 5 <= chi <= 7."
}