χR(∣cat⟩⊗7) ≥ 4

magic cat orbit, m qubits.

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

verified   certificate script asserted: CERTIFIED chi(cat_7) >= 4, by an argument declared exact throughout (no floating-point margin)

Compute

5 runs; 2.8 CPU-hours; 2.5 h wall-clock; on Apple silicon laptop, one core per run; claude-opus-5-5, designed, wrote and ran the certificate and its cross-checks.

Notes

Beyond the published values as of our literature check (Oct 2026): QPG (arXiv:2106.07740) list 3 ≤ χ(cat_7), χ(cat_8) ≤ 6. Jardine's Zenodo deposits (2026-08) report evidence against rank 5 for cat_8; not used here. Compression: χ(cat_m) = χ(cm), cm(y) = cat_m(y, parity(y)) on m-1 qubits (see bounds/cat-m5-lower-3.json). The slices of c7 along its first qubit are exactly c6 and D c6, D = diag(i^[|y| even]) a diagonal Clifford (asserted). With χ(c6) = 3 (re-established in the run from the exact m=5 exclusion; bounds/cat-m6-upper-3.json), the qubit slice-and-lift lemma of verify_challenge/slice_lift.py (lifts_qubit) says a rank-3 decomposition of c7 restricts to one (dj, aj) of c6 with sum dj mu_j Pj aj = D c6. All rank-3 decompositions of c6 are listed, following the pivot-orbit pattern of slice_lift.py, from one pivot per orbit (544 orbits of the 2,423,520 five-qubit stabilizer states, generated exactly, count asserted) of the group generated by permutations of qubits 2..7 of cat_7 and R(x)R, R = (X+Y)/√ 2, each asserted to fix c7 up to one scalar. Screens run mod 2013265921; asserted norm bounds (|q| ≤ 2, |q2| ≤ 8, minors ≤ 128) make them exact, and every candidate and lift is confirmed in Gaussian-rational arithmetic. One decomposition orbit; it does not lift. Controls: c4 rank 2 does not lift, c5 rank 3 does. 460 s on one Apple silicon core; budget declared for slower CI runners. research/cat_lift/crosscheck.py reproduces the decomposition sets by brute force at c4 and c5.

Attribution

Michel de Messieres · stabrank verify_challenge · slice-and-lift in the even-weight compressed picture: the one rank-3 decomposition of compressed cat_6 up to symmetry does not extend to cat_7

Submission

JSON
{
  "schema_version": "0.1",
  "orbit": "cat",
  "m": 7,
  "direction": "lower",
  "rank": 4,
  "certificate": {
    "script": "verify_challenge/cert_cat_m7_lift.py",
    "exact": true,
    "budget_s": 1800,
    "expect": "CERTIFIED chi(cat_7) >= 4"
  },
  "provenance": {
    "author": "Michel de Messieres",
    "reference": "stabrank verify_challenge",
    "date": "2026-10-16",
    "github": [
      "MicheldeMessieres"
    ],
    "method": "slice-and-lift in the even-weight compressed picture: the one rank-3 decomposition of compressed cat_6 up to symmetry does not extend to cat_7",
    "compute": {
      "cpu_hours": 2.8,
      "wall_clock_hours": 2.5,
      "runs": 5,
      "hardware": "Apple silicon laptop, one core per run",
      "llm": [
        {
          "model": "claude-opus-5-5",
          "role": "designed, wrote and ran the certificate and its cross-checks"
        }
      ]
    }
  },
  "notes": "Beyond the published values as of our literature check (Oct 2026): QPG (arXiv:2106.07740) list 3 <= chi(cat_7), chi(cat_8) <= 6. Jardine's Zenodo deposits (2026-08) report evidence against rank 5 for cat_8; not used here. Compression: chi(cat_m) = chi(c_m), c_m(y) = cat_m(y, parity(y)) on m-1 qubits (see bounds/cat-m5-lower-3.json). The slices of c_7 along its first qubit are exactly c_6 and D c_6, D = diag(i^[|y| even]) a diagonal Clifford (asserted). With chi(c_6) = 3 (re-established in the run from the exact m=5 exclusion; bounds/cat-m6-upper-3.json), the qubit slice-and-lift lemma of verify_challenge/slice_lift.py (lifts_qubit) says a rank-3 decomposition of c_7 restricts to one (d_j, a_j) of c_6 with sum d_j mu_j P_j a_j = D c_6. All rank-3 decompositions of c_6 are listed, following the pivot-orbit pattern of slice_lift.py, from one pivot per orbit (544 orbits of the 2,423,520 five-qubit stabilizer states, generated exactly, count asserted) of the group generated by permutations of qubits 2..7 of cat_7 and R(x)R, R = (X+Y)/sqrt 2, each asserted to fix c_7 up to one scalar. Screens run mod 2013265921; asserted norm bounds (|q| <= 2, |q2| <= 8, minors <= 128) make them exact, and every candidate and lift is confirmed in Gaussian-rational arithmetic. One decomposition orbit; it does not lift. Controls: c_4 rank 2 does not lift, c_5 rank 3 does. 460 s on one Apple silicon core; budget declared for slower CI runners. research/cat_lift/crosscheck.py reproduces the decomposition sets by brute force at c_4 and c_5."
}