χR(∣S⟩⊗5) ≥ 5

Strange 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.

← back to the board

Verification

reproduced   certificate script asserted: CERTIFIED chi(S^5) >= 5

Compute

3 runs; 2.7 CPU-hours; 2.6 h wall-clock; on Apple silicon laptop at low priority; the first enumeration ran 2.2 hours on one core, the certificate takes about two minutes on four.

Notes

χ(S^3) = χ(S^4) = 4 exactly. A rank-4 decomposition of |S⟩⊗5 sliced along one qutrit gives, at the |1⟩ slice, a rank-4 decomposition of |S⟩⊗4, itself a lift of one of |S⟩⊗3, with terms related slice to slice by a Pauli and cube roots of unity; the slice-0 equation is homogeneous because |S⟩ has no |0⟩ component. |S⟩⊗3 has 2160 rank-4 decompositions, listed through five representatives under its unitary symmetries (fifteen before the partner-index restriction; same orbits), they lift to 27 decompositions of |S⟩⊗4, and none of those extends to |S⟩⊗5: every Pauli assignment misses the slice-2 equation by at least 1e-4 (the closest one by 0.25). The first enumeration, without the stabilizer reduction of the partner loop, took 2.2 hours and agreed orbit for orbit with the reduced one. Puts the cell at 5 ≤ χ ≤ 8.

Attribution

this repository · stabrank slice_lift.py · slice-and-lift twice: every rank-4 decomposition of |S⟩⊗3 up to symmetry, lifted to |S⟩⊗4, none extends to |S⟩⊗5

Submission

JSON
{
  "schema_version": "0.1",
  "orbit": "S",
  "m": 5,
  "direction": "lower",
  "rank": 5,
  "certificate": {
    "script": "verify_challenge/cert_s_m5_lift.py",
    "expect": "CERTIFIED chi(S^5) >= 5"
  },
  "provenance": {
    "author": "this repository",
    "reference": "stabrank slice_lift.py",
    "method": "slice-and-lift twice: every rank-4 decomposition of |S>^3 up to symmetry, lifted to |S>^4, none extends to |S>^5",
    "date": "2026-09-18",
    "github": [
      "vprusso"
    ],
    "compute": {
      "cpu_hours": 2.7,
      "wall_clock_hours": 2.6,
      "runs": 3,
      "hardware": "Apple silicon laptop at low priority; the first enumeration ran 2.2 hours on one core, the certificate takes about two minutes on four"
    }
  },
  "notes": "chi(S^3) = chi(S^4) = 4 exactly. A rank-4 decomposition of |S>^5 sliced along one qutrit gives, at the |1> slice, a rank-4 decomposition of |S>^4, itself a lift of one of |S>^3, with terms related slice to slice by a Pauli and cube roots of unity; the slice-0 equation is homogeneous because |S> has no |0> component. |S>^3 has 2160 rank-4 decompositions, listed through five representatives under its unitary symmetries (fifteen before the partner-index restriction; same orbits), they lift to 27 decompositions of |S>^4, and none of those extends to |S>^5: every Pauli assignment misses the slice-2 equation by at least 1e-4 (the closest one by 0.25). The first enumeration, without the stabilizer reduction of the partner loop, took 2.2 hours and agreed orbit for orbit with the reduced one. Puts the cell at 5 <= chi <= 8."
}