χR(∣S⟩⊗4) ≤ 4

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.

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Verification

lean   strange_m4_stabRank_le_four in LeanProofs.TensorStabRank, machine-checked by Lean

Implied per-copy exponent γ ≤ 0.3155 against a published 0.3155, no improvement.

Lean proof

This bound is backed by a machine-checked theorem: strange_m4_stabRank_le_four in LeanProofs.TensorStabRank. The recorded lake build confirms it compiles.

Notes

tight Witness: the four products of the two m=2 terms, verified here in exact arithmetic.

Attribution

Labib and Russo · arXiv:2605.28586 · literature

Submission

JSON
{
  "schema_version": "0.1",
  "orbit": "S",
  "m": 4,
  "direction": "upper",
  "rank": 4,
  "provenance": {
    "author": "Labib and Russo",
    "reference": "arXiv:2605.28586",
    "method": "literature",
    "date": "2026-05-27",
    "github": [
      "FarLab",
      "vprusso"
    ]
  },
  "notes": "tight Witness: the four products of the two m=2 terms, verified here in exact arithmetic.",
  "witness": {
    "terms": [
      {
        "k": 4,
        "x0": [
          0,
          0,
          0,
          0
        ],
        "W": [
          [
            1,
            0,
            0,
            0
          ],
          [
            0,
            1,
            0,
            0
          ],
          [
            0,
            0,
            1,
            0
          ],
          [
            0,
            0,
            0,
            1
          ]
        ],
        "Q": [
          [
            1,
            1,
            0,
            0
          ],
          [
            0,
            1,
            0,
            0
          ],
          [
            0,
            0,
            1,
            1
          ],
          [
            0,
            0,
            0,
            1
          ]
        ],
        "l": [
          0,
          0,
          0,
          0
        ]
      },
      {
        "k": 4,
        "x0": [
          0,
          0,
          0,
          0
        ],
        "W": [
          [
            1,
            0,
            0,
            0
          ],
          [
            0,
            1,
            0,
            0
          ],
          [
            0,
            0,
            1,
            0
          ],
          [
            0,
            0,
            0,
            1
          ]
        ],
        "Q": [
          [
            1,
            1,
            0,
            0
          ],
          [
            0,
            1,
            0,
            0
          ],
          [
            0,
            0,
            1,
            2
          ],
          [
            0,
            0,
            0,
            1
          ]
        ],
        "l": [
          0,
          0,
          0,
          0
        ]
      },
      {
        "k": 4,
        "x0": [
          0,
          0,
          0,
          0
        ],
        "W": [
          [
            1,
            0,
            0,
            0
          ],
          [
            0,
            1,
            0,
            0
          ],
          [
            0,
            0,
            1,
            0
          ],
          [
            0,
            0,
            0,
            1
          ]
        ],
        "Q": [
          [
            1,
            2,
            0,
            0
          ],
          [
            0,
            1,
            0,
            0
          ],
          [
            0,
            0,
            1,
            1
          ],
          [
            0,
            0,
            0,
            1
          ]
        ],
        "l": [
          0,
          0,
          0,
          0
        ]
      },
      {
        "k": 4,
        "x0": [
          0,
          0,
          0,
          0
        ],
        "W": [
          [
            1,
            0,
            0,
            0
          ],
          [
            0,
            1,
            0,
            0
          ],
          [
            0,
            0,
            1,
            0
          ],
          [
            0,
            0,
            0,
            1
          ]
        ],
        "Q": [
          [
            1,
            2,
            0,
            0
          ],
          [
            0,
            1,
            0,
            0
          ],
          [
            0,
            0,
            1,
            2
          ],
          [
            0,
            0,
            0,
            1
          ]
        ],
        "l": [
          0,
          0,
          0,
          0
        ]
      }
    ],
    "coeffs": [
      "3/8 + 3*sqrt(3)*I/8",
      "-3/8 - 3*sqrt(3)*I/8",
      "-3/8 - 3*sqrt(3)*I/8",
      "3/8 + 3*sqrt(3)*I/8"
    ]
  },
  "lean": {
    "module": "LeanProofs.TensorStabRank",
    "theorem": "strange_m4_stabRank_le_four"
  }
}