χR(∣qubit_sqrtT⟩⊗2) ≥ 3

√T orbit, qubit.

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

verified   certificate script asserted: CERTIFIED chi(qubit_sqrtT^2) >= 3, by an argument declared exact throughout (no floating-point margin)

Compute

1 run; 0.003 CPU-hours; 0.003 h wall-clock; on Apple silicon laptop, one core; about 10 s for the certificate; claude-fable-5-1, found the descent argument and wrote the certificate.

Notes

Settles χ(|sqrtT⟩⊗2) = 3 with the three-term witness. Galois descent: a stabilizer span containing |sqrtT⟩⊗2 contains its four conjugates (1, zeta^k)^2, k = 1, 5, 9, 13, four distinct points on the conic of product states, which span the three-dimensional symmetric subspace, so three terms are needed. Exact: the rank is computed in Q(zeta_16) and no margin stands under the bound. The numerical rank-2 exclusion over the 60 two-qubit states (closest approach 0.986 to the parallel threshold) agrees.

Attribution

Vincent Russo · stabrank verify_challenge · Galois descent over Z[i]: the four conjugates of the target under Gal(Q(zeta_16)/Q(i)) span a space of dimension min(4, m + 1); no search

Submission

JSON
{
  "schema_version": "0.1",
  "orbit": "qubit_sqrtT",
  "m": 2,
  "direction": "lower",
  "rank": 3,
  "certificate": {
    "script": "verify_challenge/cert_sqrtt_galois.py",
    "expect": "CERTIFIED chi(qubit_sqrtT^2) >= 3",
    "exact": true
  },
  "provenance": {
    "author": "Vincent Russo",
    "reference": "stabrank verify_challenge",
    "method": "Galois descent over Z[i]: the four conjugates of the target under Gal(Q(zeta_16)/Q(i)) span a space of dimension min(4, m + 1); no search",
    "date": "2026-10-21",
    "github": [
      "vprusso"
    ],
    "compute": {
      "cpu_hours": 0.003,
      "wall_clock_hours": 0.003,
      "runs": 1,
      "hardware": "Apple silicon laptop, one core; about 10 s for the certificate",
      "llm": [
        {
          "model": "claude-fable-5-1",
          "role": "found the descent argument and wrote the certificate"
        }
      ]
    }
  },
  "notes": "Settles chi(|sqrtT>^2) = 3 with the three-term witness. Galois descent: a stabilizer span containing |sqrtT>^2 contains its four conjugates (1, zeta^k)^2, k = 1, 5, 9, 13, four distinct points on the conic of product states, which span the three-dimensional symmetric subspace, so three terms are needed. Exact: the rank is computed in Q(zeta_16) and no margin stands under the bound. The numerical rank-2 exclusion over the 60 two-qubit states (closest approach 0.986 to the parallel threshold) agrees."
}