χR(∣qubit_sqrtT⟩⊗4) ≥ 4

√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^4) >= 4, 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

Puts the cell at 4 ≤ χ(|sqrtT⟩⊗4) ≤ 5. Galois descent: the four conjugates of |sqrtT⟩⊗4 under Gal(Q(zeta_16)/Q(i)) are independent (a Vandermonde on the weights 0 to 3), so any stabilizer span containing the target has dimension at least 4. Exact, computed as a rank over Q(zeta_16). The numerical rank-3 exclusion over the 36,720 four-qubit states (research survey, 2026-10-17: 1,548 pivots, 1,099,080 collinear candidates, closest approach 0.974) agrees. Rank 4 here would give γ 1/2, matching the published exponent at a single cell; rank 5 gives 0.5805.

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": 4,
  "direction": "lower",
  "rank": 4,
  "certificate": {
    "script": "verify_challenge/cert_sqrtt_galois.py",
    "expect": "CERTIFIED chi(qubit_sqrtT^4) >= 4",
    "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": "Puts the cell at 4 <= chi(|sqrtT>^4) <= 5. Galois descent: the four conjugates of |sqrtT>^4 under Gal(Q(zeta_16)/Q(i)) are independent (a Vandermonde on the weights 0 to 3), so any stabilizer span containing the target has dimension at least 4. Exact, computed as a rank over Q(zeta_16). The numerical rank-3 exclusion over the 36,720 four-qubit states (research survey, 2026-10-17: 1,548 pivots, 1,099,080 collinear candidates, closest approach 0.974) agrees. Rank 4 here would give gamma 1/2, matching the published exponent at a single cell; rank 5 gives 0.5805."
}