χ_R(∣T3⟩^⊗3) ≥ 6

T₃ 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(T3^3) >= 6

Notes

Pairs with the rank-8 upper bound to put this cell at 6 ≤ χ ≤ 8. Every rank-5 decomposition would put five stabilizer states on one projective line off the Galois space; the 45 candidate groups are enumerated and each candidate is rejected twice, by least squares and then in exact arithmetic over Q(ω₉), where adjoining the target must raise the rank of the five states. The harvest of candidate lines is numerical and rests on the parallelism margin the certificate documents; the rejection of what it harvests is exact. The bound therefore still rests on the harvest margin, which is why it stays at the reproduced tier. The groups do exist, so the argument cannot be closed by observing that none do. Grouping is done on a random projection to C^6, which can only propose extra candidates, never lose one, since a linear map preserves collinearity.

Attribution

this repository · stabrank PR #16 · Galois descent to cut the search to finitely many candidate lines, then exhaustion

Submission

JSON

{
  "schema_version": "0.1",
  "orbit": "T3",
  "m": 3,
  "direction": "lower",
  "rank": 6,
  "certificate": {
    "script": "verify_challenge/cert_t3m3_rank6.py",
    "expect": "CERTIFIED chi(T3^3) >= 6"
  },
  "provenance": {
    "author": "this repository",
    "reference": "stabrank PR #16",
    "method": "Galois descent to cut the search to finitely many candidate lines, then exhaustion",
    "date": "2026-09-07",
    "github": [
      "vprusso"
    ]
  },
  "notes": "Pairs with the rank-8 upper bound to put this cell at 6 <= chi <= 8. Every rank-5 decomposition would put five stabilizer states on one projective line off the Galois space; the 45 candidate groups are enumerated and each candidate is rejected twice, by least squares and then in exact arithmetic over Q(w9), where adjoining the target must raise the rank of the five states. The harvest of candidate lines is numerical and rests on the parallelism margin the certificate documents; the rejection of what it harvests is exact. The bound therefore still rests on the harvest margin, which is why it stays at the reproduced tier. The groups do exist, so the argument cannot be closed by observing that none do. Grouping is done on a random projection to C^6, which can only propose extra candidates, never lose one, since a linear map preserves collinearity."
}