χ_R(∣T3⟩^⊗2) ≥ 3

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.

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Verification

lean t3_m2_stabRank_gt_two in LeanProofs.T3GaloisM, machine-checked by Lean

Lean proof

This bound is backed by a machine-checked theorem: t3_m2_stabRank_gt_two in LeanProofs.T3GaloisM. The recorded lake build confirms it compiles.

Notes

Holds for every m. With the rank-3 witness at m=2 this settles that cell exactly. Field-of-definition arguments for stabilizer rank are prior art (Lovitz and Steffan, over Q(i)); the Galois-orbit step and the Z[ω₃] instantiation are not. The certificate is exact throughout: the only computation is the rank of a 3-column matrix over Q(ω₉), taken symbolically, so no floating-point margin stands under it. Also proved in Lean against stabRank: T3GaloisM proves stabRank IsStab |T3⟩^⊗m > 2 for every m ≥ 1 (the conjugates of |T3⟩^⊗m are independent because restricting to indices with a single nonzero digit recovers the one-qutrit Vandermonde, and the Galois closure of the span is T3GaloisDescent applied on Fin (3^m)), and t3_m2_stabRank_gt_two is its m=2 instance. With the Lean rank-3 decomposition of T3M2Pointwise the cell is settled at 3 with both directions machine-checked.

Attribution

this repository · stabrank PR #16 · Galois descent over Z[ω₃]; no search

Submission

JSON

{
  "schema_version": "0.1",
  "orbit": "T3",
  "m": 2,
  "direction": "lower",
  "rank": 3,
  "certificate": {
    "script": "verify_challenge/cert_t3_galois.py",
    "expect": "CERTIFIED chi(T3^m) >= 3",
    "exact": true
  },
  "provenance": {
    "author": "this repository",
    "reference": "stabrank PR #16",
    "method": "Galois descent over Z[w3]; no search",
    "date": "2026-09-07",
    "github": [
      "vprusso"
    ]
  },
  "notes": "Holds for every m. With the rank-3 witness at m=2 this settles that cell exactly. Field-of-definition arguments for stabilizer rank are prior art (Lovitz and Steffan, over Q(i)); the Galois-orbit step and the Z[w3] instantiation are not. The certificate is exact throughout: the only computation is the rank of a 3-column matrix over Q(w9), taken symbolically, so no floating-point margin stands under it. Also proved in Lean against stabRank: T3GaloisM proves stabRank IsStab |T3>^m > 2 for every m >= 1 (the conjugates of |T3>^m are independent because restricting to indices with a single nonzero digit recovers the one-qutrit Vandermonde, and the Galois closure of the span is T3GaloisDescent applied on Fin (3^m)), and t3_m2_stabRank_gt_two is its m=2 instance. With the Lean rank-3 decomposition of T3M2Pointwise the cell is settled at 3 with both directions machine-checked.",
  "lean": {
    "module": "LeanProofs.T3GaloisM",
    "theorem": "t3_m2_stabRank_gt_two"
  }
}