χR(∣T5⟩⊗1) ≥ 3

Tâ‚… orbit, ququint.

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   t5_m1_stabRankP_gt_two in LeanProofs.T5M1StabRank, machine-checked by Lean

Lean proof

This bound is backed by a machine-checked theorem: t5_m1_stabRankP_gt_two in LeanProofs.T5M1StabRank. The recorded lake build confirms it compiles.

Notes

Settles χ(|T5⟩) = 3 with the verified rank-3 witness. Exact throughout: every amplitude of |T5⟩ and of the 30 single-ququint stabilizer states is 0 or a power of w5, so the states are enumerated from the parametrisation (five basis states and 25 full-support states with phase w5^(q y^2 + l y)), checked against 5(5 + 1) = 30 and for pairwise distinctness up to phase, and |T5⟩ lies in the span of a pair exactly when all ten 3 x 3 minors of the 5 x 3 matrix vanish. Minors are accumulated as integer coefficient vectors in Z[w5], where an element is zero if and only if its five coefficients on 1, w, ..., w^4 are equal. No floating-point margin stands anywhere under the bound; a match against the numerical dictionary is printed as a consistency check but nothing rests on it. Controls: |0⟩ + |1⟩ must be spanned by exactly the pair {|0⟩, |1⟩}, and |+⟩ must be reported as a stabilizer state.

Attribution

this repository · stabrank verify_challenge · exact: every 3 x 3 minor of [s | t | ψ] over all 435 pairs of the 30 single-ququint stabilizer states, decided in Z[w5]

Submission

JSON
{
  "schema_version": "0.1",
  "orbit": "T5",
  "m": 1,
  "direction": "lower",
  "rank": 3,
  "certificate": {
    "script": "verify_challenge/cert_t5_m1_rank2.py",
    "expect": "CERTIFIED chi(T5^1) >= 3",
    "exact": true
  },
  "provenance": {
    "author": "this repository",
    "reference": "stabrank verify_challenge",
    "method": "exact: every 3 x 3 minor of [s | t | psi] over all 435 pairs of the 30 single-ququint stabilizer states, decided in Z[w_5]",
    "date": "2026-09-21",
    "github": [
      "vprusso"
    ]
  },
  "notes": "Settles chi(|T5>) = 3 with the verified rank-3 witness. Exact throughout: every amplitude of |T5> and of the 30 single-ququint stabilizer states is 0 or a power of w_5, so the states are enumerated from the parametrisation (five basis states and 25 full-support states with phase w_5^(q y^2 + l y)), checked against 5(5 + 1) = 30 and for pairwise distinctness up to phase, and |T5> lies in the span of a pair exactly when all ten 3 x 3 minors of the 5 x 3 matrix vanish. Minors are accumulated as integer coefficient vectors in Z[w_5], where an element is zero if and only if its five coefficients on 1, w, ..., w^4 are equal. No floating-point margin stands anywhere under the bound; a match against the numerical dictionary is printed as a consistency check but nothing rests on it. Controls: |0> + |1> must be spanned by exactly the pair {|0>, |1>}, and |+> must be reported as a stabilizer state.",
  "lean": {
    "module": "LeanProofs.T5M1StabRank",
    "theorem": "t5_m1_stabRankP_gt_two"
  }
}