χR(∣N⟩⊗2) ≥ 3

Norrell 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

lean   norrell_m2_stabRank_gt_two in LeanProofs.M2Lower, machine-checked by Lean

Lean proof

This bound is backed by a machine-checked theorem: norrell_m2_stabRank_gt_two in LeanProofs.M2Lower. The recorded lake build confirms it compiles.

Notes

Pairs with the lean-certified rank-3 decomposition to settle this cell at χ = 3 exactly, and closes it as a route to the published exponent, which would need rank 2 here. The exclusion rests on a margin rather than exact arithmetic: the closest any two candidate directions come to parallel is 0.71 (N) or 0.73 (H3) against a threshold of 1 - 1e-6, so no rounding at the 1e-15 level of these inputs can close the gap. The same search finds the known rank-2 decomposition of the Strange state at m=2, which the certificate runs as a positive control and fails without. Also proved in Lean against stabRank, with no enumeration: M2Lower.norrell_m2_stabRank_gt_two (a term with at most three support points leaves the other term carrying two moduli, and two full-support terms put the three modulus classes on an equilateral triangle a + b w^j, where Pompeiu's inequality r3 ≤ r1 + r2 fails).

Attribution

this repository · stabrank PR #18 · exhaustion over all 64620 pairs of 2-qutrit stabilizer states

Submission

JSON
{
  "schema_version": "0.1",
  "orbit": "N",
  "m": 2,
  "direction": "lower",
  "rank": 3,
  "certificate": {
    "script": "verify_challenge/cert_m2_rank2.py",
    "expect": "CERTIFIED chi(N^2) >= 3"
  },
  "provenance": {
    "author": "this repository",
    "reference": "stabrank PR #18",
    "method": "exhaustion over all 64620 pairs of 2-qutrit stabilizer states",
    "date": "2026-09-12",
    "github": [
      "vprusso"
    ]
  },
  "notes": "Pairs with the lean-certified rank-3 decomposition to settle this cell at chi = 3 exactly, and closes it as a route to the published exponent, which would need rank 2 here. The exclusion rests on a margin rather than exact arithmetic: the closest any two candidate directions come to parallel is 0.71 (N) or 0.73 (H3) against a threshold of 1 - 1e-6, so no rounding at the 1e-15 level of these inputs can close the gap. The same search finds the known rank-2 decomposition of the Strange state at m=2, which the certificate runs as a positive control and fails without. Also proved in Lean against stabRank, with no enumeration: M2Lower.norrell_m2_stabRank_gt_two (a term with at most three support points leaves the other term carrying two moduli, and two full-support terms put the three modulus classes on an equilateral triangle a + b w^j, where Pompeiu's inequality r3 <= r1 + r2 fails).",
  "lean": {
    "module": "LeanProofs.M2Lower",
    "theorem": "norrell_m2_stabRank_gt_two"
  }
}