χ_R(∣T3⟩^⊗4) ≥ 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(T3^4) >= 6

Compute

1 run; 0.2 CPU-hours; 0.05 h wall-clock; on Apple silicon laptop, 6 worker processes.

Notes

The first lower bound on this cell, which previously had none and so still counted as a route to the published exponent at every rank. Puts it at 6 ≤ χ ≤ 9. The certificate re-runs the m=3 rank-6 exclusion (Galois descent cutting the search to 45 candidate lines, then exhaustion with every candidate rejected in exact arithmetic over Q(ω₉)) and applies projection monotonicity: χ(ψ ⊗ φ) ≥ χ(ψ) whenever φ has a nonzero computational amplitude, since I ⊗ ⟨x| maps a decomposition of ψ ⊗ φ to one of ψ with no more terms. Beating the published exponent 1/2 at m=4 needs rank 8, so this does not close the cell as a route; any rank from 6 to 8 at m=4 would beat it, and rank 9 is the product witness that matches it.

Attribution

this repository · stabrank cert_t3m3_rank6.py · projection monotonicity applied to the certified χ(T3^3) ≥ 6

Submission

JSON

{
  "schema_version": "0.1",
  "orbit": "T3",
  "m": 4,
  "direction": "lower",
  "rank": 6,
  "certificate": {
    "script": "verify_challenge/cert_t3_m4_from_m3.py",
    "expect": "CERTIFIED chi(T3^4) >= 6"
  },
  "provenance": {
    "author": "this repository",
    "reference": "stabrank cert_t3m3_rank6.py",
    "method": "projection monotonicity applied to the certified chi(T3^3) >= 6",
    "date": "2026-09-16",
    "github": [
      "vprusso"
    ],
    "compute": {
      "cpu_hours": 0.2,
      "wall_clock_hours": 0.05,
      "runs": 1,
      "hardware": "Apple silicon laptop, 6 worker processes"
    }
  },
  "notes": "The first lower bound on this cell, which previously had none and so still counted as a route to the published exponent at every rank. Puts it at 6 <= chi <= 9. The certificate re-runs the m=3 rank-6 exclusion (Galois descent cutting the search to 45 candidate lines, then exhaustion with every candidate rejected in exact arithmetic over Q(w9)) and applies projection monotonicity: chi(psi (x) phi) >= chi(psi) whenever phi has a nonzero computational amplitude, since I (x) <x| maps a decomposition of psi (x) phi to one of psi with no more terms. Beating the published exponent 1/2 at m=4 needs rank 8, so this does not close the cell as a route; any rank from 6 to 8 at m=4 would beat it, and rank 9 is the product witness that matches it."
}