Norrell — Stabilizer Rank Challenge

The state

| N ⟩ = \frac{| 0 ⟩ + | 1 ⟩ - 2 | 2 ⟩}{\sqrt{6}}

The Norrell state. Real amplitudes, one of them twice the others in magnitude, so unlike T3 its amplitudes are not all of equal modulus.

Shares the published exponent log₃4/3 with H₃. Its m=4 cell sits at 7; rank 6 bottoms out at √(211/4043) across five of six independent anneals, with a deeper basin at √(419/8559).

Bounds on this orbit

bound	m	rank	γ	tier	attribution
$χ_{R} ({\| N ⟩}^{\otimes 1}) \geq 2$	1	2	—	verified	@vprusso · stabrank verify_challenge
$χ_{R} ({\| N ⟩}^{\otimes 1}) \leq 2$	1	2	0.6309	verified	@vprusso · stabrank research/constructions/data
$χ_{R} ({\| N ⟩}^{\otimes 2}) \geq 3$	2	3	—	lean	@vprusso · stabrank PR #18
$χ_{R} ({\| N ⟩}^{\otimes 2}) \leq 3$	2	3	0.5000	lean	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| N ⟩}^{\otimes 3}) \geq 4$	3	4	—	reproduced	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| N ⟩}^{\otimes 3}) \leq 4$	3	4	0.4206	lean	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| N ⟩}^{\otimes 4}) \geq 3$	4	3	—	reproduced	@vprusso · stabrank rank_exclusion_codes.py
$χ_{R} ({\| N ⟩}^{\otimes 4}) \geq 5$	4	5	—	reproduced · 3600 s	@vprusso · stabrank slice_lift.py
$χ_{R} ({\| N ⟩}^{\otimes 4}) \geq 6$	4	6	—	attested · 2/171 re-run	@vprusso · stabrank research/qutrit_m4_rank5
$χ_{R} ({\| N ⟩}^{\otimes 4}) \geq 7$	4	7	—	attested · 2/1057 re-run	@vprusso · stabrank research/n4_rank6
$χ_{R} ({\| N ⟩}^{\otimes 4}) \leq 7$	4	7	0.4428	lean	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| N ⟩}^{\otimes 5}) \geq 7$	5	7	—	attested · 2/1057 re-run	@vprusso · stabrank research/n4_rank6
$χ_{R} ({\| N ⟩}^{\otimes 5}) \leq 10$	5	10	0.4192	verified	@vprusso · this repository (stabrank-research, round 28 letter lifts)
$χ_{R} ({\| N ⟩}^{\otimes 5}) \leq 12$	5	12	0.4524	verified	@vprusso · stabrank research/constructions/product_witness.py
$χ_{R} ({\| N ⟩}^{\otimes 6}) \geq 7$	6	7	—	attested · 2/1057 re-run	@vprusso · stabrank research/n4_rank6
$χ_{R} ({\| N ⟩}^{\otimes 6}) \leq 16$	6	16	0.4206	verified	@vprusso · stabrank research/constructions/product_witness.py

What would move it

The published exponent is already beaten: χ ≤ 10 at m=5 gives γ = 0.4192. Lowering it further needs χ ≤ 9 at m=5→ 0.4000.