T₃ — Stabilizer Rank Challenge

The state

| T_{3} ⟩ = \frac{| 0 ⟩ + ω_{9} | 1 ⟩ + ω_{9}^{2} | 2 ⟩}{\sqrt{3}}, ω_{9} = e^{2 π i / 9}

The qutrit T-type or face-centre state. All three amplitudes have equal modulus 1/√3 and differ only by phase, which is precisely why the subset-sum lower-bound technique cannot reach this orbit: that argument needs an exponentially increasing subsequence of amplitude moduli.

The only orbit whose amplitudes leave Q(ω₃), living instead in Q(ω₉). That extra field degree is what a Galois argument can exploit, and it is where the certified χ ≥ 6 at m=3 comes from.

Bounds on this orbit

bound	m	rank	γ	tier	attribution
$χ_{R} ({\| T_{3} ⟩}^{\otimes 1}) \geq 3$	1	3	—	lean	@vprusso · stabrank PR #19
$χ_{R} ({\| T_{3} ⟩}^{\otimes 1}) \leq 3$	1	3	1.0000	lean	Kocia and Sarovar · arXiv:2003.01130
$χ_{R} ({\| T_{3} ⟩}^{\otimes 2}) \geq 3$	2	3	—	lean	@vprusso · stabrank PR #16
$χ_{R} ({\| T_{3} ⟩}^{\otimes 2}) \leq 3$	2	3	0.5000	lean	Kocia and Sarovar · arXiv:2003.01130
$χ_{R} ({\| T_{3} ⟩}^{\otimes 3}) \geq 6$	3	6	—	reproduced	@vprusso · stabrank PR #16
$χ_{R} ({\| T_{3} ⟩}^{\otimes 3}) \geq 7$	3	7	—	verified	@vprusso · stabrank PR #41
$χ_{R} ({\| T_{3} ⟩}^{\otimes 3}) \geq 8$	3	8	—	attested · 2/459 re-run	@vprusso · stabrank research/t3_rank7
$χ_{R} ({\| T_{3} ⟩}^{\otimes 3}) \leq 8$	3	8	0.6309	lean	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| T_{3} ⟩}^{\otimes 4}) \geq 6$	4	6	—	reproduced	@vprusso · stabrank cert_t3m3_rank6.py
$χ_{R} ({\| T_{3} ⟩}^{\otimes 4}) \geq 8$	4	8	—	attested · 2/459 re-run	@vprusso · stabrank research/t3_rank7
$χ_{R} ({\| T_{3} ⟩}^{\otimes 4}) \leq 9$	4	9	0.5000	lean	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| T_{3} ⟩}^{\otimes 5}) \geq 6$	5	6	—	reproduced	@vprusso · stabrank cert_t3m3_rank6.py
$χ_{R} ({\| T_{3} ⟩}^{\otimes 5}) \geq 8$	5	8	—	attested · 2/459 re-run	@vprusso · stabrank research/t3_rank7
$χ_{R} ({\| T_{3} ⟩}^{\otimes 5}) \leq 18$	5	18	0.5262	lean	@vprusso · stabrank sector-state search
$χ_{R} ({\| T_{3} ⟩}^{\otimes 6}) \geq 8$	6	8	—	attested · 2/459 re-run	@vprusso · stabrank research/t3_rank7
$χ_{R} ({\| T_{3} ⟩}^{\otimes 6}) \leq 27$	6	27	0.5000	verified	@vprusso · stabrank research/constructions/product_witness.py

What would move it

Beating the published exponent needs χ ≤ 8 at m=4→ 0.4732.