Stabilizer Rank Challenge

Every γ below is published, except for CS and T₅, where no exponent has been published for the orbit and the single-copy product bound stands in. 2 of the 8 published exponents have been beaten here.

orbit		published $γ$ ≤	best here	progress	record held by
Strange	qutrit	0.3155	0.3155	needs χ ≤ 7 at m=6→ 0.2952	χ ≤ 2 at m=2 lean
Norrell	qutrit	0.4206	0.4192	beaten by 0.0014	χ ≤ 10 at m=5 verified
H₃	qutrit	0.4206	0.4206	needs χ ≤ 15 at m=6→ 0.4108	χ ≤ 4 at m=3 lean
T₃	qutrit	0.5000	0.5000	needs χ ≤ 8 at m=4→ 0.4732	χ ≤ 3 at m=2 lean
H-type	qubit	0.3962	0.4170	needs χ ≤ 6 at m=7→ 0.3693	χ ≤ 18 at m=10 lean
BK T-type	qubit	0.3962	0.3907	beaten by 0.0056	χ ≤ 15 at m=10 verified
√T	qubit	0.5000	0.5121	needs χ ≤ 7 at m=6→ 0.4679	χ ≤ 12 at m=7 verified
CS	two-qubit	1.0000	0.7925	beaten by 0.2075	χ ≤ 3 at m=2 verified
T₅	ququint	0.6826	0.5000	beaten by 0.1826	χ ≤ 5 at m=2 lean
magic cat	m qubits	0.3962	0.3962	needs χ ≤ 6 at m=9→ 0.3693	χ ≤ 3 at m=6 verified

Cell ledger

Best bound on $χ_{R} ({| M ⟩}^{\otimes m})$ per orbit and copy count, or per family and qubit count. Matching upper and lower bounds settle a cell.

Strange

qutrit · published

γ \leq \frac{\log_{3} 2}{2} \approx 0.3155

m = 1= 2 lean

m = 2= 2 lean

m = 3= 4 lean

m = 4= 4 lean

m = 5≥6, ≤8 lean

m = 6≥6, ≤8 lean

m = 7≥6, ≤16 verified

m = 8≥6, ≤16 verified

Norrell

qutrit · published

γ \leq \frac{\log_{3} 4}{3} \approx 0.4206

m = 1= 2 verified

m = 2= 3 lean

m = 3= 4 lean

m = 4= 7 lean

m = 5≥7, ≤10 verified

m = 6≥7, ≤16 verified

H₃

qutrit · published

γ \leq \frac{\log_{3} 4}{3} \approx 0.4206

m = 1= 2 verified

m = 2= 3 lean

m = 3= 4 lean

m = 4≥7, ≤8 lean

m = 5≥7, ≤12 verified

m = 6≥7, ≤16 verified

T₃

qutrit · published

γ \leq \frac{1}{2} = 0.5000

m = 1= 3 lean

m = 2= 3 lean

m = 3= 8 lean

m = 4≥8, ≤9 lean

m = 5≥8, ≤18 lean

m = 6≥8, ≤27 verified

H-type

qubit · published

γ \leq \frac{\log_{2} 3}{4} \approx 0.3963

m = 1= 2 verified

m = 2= 2 lean

m = 3= 3 lean

m = 4= 4 lean

m = 5= 6 lean

m = 6= 6 lean

m = 7≥6, ≤9 lean

m = 8≥6, ≤12 lean

m = 9≥6, ≤18 verified

m = 10≥6, ≤18 lean

BK T-type

qubit · published

γ \leq \frac{\log_{2} 3}{4} \approx 0.3963

m = 1= 2 verified

m = 2= 2 lean

m = 3= 3 lean

m = 4= 3 lean

m = 5= 6 lean

m = 6= 6 lean

m = 7≥6, ≤9 verified

m = 8≥6, ≤9 verified

m = 9≥6, ≤15 verified

m = 10≥6, ≤15 verified

√T

qubit · published

γ \leq \frac{1}{2} = 0.5000

m = 1= 2 verified

m = 2= 3 verified

m = 3= 4 verified

m = 4= 5 verified

m = 5= 6 verified

m = 6≥7, ≤12 verified

m = 7≥7, ≤12 verified

CS

two-qubit · baseline

γ \leq \log_{2} 2 = 1

m = 1= 2 verified

m = 2= 3 verified

m = 3≥4, ≤6 verified

T₅

ququint · baseline

γ \leq \log_{5} 3 \approx 0.6826

m = 1= 3 lean

m = 2= 5 lean

m = 3≥6, ≤15 lean

m = 4≥6, ≤25 lean

m = 5≥6, ≤75 verified

magic cat

m qubits · published

γ \leq \frac{\log_{2} 3}{4} \approx 0.3963

m = 2≤1 verified

m = 3= 2 verified

m = 4= 2 verified

m = 5= 3 verified

m = 6= 3 verified

m = 7≥5, ≤6 verified

m = 8≥5, ≤6 verified

Recently added · newest first; a literature entry is dated by its arXiv v1

χ_{R} ({| S ⟩}^{\otimes 5}) \geq 6

★Strange@vprussoattested · 3/4259 re-run2026-10-23

χ_{R} ({| S ⟩}^{\otimes 6}) \geq 6

★Strange@vprussoattested · 3/4259 re-run2026-10-23

χ_{R} ({| S ⟩}^{\otimes 7}) \geq 6

★Strange@vprussoattested · 3/4259 re-run2026-10-23

χ_{R} ({| S ⟩}^{\otimes 8}) \geq 6

★Strange@vprussoattested · 3/4259 re-run2026-10-23

χ_{R} ({| \sqrt{T} ⟩}^{\otimes 4}) \geq 5

★√T@vprussoverified2026-10-23

χ_{R} ({| \sqrt{T} ⟩}^{\otimes 5}) \geq 6

★√T@vprussoverified2026-10-23

χ_{R} ({| \sqrt{T} ⟩}^{\otimes 5}) \leq 6

★√T@vprussoverified2026-10-23

χ_{R} ({| \sqrt{T} ⟩}^{\otimes 6}) \geq 6

★√T@vprussoverified2026-10-23

χ_{R} ({| \sqrt{T} ⟩}^{\otimes 6}) \geq 7

★√T@vprussoattested · 2/859 re-run2026-10-23

χ_{R} ({| \sqrt{T} ⟩}^{\otimes 6}) \leq 12

★√T@vprussoverified2026-10-23

All submissions

bound	orbit	m	rank	γ	tier	attribution
$χ_{R} ({\| S ⟩}^{\otimes 1}) \geq 2$	Strange	1	2	—	lean	@vprusso · stabrank verify_challenge
$χ_{R} ({\| S ⟩}^{\otimes 1}) \leq 2$	Strange	1	2	0.6309	lean	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| S ⟩}^{\otimes 2}) \geq 2$	Strange	2	2	—	lean	@vprusso · stabrank PR #19
$χ_{R} ({\| S ⟩}^{\otimes 2}) \leq 2$	Strange	2	2	0.3155	lean	@FarLab @vprusso · arXiv:2605.28586, Theorem 2
$χ_{R} ({\| S ⟩}^{\otimes 3}) \geq 4$	Strange	3	4	—	reproduced	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| S ⟩}^{\otimes 3}) \leq 4$	Strange	3	4	0.4206	lean	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| S ⟩}^{\otimes 4}) \geq 4$	Strange	4	4	—	reproduced	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| S ⟩}^{\otimes 4}) \leq 4$	Strange	4	4	0.3155	lean	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| S ⟩}^{\otimes 5}) \geq 4$	Strange	5	4	—	reproduced	@vprusso · stabrank rank_exclusion.py
$χ_{R} ({\| S ⟩}^{\otimes 5}) \geq 5$	Strange	5	5	—	reproduced	@vprusso · stabrank slice_lift.py
$χ_{R} ({\| S ⟩}^{\otimes 5}) \geq 6$	Strange	5	6	—	attested	@vprusso · stabrank-research, round 33 (attacks/round33-strange-rank5)
$χ_{R} ({\| S ⟩}^{\otimes 5}) \leq 8$	Strange	5	8	0.3786	lean	@vprusso · product of board witnesses
$χ_{R} ({\| S ⟩}^{\otimes 6}) \geq 5$	Strange	6	5	—	reproduced	@vprusso · stabrank verify_challenge
$χ_{R} ({\| S ⟩}^{\otimes 6}) \geq 6$	Strange	6	6	—	attested	@vprusso · stabrank-research, round 33 (attacks/round33-strange-rank5)
$χ_{R} ({\| S ⟩}^{\otimes 6}) \leq 8$	Strange	6	8	0.3155	lean	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| S ⟩}^{\otimes 7}) \geq 5$	Strange	7	5	—	reproduced	@vprusso · stabrank verify_challenge
$χ_{R} ({\| S ⟩}^{\otimes 7}) \geq 6$	Strange	7	6	—	attested	@vprusso · stabrank-research, round 33 (attacks/round33-strange-rank5)
$χ_{R} ({\| S ⟩}^{\otimes 7}) \leq 16$	Strange	7	16	0.3605	verified	@vprusso · stabrank research/constructions/product_witness.py
$χ_{R} ({\| S ⟩}^{\otimes 8}) \geq 5$	Strange	8	5	—	reproduced	@vprusso · stabrank verify_challenge
$χ_{R} ({\| S ⟩}^{\otimes 8}) \geq 6$	Strange	8	6	—	attested	@vprusso · stabrank-research, round 33 (attacks/round33-strange-rank5)
$χ_{R} ({\| S ⟩}^{\otimes 8}) \leq 16$	Strange	8	16	0.3155	verified	@vprusso · stabrank research/constructions/product_witness.py
$χ_{R} ({\| N ⟩}^{\otimes 1}) \geq 2$	Norrell	1	2	—	verified	@vprusso · stabrank verify_challenge
$χ_{R} ({\| N ⟩}^{\otimes 1}) \leq 2$	Norrell	1	2	0.6309	verified	@vprusso · stabrank research/constructions/data
$χ_{R} ({\| N ⟩}^{\otimes 2}) \geq 3$	Norrell	2	3	—	lean	@vprusso · stabrank PR #18
$χ_{R} ({\| N ⟩}^{\otimes 2}) \leq 3$	Norrell	2	3	0.5000	lean	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| N ⟩}^{\otimes 3}) \geq 4$	Norrell	3	4	—	reproduced	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| N ⟩}^{\otimes 3}) \leq 4$	Norrell	3	4	0.4206	lean	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| N ⟩}^{\otimes 4}) \geq 3$	Norrell	4	3	—	reproduced	@vprusso · stabrank rank_exclusion_codes.py
$χ_{R} ({\| N ⟩}^{\otimes 4}) \geq 5$	Norrell	4	5	—	reproduced	@vprusso · stabrank slice_lift.py
$χ_{R} ({\| N ⟩}^{\otimes 4}) \geq 6$	Norrell	4	6	—	attested	@vprusso · stabrank research/qutrit_m4_rank5
$χ_{R} ({\| N ⟩}^{\otimes 4}) \geq 7$	Norrell	4	7	—	attested	@vprusso · stabrank research/n4_rank6
$χ_{R} ({\| N ⟩}^{\otimes 4}) \leq 7$	Norrell	4	7	0.4428	lean	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| N ⟩}^{\otimes 5}) \geq 7$	Norrell	5	7	—	attested	@vprusso · stabrank research/n4_rank6
$χ_{R} ({\| N ⟩}^{\otimes 5}) \leq 10$	Norrell	5	10	0.4192	verified	@vprusso · this repository (stabrank-research, round 28 letter lifts)
$χ_{R} ({\| N ⟩}^{\otimes 5}) \leq 12$	Norrell	5	12	0.4524	verified	@vprusso · stabrank research/constructions/product_witness.py
$χ_{R} ({\| N ⟩}^{\otimes 6}) \geq 7$	Norrell	6	7	—	attested	@vprusso · stabrank research/n4_rank6
$χ_{R} ({\| N ⟩}^{\otimes 6}) \leq 16$	Norrell	6	16	0.4206	verified	@vprusso · stabrank research/constructions/product_witness.py
$χ_{R} ({\| H_{3} ⟩}^{\otimes 1}) \geq 2$	H₃	1	2	—	verified	@vprusso · stabrank verify_challenge
$χ_{R} ({\| H_{3} ⟩}^{\otimes 1}) \leq 2$	H₃	1	2	0.6309	verified	@vprusso · stabrank research/constructions/data
$χ_{R} ({\| H_{3} ⟩}^{\otimes 2}) \geq 3$	H₃	2	3	—	lean	@vprusso · stabrank PR #18
$χ_{R} ({\| H_{3} ⟩}^{\otimes 2}) \leq 3$	H₃	2	3	0.5000	lean	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| H_{3} ⟩}^{\otimes 3}) \geq 4$	H₃	3	4	—	reproduced	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| H_{3} ⟩}^{\otimes 3}) \leq 4$	H₃	3	4	0.4206	lean	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| H_{3} ⟩}^{\otimes 4}) \geq 3$	H₃	4	3	—	reproduced	@vprusso · stabrank rank_exclusion_codes.py
$χ_{R} ({\| H_{3} ⟩}^{\otimes 4}) \geq 5$	H₃	4	5	—	reproduced	@vprusso · stabrank slice_lift.py
$χ_{R} ({\| H_{3} ⟩}^{\otimes 4}) \geq 6$	H₃	4	6	—	attested	@vprusso · stabrank research/qutrit_m4_rank5
$χ_{R} ({\| H_{3} ⟩}^{\otimes 4}) \geq 7$	H₃	4	7	—	attested	@vprusso · stabrank research/h3_m4_rank6
$χ_{R} ({\| H_{3} ⟩}^{\otimes 4}) \leq 8$	H₃	4	8	0.4732	lean	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| H_{3} ⟩}^{\otimes 5}) \geq 7$	H₃	5	7	—	attested	@vprusso · stabrank research/h3_m4_rank6
$χ_{R} ({\| H_{3} ⟩}^{\otimes 5}) \leq 12$	H₃	5	12	0.4524	verified	@vprusso · stabrank research/constructions/product_witness.py
$χ_{R} ({\| H_{3} ⟩}^{\otimes 6}) \geq 7$	H₃	6	7	—	attested	@vprusso · stabrank research/h3_m4_rank6
$χ_{R} ({\| H_{3} ⟩}^{\otimes 6}) \leq 16$	H₃	6	16	0.4206	verified	@vprusso · stabrank research/constructions/product_witness.py
$χ_{R} ({\| T_{3} ⟩}^{\otimes 1}) \geq 3$	T₃	1	3	—	lean	@vprusso · stabrank PR #19
$χ_{R} ({\| T_{3} ⟩}^{\otimes 1}) \leq 3$	T₃	1	3	1.0000	lean	Kocia and Sarovar · arXiv:2003.01130
$χ_{R} ({\| T_{3} ⟩}^{\otimes 2}) \geq 3$	T₃	2	3	—	lean	@vprusso · stabrank PR #16
$χ_{R} ({\| T_{3} ⟩}^{\otimes 2}) \leq 3$	T₃	2	3	0.5000	lean	Kocia and Sarovar · arXiv:2003.01130
$χ_{R} ({\| T_{3} ⟩}^{\otimes 3}) \geq 6$	T₃	3	6	—	reproduced	@vprusso · stabrank PR #16
$χ_{R} ({\| T_{3} ⟩}^{\otimes 3}) \geq 7$	T₃	3	7	—	verified	@vprusso · stabrank PR #41
$χ_{R} ({\| T_{3} ⟩}^{\otimes 3}) \geq 8$	T₃	3	8	—	attested	@vprusso · stabrank research/t3_rank7
$χ_{R} ({\| T_{3} ⟩}^{\otimes 3}) \leq 8$	T₃	3	8	0.6309	lean	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| T_{3} ⟩}^{\otimes 4}) \geq 6$	T₃	4	6	—	reproduced	@vprusso · stabrank cert_t3m3_rank6.py
$χ_{R} ({\| T_{3} ⟩}^{\otimes 4}) \geq 8$	T₃	4	8	—	attested	@vprusso · stabrank research/t3_rank7
$χ_{R} ({\| T_{3} ⟩}^{\otimes 4}) \leq 9$	T₃	4	9	0.5000	lean	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| T_{3} ⟩}^{\otimes 5}) \geq 6$	T₃	5	6	—	reproduced	@vprusso · stabrank cert_t3m3_rank6.py
$χ_{R} ({\| T_{3} ⟩}^{\otimes 5}) \geq 8$	T₃	5	8	—	attested	@vprusso · stabrank research/t3_rank7
$χ_{R} ({\| T_{3} ⟩}^{\otimes 5}) \leq 18$	T₃	5	18	0.5262	lean	@vprusso · stabrank sector-state search
$χ_{R} ({\| T_{3} ⟩}^{\otimes 6}) \geq 8$	T₃	6	8	—	attested	@vprusso · stabrank research/t3_rank7
$χ_{R} ({\| T_{3} ⟩}^{\otimes 6}) \leq 27$	T₃	6	27	0.5000	verified	@vprusso · stabrank research/constructions/product_witness.py
$χ_{R} ({\| H ⟩}^{\otimes 1}) \geq 2$	H-type	1	2	—	verified	@vprusso · stabrank verify_challenge
$χ_{R} ({\| H ⟩}^{\otimes 1}) \leq 2$	H-type	1	2	1.0000	verified	@vprusso · stabrank research/constructions/data
$χ_{R} ({\| H ⟩}^{\otimes 2}) \geq 2$	H-type	2	2	—	lean	@vprusso · stabrank verify_challenge
$χ_{R} ({\| H ⟩}^{\otimes 2}) \leq 2$	H-type	2	2	0.5000	lean	Bravyi, Smith, and Smolin · arXiv:1506.01396
$χ_{R} ({\| H ⟩}^{\otimes 3}) \geq 3$	H-type	3	3	—	reproduced	@vprusso · stabrank PR #18
$χ_{R} ({\| H ⟩}^{\otimes 3}) \leq 3$	H-type	3	3	0.5283	lean	Bravyi, Smith, and Smolin · arXiv:1506.01396
$χ_{R} ({\| H ⟩}^{\otimes 4}) \geq 3$	H-type	4	3	—	reproduced	@vprusso · stabrank PR #18
$χ_{R} ({\| H ⟩}^{\otimes 4}) \geq 4$	H-type	4	4	—	reproduced	@vprusso · stabrank rank_exclusion.py
$χ_{R} ({\| H ⟩}^{\otimes 4}) \leq 4$	H-type	4	4	0.5000	lean	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| H ⟩}^{\otimes 5}) \geq 4$	H-type	5	4	—	reproduced	@vprusso · stabrank rank_exclusion.py
$χ_{R} ({\| H ⟩}^{\otimes 5}) \geq 5$	H-type	5	5	—	reproduced	@vprusso · stabrank slice_lift.py
$χ_{R} ({\| H ⟩}^{\otimes 5}) \geq 6$	H-type	5	6	—	attested	@vprusso · stabrank research/h5_rank5
$χ_{R} ({\| H ⟩}^{\otimes 5}) \leq 6$	H-type	5	6	0.5170	lean	Bravyi, Smith, and Smolin · arXiv:1506.01396
$χ_{R} ({\| H ⟩}^{\otimes 6}) \geq 4$	H-type	6	4	—	reproduced	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| H ⟩}^{\otimes 6}) \geq 6$	H-type	6	6	—	attested	@vprusso · stabrank research/h6_rank5
$χ_{R} ({\| H ⟩}^{\otimes 6}) \leq 6$	H-type	6	6	0.4308	lean	Qassim, Pashayan, and Gosset · arXiv:2106.07740
$χ_{R} ({\| H ⟩}^{\otimes 6}) \leq 7$	H-type	6	7	0.4679	verified	Bravyi, Smith, and Smolin · arXiv:1506.01396
$χ_{R} ({\| H ⟩}^{\otimes 7}) \geq 6$	H-type	7	6	—	attested	@vprusso · stabrank research/h6_rank5
$χ_{R} ({\| H ⟩}^{\otimes 7}) \leq 9$	H-type	7	9	0.4528	lean	Kissinger, van de Wetering, and Vilmart · arXiv:2202.09202
$χ_{R} ({\| H ⟩}^{\otimes 8}) \geq 6$	H-type	8	6	—	attested	@vprusso · stabrank research/h6_rank5
$χ_{R} ({\| H ⟩}^{\otimes 8}) \leq 12$	H-type	8	12	0.4481	lean	Kissinger, van de Wetering, and Vilmart · arXiv:2202.09202
$χ_{R} ({\| H ⟩}^{\otimes 9}) \geq 6$	H-type	9	6	—	attested	@vprusso · stabrank research/h6_rank5
$χ_{R} ({\| H ⟩}^{\otimes 9}) \leq 18$	H-type	9	18	0.4633	verified	@vprusso · arXiv:2106.07740, arXiv:2202.09202
$χ_{R} ({\| H ⟩}^{\otimes 10}) \geq 6$	H-type	10	6	—	attested	@vprusso · stabrank research/h6_rank5
$χ_{R} ({\| H ⟩}^{\otimes 10}) \leq 18$	H-type	10	18	0.4170	lean	Kissinger, van de Wetering, and Vilmart · arXiv:2202.09202
$χ_{R} ({\| T ⟩}^{\otimes 1}) \geq 2$	BK T-type	1	2	—	verified	@vprusso · stabrank verify_challenge
$χ_{R} ({\| T ⟩}^{\otimes 1}) \leq 2$	BK T-type	1	2	1.0000	verified	@vprusso · stabrank research/constructions/data
$χ_{R} ({\| T ⟩}^{\otimes 2}) \geq 2$	BK T-type	2	2	—	lean	@vprusso · stabrank verify_challenge
$χ_{R} ({\| T ⟩}^{\otimes 2}) \leq 2$	BK T-type	2	2	0.5000	lean	@vprusso · stabrank annealer
$χ_{R} ({\| T ⟩}^{\otimes 3}) \geq 3$	BK T-type	3	3	—	reproduced	@vprusso · stabrank rank_exclusion.py
$χ_{R} ({\| T ⟩}^{\otimes 3}) \leq 3$	BK T-type	3	3	0.5283	lean	@vprusso · stabrank annealer
$χ_{R} ({\| T ⟩}^{\otimes 4}) \geq 3$	BK T-type	4	3	—	reproduced	@vprusso · stabrank PR #18
$χ_{R} ({\| T ⟩}^{\otimes 4}) \leq 3$	BK T-type	4	3	0.3962	lean	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| T ⟩}^{\otimes 5}) \geq 3$	BK T-type	5	3	—	reproduced	@vprusso · stabrank rank_exclusion.py
$χ_{R} ({\| T ⟩}^{\otimes 5}) \geq 4$	BK T-type	5	4	—	reproduced	@vprusso · stabrank slice_lift.py
$χ_{R} ({\| T ⟩}^{\otimes 5}) \geq 5$	BK T-type	5	5	—	attested	@vprusso · stabrank research/t5_rank4
$χ_{R} ({\| T ⟩}^{\otimes 5}) \geq 6$	BK T-type	5	6	—	attested	@vprusso · stabrank research/t5_rank5
$χ_{R} ({\| T ⟩}^{\otimes 5}) \leq 6$	BK T-type	5	6	0.5170	lean	@vprusso · product of board witnesses
$χ_{R} ({\| T ⟩}^{\otimes 6}) \geq 4$	BK T-type	6	4	—	reproduced	@vprusso · stabrank verify_challenge
$χ_{R} ({\| T ⟩}^{\otimes 6}) \geq 5$	BK T-type	6	5	—	attested	@vprusso · stabrank research/t5_rank4
$χ_{R} ({\| T ⟩}^{\otimes 6}) \geq 6$	BK T-type	6	6	—	attested	@vprusso · stabrank research/t5_rank5
$χ_{R} ({\| T ⟩}^{\otimes 6}) \leq 6$	BK T-type	6	6	0.4308	lean	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| T ⟩}^{\otimes 7}) \geq 6$	BK T-type	7	6	—	attested	@vprusso · stabrank research/t5_rank5
$χ_{R} ({\| T ⟩}^{\otimes 7}) \leq 9$	BK T-type	7	9	0.4528	verified	@vprusso · stabrank research/constructions/product_witness.py
$χ_{R} ({\| T ⟩}^{\otimes 8}) \geq 6$	BK T-type	8	6	—	attested	@vprusso · stabrank research/t5_rank5
$χ_{R} ({\| T ⟩}^{\otimes 8}) \leq 9$	BK T-type	8	9	0.3962	verified	@vprusso · stabrank research/constructions/product_witness.py
$χ_{R} ({\| T ⟩}^{\otimes 9}) \geq 6$	BK T-type	9	6	—	attested	@vprusso · stabrank research/t5_rank5
$χ_{R} ({\| T ⟩}^{\otimes 9}) \leq 15$	BK T-type	9	15	0.4341	verified	@vprusso · arxiv:2610.17500 (the m=10 identity) and stabrank bounds/qubit_T-m10-upper-15.json
$χ_{R} ({\| T ⟩}^{\otimes 10}) \geq 6$	BK T-type	10	6	—	attested	@vprusso · stabrank research/t5_rank5
$χ_{R} ({\| T ⟩}^{\otimes 10}) \leq 15$	BK T-type	10	15	0.3907	verified	@sjbevins · arxiv:2610.17500
$χ_{R} ({\| T ⟩}^{\otimes 10}) \leq 18$	BK T-type	10	18	0.4170	verified	@vprusso · stabrank research/constructions/product_witness.py
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 1}) \geq 2$	√T	1	2	—	verified	@vprusso · stabrank verify_challenge
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 1}) \leq 2$	√T	1	2	1.0000	verified	@vprusso · this repository (stabrank-research, round 31 new-orbit survey)
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 2}) \geq 3$	√T	2	3	—	verified	@vprusso · stabrank verify_challenge
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 2}) \leq 3$	√T	2	3	0.7925	verified	@vprusso · this repository (stabrank-research, round 31 new-orbit survey)
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 3}) \geq 4$	√T	3	4	—	verified	@vprusso · stabrank verify_challenge
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 3}) \leq 4$	√T	3	4	0.6667	verified	@vprusso · this repository (stabrank-research, round 31 new-orbit survey)
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 4}) \geq 4$	√T	4	4	—	verified	@vprusso · stabrank verify_challenge
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 4}) \geq 5$	√T	4	5	—	verified	@vprusso · this repository (stabrank-research, round 33)
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 4}) \leq 5$	√T	4	5	0.5805	verified	@vprusso · this repository (stabrank-research, round 31 new-orbit survey)
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 5}) \geq 6$	√T	5	6	—	verified	@vprusso · this repository (stabrank-research, round 33)
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 5}) \leq 6$	√T	5	6	0.5170	verified	@vprusso · this repository (stabrank-research, round 33)
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 6}) \geq 6$	√T	6	6	—	verified	@vprusso · this repository (stabrank-research, round 33)
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 6}) \geq 7$	√T	6	7	—	attested	@vprusso · this repository (stabrank-research, round 33)
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 6}) \leq 12$	√T	6	12	0.5975	verified	@vprusso · stabrank research/constructions/product_witness.py
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 7}) \geq 7$	√T	7	7	—	attested	@vprusso · this repository (stabrank-research, round 33)
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 7}) \leq 12$	√T	7	12	0.5121	verified	@vprusso · stabrank autoresearch/run.py on Perlmutter (round 33 of stabrank-research, job 60161156)
$χ_{R} ({\| C S ⟩}^{\otimes 1}) \geq 2$	CS	1	2	—	verified	@vprusso · stabrank verify_challenge
$χ_{R} ({\| C S ⟩}^{\otimes 1}) \leq 2$	CS	1	2	1.0000	verified	@vprusso · stabrank docs/notes/new_orbits_design.md
$χ_{R} ({\| C S ⟩}^{\otimes 2}) \geq 3$	CS	2	3	—	verified	@vprusso · stabrank verify_challenge
$χ_{R} ({\| C S ⟩}^{\otimes 2}) \leq 3$	CS	2	3	0.7925	verified	@vprusso · stabrank docs/notes/new_orbits_design.md
$χ_{R} ({\| C S ⟩}^{\otimes 3}) \geq 3$	CS	3	3	—	verified	@vprusso · stabrank verify_challenge
$χ_{R} ({\| C S ⟩}^{\otimes 3}) \geq 4$	CS	3	4	—	reproduced	@vprusso · stabrank slice_lift.py
$χ_{R} ({\| C S ⟩}^{\otimes 3}) \leq 6$	CS	3	6	0.8617	verified	@vprusso · stabrank research/constructions/product_witness.py
$χ_{R} ({\| T_{5} ⟩}^{\otimes 1}) \geq 3$	T₅	1	3	—	lean	@vprusso · stabrank verify_challenge
$χ_{R} ({\| T_{5} ⟩}^{\otimes 1}) \leq 3$	T₅	1	3	0.6826	lean	@vprusso · stabrank docs/notes/new_orbits_design.md
$χ_{R} ({\| T_{5} ⟩}^{\otimes 2}) \geq 5$	T₅	2	5	—	reproduced	@vprusso · stabrank docs/notes/new_orbits_design.md
$χ_{R} ({\| T_{5} ⟩}^{\otimes 2}) \leq 5$	T₅	2	5	0.5000	lean	@vprusso · stabrank docs/notes/t5q_m2_rank5_exclusion.md
$χ_{R} ({\| T_{5} ⟩}^{\otimes 2}) \leq 8$	T₅	2	8	0.6460	lean	@vprusso · stabrank autoresearch/run.py
$χ_{R} ({\| T_{5} ⟩}^{\otimes 3}) \geq 5$	T₅	3	5	—	reproduced	@vprusso · stabrank verify_challenge
$χ_{R} ({\| T_{5} ⟩}^{\otimes 3}) \geq 6$	T₅	3	6	—	attested	@vprusso · stabrank docs/notes/t5_m3_lift_design.md
$χ_{R} ({\| T_{5} ⟩}^{\otimes 3}) \leq 15$	T₅	3	15	0.5609	lean	@vprusso · stabrank research/constructions/product_witness.py
$χ_{R} ({\| T_{5} ⟩}^{\otimes 4}) \geq 5$	T₅	4	5	—	reproduced	@vprusso · stabrank verify_challenge
$χ_{R} ({\| T_{5} ⟩}^{\otimes 4}) \geq 6$	T₅	4	6	—	attested	@vprusso · stabrank research/t5_m3_lift
$χ_{R} ({\| T_{5} ⟩}^{\otimes 4}) \leq 25$	T₅	4	25	0.5000	lean	@vprusso · stabrank research/constructions/product_witness.py
$χ_{R} ({\| T_{5} ⟩}^{\otimes 5}) \geq 6$	T₅	5	6	—	attested	@vprusso · stabrank research/t5_m3_lift
$χ_{R} ({\| T_{5} ⟩}^{\otimes 5}) \leq 75$	T₅	5	75	0.5365	verified	@vprusso · stabrank research/constructions/product_witness.py
$χ_{R} (\| {cat}_{2} ⟩) \leq 1$	magic cat	2	1	—	verified	Qassim, Pashayan, and Gosset · arXiv:2106.07740
$χ_{R} (\| {cat}_{3} ⟩) \geq 2$	magic cat	3	2	—	verified	@vprusso · stabrank verify_challenge
$χ_{R} (\| {cat}_{3} ⟩) \leq 2$	magic cat	3	2	1.0000	verified	Qassim, Pashayan, and Gosset · arXiv:2106.07740
$χ_{R} (\| {cat}_{4} ⟩) \geq 2$	magic cat	4	2	—	verified	@vprusso · stabrank verify_challenge
$χ_{R} (\| {cat}_{4} ⟩) \leq 2$	magic cat	4	2	0.5000	verified	Qassim, Pashayan, and Gosset · arXiv:2106.07740
$χ_{R} (\| {cat}_{5} ⟩) \geq 3$	magic cat	5	3	—	verified	@MicheldeMessieres · stabrank verify_challenge
$χ_{R} (\| {cat}_{5} ⟩) \leq 3$	magic cat	5	3	0.5283	verified	Qassim, Pashayan, and Gosset · arXiv:2106.07740
$χ_{R} (\| {cat}_{6} ⟩) \geq 3$	magic cat	6	3	—	verified	@MicheldeMessieres · stabrank verify_challenge
$χ_{R} (\| {cat}_{6} ⟩) \leq 3$	magic cat	6	3	0.3962	verified	Qassim, Pashayan, and Gosset · arXiv:2106.07740
$χ_{R} (\| {cat}_{7} ⟩) \geq 3$	magic cat	7	3	—	cited	Qassim, Pashayan, and Gosset · arXiv:2106.07740
$χ_{R} (\| {cat}_{7} ⟩) \geq 4$	magic cat	7	4	—	verified	@MicheldeMessieres · stabrank verify_challenge
$χ_{R} (\| {cat}_{7} ⟩) \geq 5$	magic cat	7	5	—	attested	@vprusso · stabrank-research, round 31 (attacks/round31-J-lower-bounds)
$χ_{R} (\| {cat}_{7} ⟩) \leq 6$	magic cat	7	6	0.5170	verified	Qassim, Pashayan, and Gosset · arXiv:2106.07740
$χ_{R} (\| {cat}_{8} ⟩) \geq 3$	magic cat	8	3	—	cited	Qassim, Pashayan, and Gosset · arXiv:2106.07740
$χ_{R} (\| {cat}_{8} ⟩) \geq 4$	magic cat	8	4	—	verified	@MicheldeMessieres · stabrank verify_challenge
$χ_{R} (\| {cat}_{8} ⟩) \geq 5$	magic cat	8	5	—	attested	@vprusso · stabrank-research, round 31 (attacks/round31-J-lower-bounds)
$χ_{R} (\| {cat}_{8} ⟩) \leq 6$	magic cat	8	6	0.4308	verified	Qassim, Pashayan, and Gosset · arXiv:2106.07740