T₅ — Stabilizer Rank Challenge

The state

| T_{5} ⟩ = \frac{1}{\sqrt{5}} \sum_{x \in 𝔽_{5}} ω_{5}^{x^{3}} | x ⟩, ω_{5} = e^{2 π i / 5}

The ququint T-type state, 5^−1/2 ∑_x ω₅^x³|x⟩: the qudit π/8 gate of Howard and Vala applied to |+⟩. Every cubic phase over F₅ is Clifford-equivalent to x³, since the quadratic and linear parts are Clifford phases and x → ax scales the cubic coefficient by a³, which ranges over all of F₅^*; so the Howard-Vala family, the Campbell-Anwar-Browne M state and the |XV_s, 1⟩ state of Jain and Prakash are one Clifford orbit, this one. Its amplitudes lie in Q(ω₅), the same field as the stabilizer states, so the Galois argument that works for T₃ has no analogue here.

χ(T₅) = 3: rank 2 is excluded exactly over the 30 single-ququint stabilizer states and a three-term decomposition (two basis states and one full-support state) verifies symbolically, so χ is below the local dimension, unlike T₃ where χ = 3 = p. At two copies rank 3 and rank 4 are excluded over the 3,900 two-ququint states and the annealer found a rank-8 decomposition, so 5 ≤ χ(T₅^⊗2) ≤ 8 against the product bound 9. No exponent has been published for any p = 5 state; the baseline log₅3 = 0.6826 is the single-copy product bound, and the rank-8 cell already sits below it at 0.6460.

Bounds on this orbit

bound	m	rank	γ	tier	attribution
$χ_{R} ({\| T_{5} ⟩}^{\otimes 1}) \geq 3$	1	3	—	lean	@vprusso · stabrank verify_challenge
$χ_{R} ({\| T_{5} ⟩}^{\otimes 1}) \leq 3$	1	3	0.6826	lean	@vprusso · stabrank docs/notes/new_orbits_design.md
$χ_{R} ({\| T_{5} ⟩}^{\otimes 2}) \geq 5$	2	5	—	reproduced	@vprusso · stabrank docs/notes/new_orbits_design.md
$χ_{R} ({\| T_{5} ⟩}^{\otimes 2}) \leq 5$	2	5	0.5000	lean	@vprusso · stabrank docs/notes/t5q_m2_rank5_exclusion.md
$χ_{R} ({\| T_{5} ⟩}^{\otimes 2}) \leq 8$	2	8	0.6460	lean	@vprusso · stabrank autoresearch/run.py
$χ_{R} ({\| T_{5} ⟩}^{\otimes 3}) \geq 5$	3	5	—	reproduced	@vprusso · stabrank verify_challenge
$χ_{R} ({\| T_{5} ⟩}^{\otimes 3}) \geq 6$	3	6	—	attested · 2/74 re-run	@vprusso · stabrank docs/notes/t5_m3_lift_design.md
$χ_{R} ({\| T_{5} ⟩}^{\otimes 3}) \leq 15$	3	15	0.5609	lean	@vprusso · stabrank research/constructions/product_witness.py
$χ_{R} ({\| T_{5} ⟩}^{\otimes 4}) \geq 5$	4	5	—	reproduced	@vprusso · stabrank verify_challenge
$χ_{R} ({\| T_{5} ⟩}^{\otimes 4}) \geq 6$	4	6	—	attested · 2/74 re-run	@vprusso · stabrank research/t5_m3_lift
$χ_{R} ({\| T_{5} ⟩}^{\otimes 4}) \leq 25$	4	25	0.5000	lean	@vprusso · stabrank research/constructions/product_witness.py
$χ_{R} ({\| T_{5} ⟩}^{\otimes 5}) \geq 6$	5	6	—	attested · 2/74 re-run	@vprusso · stabrank research/t5_m3_lift
$χ_{R} ({\| T_{5} ⟩}^{\otimes 5}) \leq 75$	5	75	0.5365	verified	@vprusso · stabrank research/constructions/product_witness.py

What would move it

The baseline exponent (single-copy product bound; no published exponent) is already beaten: χ ≤ 5 at m=2 gives γ = 0.5000. Lowering it further needs χ ≤ 24 at m=4→ 0.4937.