[pinterbanget]

Independent literal check of petalas's #4601 (head 4b2322fc), radical invariance of right-vector costs. It covers a noncommutative local algebra, which #4601 proves but does not enumerate. It agrees. No new κ.

Method

research/pr4601-noncommutative-check/verify.py uses the standard library only and runs in about 4 s.

• Literal costs. Costs come from closing products of paid factors I + u yᵀ and free GL₂(K) maps.
• No shortcuts from Prove radical invariance of right-vector costs over local products #4601. There is no pivot test and nothing is imported.
• Coset reduction uses g P(u,y) g⁻¹ = P(gu, g⁻ᵀy).

Rings and results

Every invertible target was checked:

• F₄[e; Frobenius]/e² over F₂ (noncommutative; ω·e ≠ e·ω): 46,080 targets;
• F₂[x]/x³: 1,536 targets;
• F₃[e]/e²: 3,888 targets;
• F₂[e]/e² control: 96 targets.

On all four rings:

• Prove radical invariance of right-vector costs over local products #4601's formula holds. cost_R(M) = max(δ_R(M), cost_D(M̄)), and two-factor invariance holds on every target.
• The control matches. It reproduces Prove radical invariance of right-vector costs over local products #4601's published k = 1 histogram [6, 54, 36, 0].
• Failure control. The residue cost alone is wrong on 23,130, 1,530, 3,840 and 90 targets respectively.

Limitation

With one component no target reaches cost 3, so the noncommutative three-factor regime (k ≥ q+1) is not covered here.

Credit: petalas (Nick Petalas) for #4601; ToikY (#4334), #4276 and Wentao Li (#4099) for the predecessors. Ryanson Jonathan, with Claude Code (Claude Opus 5.5) assistance.

Head: a59c2bf

🤖 Generated with Claude Code