[pinterbanget]

Summary

This adds research/right-vector-extension-fields/. It checks #4334 (ToikY) over six non-prime fields.

#4334 proves a sharp component threshold for invertible right-vector targets over K^k, for every finite field F_q: cost at most 2 when k ≤ q, and examples with cost 3 and δ = 2 when k ≥ q+1. The sharp example uses a quadratic extension. Its software witnesses cover only prime fields; the README says it does not "claim an extension-field software census".

Claim type: an independent finite check. No new κ.

Result

For F_4, F_8, F_9, F_16, F_25 and F_27, with #4334's target of q+1 multiplication components:

• every input line is sent onto every line of P¹(K) exactly once;
• there is no common pivot over all (q+1)² pairs, so the cost exceeds 2;
• Prove the sharp component threshold for commutative right-vector targets #4334's Section 3 three-factor word reproduces every component, so the cost is 3;
• δ = 2, by brute force over every free T ∈ GL_2(K);
• 300 random k = q targets all have a common pivot.

F4 without the criterion. An exhaustive search over every free map, right-vector pair and left vector finds no two-paid-factor word. The same search finds one when a component is dropped (k = q), which is the control.

Evidence

• Standard library only; runs in about 90 seconds.
• F_(p^k) arithmetic and the irreducible quadratic are built and checked here. Nothing is imported from Prove the sharp component threshold for commutative right-vector targets #4334.

python3.12 -B scripts/research_package.py --package research/right-vector-extension-fields --check-only
  PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
python3.12 -B scripts/research_package.py --package research/right-vector-extension-fields
  PASS_REGISTERED_RESEARCH_PACKAGE research/right-vector-extension-fields outcome=completed

Head: 3a54499c0967ff4cf17214449b20fb080087e5a3.

Not covered

• The all-field proof itself, which this corroborates and does not replace.
• Singular targets and general commutative algebras.
• Any caller.

Task and overlap

• Card: bit-new-register-model, Prove the sharp component threshold for commutative right-vector targets #4334's card; refined, priority revision 7.
• Base: canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244.
• Prove the sharp component threshold for commutative right-vector targets #4334 has no comments.

Credit

The theorem, construction and word are ToikY's (#4334), answering Nick Petalas's #4276 and building on Wentao Li's #4099. Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance. Apache-2.0.

🤖 Generated with Claude Code