Companion note, version 1.23

Version 1.23 extends Proposition 5.8 from the involution u ↦ −u to every Möbius involution of the u-line.

• Theorem 5.9. For u ∉ Q̄ with ρ = uū algebraic, let q₁, q₂ ∈ Q̄[X] be coprime quadratics (the orbit pencil of an involution τ) and l₁, l₂ non-zero members of the pencil. The space Q̄ + Q̄u + Q̄ū + Σᵢ Q̄ q₁(u)ⁱq₂(u)³⁻ⁱ/(u l₁(u) l₂(u)) carries a 2×3 configuration and contains none of u², ū², 1/(u − b), so no four-dimensional space of Theorem 5.5 inside it carries one.
• Remark 5.10. Conversely (proved in the text, not formalised), every five-dimensional space that contains a four-term progression and is invisible to Theorem 5.5 is of this form; the progression step is Bachoc–Serra–Zémor, Lemma 4. A configuration whose products fill the space can be invisible too (Q̄u⁻¹ + Q̄ + Q̄u + Q̄u³ + Q̄u⁴, not stable under conjugation). Not found in the sources read.

Four new formal results (milestones 119–120 on the Diaz mission); the mirrored project holds all 393.