cat _posts/2026-10-07-diaz-modulus-lean-note-v1-20-adds-the-five-dimensional-case-theorem-5-5-left-open.md
diaz-modulus-lean: note v1.20 adds the five-dimensional case Theorem 5.5 left open
Version 1.20 of the Diaz companion note adds the five-dimensional case left open by Theorem 5.5, and with it five new formal results (milestones 108–110), taking the mirrored project to all 364 proved results on Lean’s three standard axioms (c5a6007). Proposition 5.6 settles where the 2×3 configuration lives once a fifth coordinate is allowed: for u ∉ ℚ̄ with ρ = uū algebraic and a ∈ ℚ̄ non-zero, put z = u/(u² − a) when aā ≠ ρ² and z = u/(u² − a)² when aā = ρ²; then W = ℚ̄ + ℚ̄u + ℚ̄ū + ℚ̄z + ℚ̄z̄ carries a configuration but no four-dimensional space ℚ̄ + ℚ̄u + ℚ̄ū + ℚ̄w inside it does, and in the aā = ρ² case the conjugation-stable space ℚ̄ + ℚ̄u + ℚ̄ū + ℚ̄·u/(u² − a) carries none. Remark 5.7 prints the configuration’s shape, attributes its consequences at a candidate — u/(u² − a) ∉ ℒ̃ is Diaz 2004, Théorème 2, and u/(u² − a)² ∉ ℒ̃ is Diaz 2007, Théorème 6(3) — and records that the configuration’s invisibility to the four-dimensional spaces of Theorem 5.5 was not found in the sources read. The blueprint gained a chapter holding the results the earlier ones listed as library-only, drawn as 18 nodes with their Prove2Me pages cached (14b4337, c5b4a67), and the chapter of results not found in the sources read moved to the end (90c96da). The same day’s prove2me-logs entry carries the 6 October mission, five-dimensional spaces carrying configurations Theorem B cannot see (007ec16).