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Companion note, version 1.22

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@carlok carlok released this 09 Oct 10:09

Version 1.22 adds two short results and makes one remark formal.

  • Proposition 5.8. For u ∉ Q̄ with ρ = uū algebraic and distinct non-zero algebraic a₁, a₂, the space Q̄ + Q̄u + Q̄ū + Q̄u/(u² − a₁) + Q̄u/(u² − a₂) carries a 2×3 configuration, but no four-dimensional Q̄ + Q̄u + Q̄ū + Q̄w inside it does; no conjugation condition is needed, and the third family of Remark 5.7 is the case a₁ā₁ = a₂ā₂ = ρ². At a candidate, under Roy's theorem, the two terms are not both in ℒ̃, and in the third family their sum is not: one substitution in Diaz 2007, Théorème 7(2) (Fischler 2001, Lemma 6.1), not claimed.
  • Remark 2.3 is now formal: under the case n = 2 of Kirby's weak Schanuel conjecture, every candidate has Im u ∈ πℚ, and Diaz's conjecture is equivalent to the single relation of Theorem 3.5 (t² + π² transcendental for real t ≠ 0 with eᵗ algebraic).

Seven new formal results (milestones 116–118 on the Diaz mission); the mirrored project holds all 389.