Companion note, version 1.6

The Diaz modulus conjecture: a machine-checked spine and the shape of its obstruction, version 1.6.

Attributions, after a reading of the literature.

This version follows a reading of the published work of Diaz, Roy, Waldschmidt and Dasgupta–Kakde, books included. Several results of the note turn out to be classical, or special cases of theirs, and each is now attributed where it appears. No formal statement changed.

Corrections

• The boundary section (§4) was wrong. Over the algebraic span of 1 and the logarithms, the strict rank inequality alone would settle Diaz's conjecture: a candidate violates it on the matrix H(u, r) itself, and the constant 1 turns the affine conic into a homogeneous quadric. Homogeneity is an obstacle only on the unconditional route over the logarithms themselves, in transcendence degree one. Version 1.5 also said the inhomogeneous direction was not being worked on. In fact Waldschmidt proved a 2×3 inhomogeneous result in 2005 and conjectured the 2×2 case, and that conjecture implies Diaz's.
• Item (2) of §7 is now Waldschmidt's strong five exponentials conjecture (1988), which he notes is weaker than the strong four exponentials conjecture and which already gives Diaz's conjecture and (S). The rank inequality that stood there contains the strong four exponentials conjecture for 2×2 matrices, so it was no weaker than item (3).
• The strong four exponentials conjecture (1.2) was proposed by Roy (1992), not Waldschmidt. The instantiation behind Theorem 2.1 is Diaz's, as Waldschmidt's book records (2000, p. 399).
• Now attributed:
  • Corollary 4.5 (e^{π²} or e^{iπ³}) to Waldschmidt 1973 and Brownawell 1974.
  • Proposition 4.4 to Waldschmidt 1974.
  • Theorem 3.12 to Diaz 2004.
  • Remark 2.2 to Diaz 2004, Proposition 1.
  • Theorem 6.4 and Corollaries 6.5–6.6 to Diaz 2007 and Waldschmidt 2005.
  • The partners of (S) to Waldschmidt 1973.
  • Stronger known forms of Theorem 3.6 and Corollary 3.7 are cited: Waldschmidt 1974, Diaz 1997.
• The Dasgupta–Kakde references now give the published papers.

The node texts on the platform that repeated any of these claims have been corrected the same way.

Checks at release

• Appendix A: 72 identifiers checked against the platform by script, 0 mismatches.
• Library: 212 of 212 proved results mirrored; the only axioms are propext, Classical.choice and Quot.sound.

The PDF attached here is built from tex/diaz_prove2me.tex at this tag. Earlier versions remain under note-v1.0 to note-v1.5.