[Revisit 3] Project Alchemy Revisited: A Map and Seven Gates for Mixing Epitaxial Oxides

Project Alchemy Revisited turns Angie's Mixing Rule into a method. No single rare-earth oxide matches Si(111) exactly; a mixture of two can. Instead of choosing answers first, this revision places all thirteen C-type (bixbyite) sesquioxides on one map (lattice constant at the 650 °C growth temperature against cation radius) and passes every binary mixture through seven gates: lattice at growth, thermal history, phase stability at GaN growth, valence stability, cation-size disorder, critical thickness, and an application filter. Of 78 pairs, 36 cross the Si target and 18 survive phase, size and valence. Every claim is tagged IMPLEMENTABLE, TESTABLE or CONCEPT; three scripts regenerate every number. Four results. (1) Lock-in: the lattice constant is linear in cation radius (rms 0.024 Å), so matching Si pins the mean radius near 0.941 Å, and with it the C-type stability limit of the survivors at an estimated 1184–1241 °C, only 134–191 K above GaN growth. (2) Thermal history: all C-type oxides expand about twice as fast as Si, so no mixture matches at more than one temperature. Misfit dislocations glide only above a freeze-out temperature (526–839 °C), so coherence is decided over the mobile window up to GaN growth at 1050 °C, not between growth and room temperature. (3) Choices: the (Gd,Eu) base is stopped by the europium valence gate (Eu₂O₃ can be reduced to EuO in a GaN reactor); (Gd,Sm) and (Gd,Nd) pass every general gate. (4) An exam against the literature: the method places the reported Si-matched PrYO₃ within 0.03 of its grown composition and predicts the Eu₂O₃ phase transition within 33 K. A short extension carries the map to ScAlN on GaN, the starting point of Project Proteus. The first edition's three paths become verdicts on the map: Eu₂O₃ is stopped, PrYO₃ passes, and the digital alloy needs thinner layers than proposed. Formal paper in English (authoritative), Traditional Chinese and Japanese; primers for middle-school readers in the same three languages; full code and sources. Not peer-reviewed. Defensive publication. Text CC BY 4.0, code Apache 2.0. Relation to the first edition (June 2026, DOI 10.5281/zenodo.20624822): this is a rewrite as a method paper. The first edition's balance-point rule omitted the 1050 °C nitride step and is replaced by balance over the mobile window.