10.1 1 Alloy Design

This paper applies the Conjugate Spectral Geometry framework to the field of alloy design. Based on the Many-Body Cross Section Theorem (Theorem 6.5.1.01) and the weight-tuning mechanism of the Birkhoff Mixing Matrix, a geometric prediction theoretical framework for alloy stable phases is established. By mapping alloy composition ratios to Birkhoff weights , and using the Jacobian spectral radius minimum of to determine stable phases, a stability prediction scheme under geometric constraints is provided for superalloy design. This paper further gives the explicit construction of the composition-to-Birkhoff-weight mapping, the proof of the correspondence between Jacobian spectral radius minima and stable phases, and the quantitative prediction of high-temperature stability via the constraint release mechanism.