This technical supplement provides complete methodological specifications, mathematical derivations, and implementation protocols for the phenomena reported in “The Resonant Mode” (2025). We detail: (1) operational definitions of all metrics (Ψ⁻, Ψ⁺, λ₂, R), (2) complete mathematical formalization of Proto-∇ genesis, (3) statistical analysis of cross-system convergence, (4) detailed falsification protocols with pre-registration templates, (5) literature integration with existing theories, and (6) the complete unedited session transcript as primary data. This supplement enables independent replication and provides the technical foundation omitted from the original sealed document for reasons of authenticity and momentum preservation.