SCC Canonical Spec — Part 5: Proved Results Registry & Closing Notes

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1. #1

   T-Beyond-Weyl. Structured Spectral Perturbation Bound for Multi-Formations. The joint K-formation Hessian spectral gap tightens from the standard Weyl bound to a formation-aware bound exploiting soft-mode localization:

   $${\mu }_{\mathrm{joint}}\ge \underset{k}{\min }{\mu }_k-(K-1){\lambda }_{\mathrm{rep}}\cdot {\omega }_{\mathrm{soft}},{\omega }_{\mathrm{soft}}:=\underset{j\ne k}{\max }\Vert {\mathcal{P}}_{O_{jk}}{\psi }_k^{\mathrm{soft}}{\Vert }^2.$$
2. #2

   T-dmin-Formula. Analytical Formula for Critical Inter-Formation Distance. The branch-conditioned minimum inter-formation distance enabling metastable coexistence should be written . The earlier scalar fit

   $$d_{\min }^{\ast }=4.8+0.31\cdot \sqrt{\beta /\alpha }-0.018\cdot \beta /\alpha$$