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:

   $$\mu_{\mathrm{joint}} \geq \min_k \mu_k - (K-1)\lambda_{\mathrm{rep}} \cdot \omega_{\mathrm{soft}}, \qquad \omega_{\mathrm{soft}} := \max_{j \neq k} \|\mathcal{P}_{O_{jk}} \psi_k^{\mathrm{soft}}\|^2.$$
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

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