T-L1-F — Hard-Bar / Active-Count Bridge under L1-J Regime

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1. #1

   Let be a finite graph and a shared-pool multi-formation state. Let

   $$U(\mathbf u) = \sum_{j=1}^{K_{\mathrm{field}}} u^{(j)},\qquad A^\varepsilon(\mathbf u) = \{j : m_j(\mathbf u) > \varepsilon\},\qquad K_{\mathrm{act}}^\varepsilon = |A^\varepsilon|,$$
2. #2

   and

   $$K_{\mathrm{bar}}^{\ell_{\min}}(U;G) = \#\{[d,b] \in \mathrm{Bars}_0^{\mathrm{term}}(U;G) : b - d \ge \ell_{\min}\}$$
3. #3

   Theorem (T-L1-F). Under the L1-J regime hypothesis package – ,

   $$\boxed{\;K_{\mathrm{bar}}^{\ell_{\min}}(U(\mathbf u);G) = K_{\mathrm{act}}^\varepsilon(\mathbf u),\;}$$
4. #4

   and the map

   $$\mathcal A_{\mathrm{bar}} : A^\varepsilon(\mathbf u) \to \mathrm{Bars}_0^{\mathrm{term}}(U;G), \qquad \mathcal A_{\mathrm{bar}}(j) = \text{the unique dominant bar with birth in } N_j^r,$$