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

   $$U(\mathbf{u})=\sum _{j=1}^{K_{\mathrm{field}}}{u}^{(j)},A^{\epsilon }(\mathbf{u})=\{j:m_j(\mathbf{u})>\epsilon \},K_{\mathrm{act}}^{\epsilon }=|A^{\epsilon }|,$$
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 – ,

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

   and the map

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