Conclusion

Displayed formulas in this passage

• unlabeled · line 1892

  $$\begin{array}{c}\ker {\mathsf{K}}_{\mathrm{eq}}=\ker {\mathsf{K}}_{\mathrm{act}}={\mathcal{C}}_{\mathrm{sol}},\mathbf{P}\text{governs separation},\\ {{\mathcal{A}}^{\mathrm{reg}}}_{\mathrm{off},H}/{\mathcal{N}}_{\mathrm{reg}}[[\hslash ,\kappa ]]\cong {\mathcal{A}}_0(M)[[\kappa ]]\hookrightarrow {\mathcal{E}}_H,\\ {\mathcal{E}}_H=({\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}[[\hslash ,\kappa ]],{\star }_H)/{\mathcal{N}}_{\mathrm{sol}}[[\hslash ,\kappa ]],\\ {\alpha }_d(F{\star }_{H}G)={\alpha }_{d}F{\star }_{H^{\prime }}{\alpha }_{d}G,{\mathcal{S}}_{H^{\prime }}={\bar{\alpha }}_d({\mathcal{S}}_H)={\widehat{\mathrm{Alg}}}_{{\star }_{H^{\prime }}}(\text{unchanged named seeds}),\\ f\in {\mathfrak{O}}_{\mathrm{G}}(g)⟹{\mathcal{C}}_3(f,g){|}_{{\mathcal{C}}_{\mathrm{sol}}}\notin {{\mathcal{F}}^{\mathrm{sol}}}_{\mathrm{mc},\mathrm{pol}}\text{under Input theorem }.\end{array}$$