<aside> 🟣

NEW MATHEMATICAL CITIZEN · E0 CONDITIONAL DYNAMICAL THEOREM

This citizen records the exact open-system construction that pumps population from the 78-dimensional complement into the 47-dimensional kernel.

</aside>

Canonical identity

Let $P=P_{47}$, $Q=I-P$, and choose jump operators

$J_{ab}=\sqrt{\gamma_{ab}}\,|e_a\rangle\langle f_b|$

with $\{e_a\}$ an orthonormal basis of $E_{47}$ and $\{f_b\}$ an orthonormal basis of $Q\mathcal H$.

For the Lindblad generator

$\mathcal L(\rho)=-i[H,\rho]+\sum_{a,b}\left(J_{ab}\rho J_{ab}^\dagger-\tfrac12\{J_{ab}^\dagger J_{ab},\rho\}\right)$

with $[H,P]=0$ and $\sum_a\gamma_{ab}\ge\gamma_{\min}>0$ for every complement basis direction,

$q(t)=\operatorname{Tr}(Q\rho_t)$

obeys

$\boxed{q(t)\le e^{-\gamma_{\min}t}q(0)}$

and therefore

$\boxed{\operatorname{Tr}(Q\rho_t)\to0,\qquad \rho_\infty=P\rho_\infty P.}$

Logical distinction

$L=K$ alone produces spectral dephasing and does not generally pump all population into $E_{47}$. Active shunting requires complement-to-kernel jumps such as $PA_\mu Q$.

Evidence class