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Case file registered at the Verification Precinct
This open-system Lindblad construction is a featured Purple-Line case file at The Verification Precinct, with exact theorem, QuTiP simulation, certificate correction, and unresolved 125-dimensional lift recorded separately.
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Page summary: an exact two-state Lindblad model whose stationary coherent-sector population is fixed by the declared rates $47$ and $78$. The model repairs the previous unitary non-contraction, but it is a coarse-grained $2\times2$ open-system reduction, not yet a full $125$-dimensional realization of the canonical $E_{47}\oplus E_{47}^{\perp}$ carrier.
| Component | Route | Status |
|---|---|---|
| $\Omega_c=47/125$ | 🔴 Red Line | E0 / S0 exact ratio |
| Two-state Lindblad stationary distribution | 🔴 + 🟣 | E0 exact theorem for the declared model |
| QuTiP integration | 🔵 + 🟣 | E2 simulation, numerically testable |
| Full $125$-dimensional $47\oplus78$ channel | 🔭 Observatory | not implemented |
| Physical universality of the rates $47,78$ | 🔭 Observatory | open constitutive choice |
Let $|0\rangle$ denote the aggregate coherent sector and $|1\rangle$ the aggregate damped complement. Define
$C_{\mathrm{op}}=|0\rangle\langle0|.$
The jump operators are
$J_{10}=\sqrt{47}\,|0\rangle\langle1|,\qquad J_{01}=\sqrt{78}\,|1\rangle\langle0|.$
With $H=0$, the Lindblad equation is
$\dot\rho=\sum_{a\in\{10,01\}}\left(J_a\rho J_a^\dagger-\frac12\{J_a^\dagger J_a,\rho\}\right).$
This is a legitimate completely positive trace-preserving Markov semigroup on a two-level system.
Write
$p(t)=\rho_{00}(t)=\langle C_{\mathrm{op}}\rangle_t.$
Then
$\dot p=47(1-p)-78p=47-125p.$
Therefore
$p(t)=\frac{47}{125}+\left(p(0)-\frac{47}{125}\right)e^{-125t}.$