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STATUS · 2026-08-19: E0 operator-algebra theorem + E1 Python certificate. Canonical Drive monograph: E47 Projector-Induced Conditional-Expectation and Cross-Dynamical Closure Theorem.
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Let $P=P_{47}$ and $Q=I-P$ with ranks $47$ and $78$. Define
$$ \mathcal D_P(X)=PXP+QXQ. $$
This map is completely positive, trace preserving, unital, self-adjoint in Hilbert-Schmidt inner product, and idempotent.
$$ \operatorname{Fix}(\mathcal D_P)=\operatorname{Comm}(P)\cong M_{47}(\mathbb C)\oplus M_{78}(\mathbb C), $$
so
$$ \dim_\mathbb C\operatorname{Fix}(\mathcal D_P)=47^2+78^2=8293. $$
The transverse coherence space has dimension
$$ 2(47)(78)=7332, $$
and
$$ 125^2=15625=8293+7332. $$
The superoperator spectrum is $\{1^{(8293)},0^{(7332)}\}$.
If $[G,P]=0$, then $e^{-itG}$ preserves $\operatorname{Ran}P$. If $A\succeq0$ and $[A,P]=0$, then $e^{-tA}$ preserves the same sector. Projected nonlinear flows $\dot u=PF(u)$ are tangent to $\operatorname{Ran}P$.
Machine boundary: canonical-basis channel residuals are zero; basis-covariance closes at floating precision. The finite-step fluid energy derivative carries the expected discretization error while the continuous energy-skew identity is algebraic.