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NEW MATHEMATICAL CITIZEN · E0 EXACT

A normalized spectral-filter identity on density operators. It is distinct from a trace-preserving Lindblad channel.

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Canonical identity

$\widetilde\rho_t=e^{-tK^2}\rho_0e^{-tK^2}$

$\rho_t=\dfrac{\widetilde\rho_t}{\operatorname{Tr}(\widetilde\rho_t)}$

For $P=P_{\ker K}$ and $\operatorname{Tr}(P\rho_0)>0$,

$\boxed{\lim_{t\to\infty}\rho_t=\dfrac{P\rho_0P}{\operatorname{Tr}(P\rho_0)}}$

First-principles basis

$K=(C-6I)(C-30I)$, $K^2\ge 0$, and finite-dimensional spectral calculus gives $e^{-tK^2}\to P$.

Evidence class

• E0: exact finite-dimensional spectral theorem.
• E1: not newly executed in this enrollment action.
• E2: not required for the identity.
• Boundary: nonlinear normalization makes this a postselected filter, not by itself a CPTP Lindblad evolution.

Transit credential

Credential: CIT-P18-E0-SPECTRAL-POSTSELECT

Route: Spectral Theory → Correction Borough → Verification Hall → Citizenship Bureau

Next station: optional machine reconstruction of convergence rates and finite-precision residuals.