What closes after projection

Projection can lose information even when the source is exactly known. In a finite positive carrier the dissipative closure defect measures that loss, and regularity of the coherence symbol turns it into a bound on coherent leakage.

11. Observable quotients, memory and two-face closure

A field description must specify how it is accessed. Let $\mathcal X$ be a candidate state space, $\mathcal B$ a probe family and $R:\mathcal X\times\mathcal B\to\mathcal V$ its response map. Define

$$x\sim_{\mathcal B}y\quad\Longleftrightarrow\quad R(x,b)=R(y,b)\ \text{for all }b\in\mathcal B.$$

The quotient retains exactly the distinctions available through those probes. Dissipative probes may sample $S_t$, $\Lambda_T$ or $\mathcal D_T$, while coherent probes may sample $U_s$, $\Gamma_T$ or $\mathfrak B_T$. Complete operator recovery, scalar-profile identifiability and finite window incidence have different information content. The factorization questions of Shadow Theory apply to each target with its declared probes [22, 20].

11.1 Compression and exact projected dynamics

In finite dimension let $J:\mathbb C^k\to\mathsf H$ be an isometry, $P=JJ^*$ and $Q=I-P$. For a bounded linear generator $G$, the source equation $\dot y=Gy$ has resolved component $x=Py$ and unresolved component $z=Qy$. Solving the latter by variation of constants yields

$$\begin{aligned}\dot x(t)={}&PGP x(t)+PGQ e^{tQGQ}z(0)\qquad\text{(12)}\\ &+\int_0^t PGQ e^{(t-s)QGQ}QGP x(s)\,ds.\end{aligned}$$

The restricted exponentials act on $Q\mathsf H$. This identity retains the unresolved preparation and its dynamical return. It applies to the finite combined CSCF generator $G=-K-i h(K)$ as well as to $G=-K$. Dropping the last two terms requires an invariant-subspace or approximation argument. In the general quotient formulation, an induced evolution under a surjective linear readout $R$ exists precisely when its kernel is invariant under the source evolution [20, Thm. 5.2].

The response-kernel viewpoint also covers pairings such as $\mathop{\mathrm{Tr}}(\rho O)$, boundary input/output operators and scattering maps. Such a relational description is compatible with CSCF when its kernels are specified functions or compressions of $K$. A low-rank point-cloud approximation alone does not supply a physical carrier identification; its extraction and target must be independently specified. Here the empirical carriers are the actual finite operators declared in Section 10.

11.2 An exact dissipative closure certificate

Set

$$A(t)=J^*e^{-tK}J,\qquad B(t)=Qe^{-tK}J,\qquad M(t)=A(2t)-A(t)^2.\tag{13}$$

Theorem 11.1. Gram defect and closure.

For finite selfadjoint $K\ge0$ and all $t,s\ge0$,

$$A(t+s)-A(t)A(s)=B(t)^*B(s),\qquad M(t)=B(t)^*B(t)\ge0.$$

At any fixed $\tau>0$, $M(\tau)=0$ if and only if $[P,K]=0$. In that case the compressed dissipative evolution closes exactly, and the same subspace reduces $h(K)$ and every coherent flow $U_s$.

Proof. Insert $I=P+Q$ between the commuting selfadjoint operators $e^{-tK}$ and $e^{-sK}$. The $P$ term gives $A(t)A(s)$ and the $Q$ term gives $B(t)^*B(s)$. If $M(\tau)=0$, then $B(\tau)=0$, so $P\mathsf H$ is invariant under $e^{-\tau K}$. Selfadjointness also makes its orthogonal complement invariant. Therefore $P$ commutes with that operator and with $K=-\tau^{-1}\log(e^{-\tau K})$. The converse follows directly. Functional calculus then implies $[P,h(K)]=[P,U_s]=0$. ◻

For reversible finite-state Markov dynamics this is the equilibrium-weighted certificate of the effective-autonomy study [1, Sec. 6]. Weighting is part of the correspondence: for row generator $Q_{\rm M}$, strictly positive stationary law $\pi$ and partition incidence matrix $C_{\rm part}$ with nonempty blocks,

$$K=-D_\pi^{1/2}Q_{\rm M}D_\pi^{-1/2},\qquad J=D_\pi^{1/2}C_{\rm part}D_{\bar\pi}^{-1/2},\quad \bar\pi=\pi C_{\rm part}.$$

Here $\pi$ is written as a row vector in $\bar\pi=\pi C_{\rm part}$, and $D_\pi$ and $D_{\bar\pi}$ are diagonal probability matrices. Then $J^*J=I$. Unweighted matrices for a nonuniform equilibrium generally do not satisfy the symmetric formula. The coherent invariance statement concerns a Hilbert subspace; its reduced unitary is not thereby a stochastic transition matrix.

11.3 Quantitative transfer to the coherent face

Theorem 11.2. Closure-to-coherence transfer.

Suppose $0\le K\le\Lambda I$ in finite dimension, and let $h$ be real and Lipschitz on $\mathop{\mathrm{\sigma}}K$, with constant $L_h$. For $\tau>0$ and $t\in\mathbb R$,

$$\left\lVert Qe^{-it h(K)}J\right\rVert_F\le \min\left\{\sqrt{k},\ \frac{|t|L_h e^{\Lambda\tau}}{\tau} \sqrt{\mathop{\mathrm{Tr}}M(\tau)}\right\}.\tag{14}$$

Here $\left\lVert\cdot\right\rVert_F$ denotes the Frobenius norm. In particular,

$$\left\lVert Qh(K)J\right\rVert_F\le L_h\left\lVert QKJ\right\rVert_F, \quad \tau e^{-\Lambda\tau}\left\lVert QKJ\right\rVert_F \le\sqrt{\mathop{\mathrm{Tr}}M(\tau)}\le\tau\left\lVert QKJ\right\rVert_F.\tag{15}$$

Proof. In an orthonormal eigenbasis of $K$, the entries of $[P,f(K)]$ are $P_{ij}(f(\lambda_j)-f(\lambda_i))$. For Hermitian $F$, its two off-diagonal blocks relative to $P$ have equal Frobenius norm, giving

$$\left\lVert[P,F]\right\rVert_F^2=2\left\lVert QFJ\right\rVert_F^2.$$

The same equality holds for $F=e^{-it h(K)}$: it is normal and diagonal in this eigenbasis, or directly its two block norms agree by unitarity. The scalar bounds

$$\begin{aligned} |h(\lambda)-h(\mu)|&\le L_h|\lambda-\mu|,\\ |e^{-it h(\lambda)}-e^{-it h(\mu)}|&\le |t|L_h|\lambda-\mu|,\\ \tau e^{-\Lambda\tau}|\lambda-\mu| &\le |e^{-\tau\lambda}-e^{-\tau\mu}| \le\tau|\lambda-\mu| \end{aligned}$$

follow from Lipschitz continuity, the unit-circle chord estimate and the mean-value theorem. Summing squared commutator entries proves (15) and bounds the unitary leakage by $|t|L_h\left\lVert QKJ\right\rVert_F$. Theorem 11.1 identifies $\left\lVert Qe^{-\tau K}J\right\rVert_F^2$ with $\mathop{\mathrm{Tr}}M(\tau)$. Finally unitarity and $J^*J=I$ give the ceiling $\sqrt{k}$. ◻

The proof uses a Hilbert–Schmidt estimate, not an operator-Lipschitz claim for arbitrary functions. For $h(\lambda)=(2/\pi)\arctan(\lambda/\lambda_0)$, one may take $L_h=2/(\pi\lambda_0)$. The factor $e^{\Lambda\tau}$ explains why late-time near-stationarity can yield a weak inverse estimate. A useful measurement therefore declares bandwidth and lag as well as the symbol.

If an estimated Hermitian defect $\widehat M$ obeys $\left\lVert\widehat M-M\right\rVert_{\rm op}\le\delta$, then $\mathop{\mathrm{Tr}}M\le\mathop{\mathrm{Tr}}\widehat M+k\delta$. Thus replacing the square root in (14) by $\sqrt{\max\{0,\mathop{\mathrm{Tr}}\widehat M+k\delta\}}$ gives a valid conservative bound whenever the stated error certificate holds. This separates measurement precision from an exact closure identity.