Abstract

Abstract

We develop the observable and dynamical layer of Shadow Theory. A bounded observation operator $K:\mathcal{H}_{\mathrm{src}}\to\mathcal{H}_{\mathrm{obs}}$ identifies source states differing by $\ker K$ and produces an exact observable quotient $\mathcal{H}_{K}=\mathcal{H}_{\mathrm{src}}/\ker K$, canonically and boundedly bijective onto $\operatorname{Ran}K$. We prove that this bijection has bounded inverse in the norm inherited from $\mathcal{H}_{\mathrm{obs}}$ exactly when $\operatorname{Ran}K$ is closed, and we keep sharply apart three distinct phenomena: exact quotient loss ($\ker K\neq\{0\}$), controlled low-rank approximation (singular-value truncation, with the operator-norm optimality $\lVert K-K_r\rVert_{\mathrm{op}}=\sigma_{r+1}$), and unstable inversion (failure of $K$ to be bounded below on $(\ker K)^{\perp}$, equivalently nonclosed range). Compactness is exactly what supplies the controlled finite-rank approximation, but by itself it implies neither a nontrivial kernel nor unstable inversion.

For a strongly continuous source semigroup $U(t)$ we prove that an induced evolution on $\operatorname{Ran}K$ exists if and only if $U(t)\ker K\subseteq\ker K$ for all $t\ge 0$; failure produces initially indistinguishable source states with later distinct readouts, so that no single-valued autonomous law on the instantaneous observable state can reproduce all source trajectories. For an orthogonal resolved/unresolved splitting with bounded generator $L$ we derive the exact projected equation $$\dot x(t)=PLP\,x(t)+PLQ\,e^{tQLQ}z(0) +\int_{0}^{t}PLQ\,e^{(t-s)QLQ}QLP\,x(s)\,ds ,$$ whose second and third terms are, respectively, a deterministic unresolved-initial-state term and an exact memory kernel; no stochastic, Markovian, or timescale approximation is made anywhere. Autonomous first-order closure valid for every source initial state is proved equivalent to $PLQ=0$, i.e. to invariance of $\ker P$ under $L$.

We then construct the minimal dynamical completion: the unresolved response map $z_0\mapsto PLQ\,e^{\cdot\,QLQ}z_0$ descends to an injective map on the Hilbert quotient $E_{\mathrm{dyn}}=\mathcal{H}_{Q}/N_Q$, which is universal among surjective extensions determining all resolved responses, and which in finite dimensions is computed by a Kalman-type observability matrix. A fiberwise criterion for smooth surjective submersions gives the nonlinear analogue. Finally, eliminating a genuine source sector from a bounded quadratic action yields Schur-complement effective operators $A-BC^{-1}B^{*}$, whose correction can be nonlocal when the composed operator $BC^{-1}B^{*}$ retains nonlocal Green-operator directions; and stationary elimination in a diffeomorphism-invariant action yields a covariant effective stress-energy defined as a variational response, together with total, not sectorwise, covariant conservation. Throughout, the results are necessary-and-sufficient criteria and exact reduction identities for models in which a source/readout architecture is declared; no empirical claim is made.

¹