Introduction

Introduction

The problem

Suppose a physical model is specified on a state space $\mathcal{H}_{\mathrm{src}}$ (the source), while every measurement one can actually perform is the image of a fixed bounded linear map $K:\mathcal{H}_{\mathrm{src}}\to\mathcal{H}_{\mathrm{obs}}$ (the readout). Two questions then arise, and they are not the same question.

1. Statics. Which source quantities are functions of the readout alone?

2. Dynamics. If the source evolves, does the readout evolve autonomously, i.e. is there a law of motion on the observable state alone that reproduces the readout of every source trajectory?

An affirmative answer to (Q1) for a given functional is a purely kinematic statement about the fibers of $K$. An affirmative answer to (Q2) is a compatibility statement between the fibers of $K$ and the flow. The central point of this paper is that (Q2) can fail while the quotient of (Q1) remains a perfectly coherent, complete Hilbert space with a perfectly well-defined observable state. A theory can be an exact quotient theory and nevertheless fail to be an autonomous presentation of its own observable degrees of freedom. When that happens, the exact projection of the source dynamics is not a first-order autonomous law but an integro-differential law carrying two extra structures: dependence on unresolved initial data, and dependence on the resolved history through a memory kernel.

Four distinctions that must not be merged

Much informal discussion of “information loss” under coarse-graining conflates phenomena that are mathematically independent. We insist on four:

1. Exact nonidentifiability. $\ker K\neq\{0\}$. Distinct source states are identically equal in readout, for all time-zero purposes. No amount of numerical precision recovers them.

2. Unstable inversion. $K$ may be injective yet have unbounded inverse on its range. In general this is the failure of $K$ to be bounded below on $(\ker K)^{\perp}$, equivalently the failure of $\operatorname{Ran}K$ to be closed (Theorem 2.3). Under the additional hypothesis that $K$ is compact with infinite-dimensional range, it takes the familiar form that the singular values tend to $0$; but singular values are available only in the compact setting, and the general criterion is the one just stated. Here the source state is uniquely determined by the readout, but not continuously.

3. Approximation error. If $K$ is compact one may truncate its singular system at rank $r$ and incur exactly $\sigma_{r+1}$ in operator norm. This is a controlled, quantified deficit, and it exists even when $K$ is injective and even when its inverse is bounded.

4. Failure of autonomous dynamics. $U(t)\ker K\not\subseteq\ker K$. This is a genuinely dynamical obstruction, not determined by the static facts (D1)–(D3).

The logical relation between (D1) and (D4) must be stated with care, and we do so once here and abide by it throughout. Since $\{0\}$ is invariant under every linear map, $\ker K=\{0\}$ forces $U(t)\ker K\subseteq\ker K$ trivially. Hence

$$\text{(D4) holds}\ \Longrightarrow\ \text{(D1) holds:} \qquad U(t_{0})\ker K\not\subseteq\ker K\ \Longrightarrow\ \ker K\neq\{0\} .\tag{1.1}$$

A nontrivial kernel is therefore necessary for failure of descent. It is not sufficient, and this is the whole point: a nontrivial kernel may be invariant under the source semigroup, in which case dynamics descend exactly and the observable theory is autonomous despite losing exact source distinctions; or it may fail to be invariant, in which case descent fails. The static fact (D1) does not decide which. So (D4) is an additional dynamical obstruction, logically stronger than (D1) and not deducible from it, but it is not logically independent of it. (D2) and (D3), by contrast, are genuinely independent of both: Example 4.5 exhibits (D2) and (D3) with (D1), and hence also (D4), failing.

Section 2 isolates (D1) and (D2), Section 4 isolates (D3), and Sections 5–6 isolate (D4). The compact injective diagonal operator of Example 4.5 exists precisely to show that (D1), (D2), (D3) can be dissociated; Example 6.6 exists to show (D4) in a completely computed finite-dimensional block, where, in accordance with (1.1), the kernel $\ker P=\mathcal{H}_{Q}$ is indeed nontrivial, and the content of the failure is that it is not $L$-invariant.

Contributions

• Exact quotient (§2). $\widetilde{K}:\mathcal{H}_{K}\to\operatorname{Ran}K$ is a bounded linear bijection; $\widetilde{K}^{-1}$ is bounded in the norm inherited from $\mathcal{H}_{\mathrm{obs}}$ if and only if $\operatorname{Ran}K$ is closed (Theorem 2.3). We are careful never to identify $\mathcal{H}_{K}$ isometrically with $\operatorname{Ran}K$ carrying the inherited norm: what is always true is that $\widetilde{K}$ transports the quotient norm to $\operatorname{Ran}K$, and the transported norm agrees topologically with the inherited one exactly in the closed-range case.

• Descent of functionals (§3). $A$ factors through $K$ iff $A$ is constant on cosets of $\ker K$ (Proposition 3.2). For continuous linear functionals this is $A|_{\ker K}=0$, and $(\ker K)^{\perp}=\overline{\operatorname{Ran}K^{*}}$. The closure is essential: an observable linear functional need not be $K^{*}$ applied to any exact observable probe when $\operatorname{Ran}K^{*}$ fails to be closed (Remark 3.5).

• Optimal reduction (§4). Schmidt’s singular expansion and the Eckart–Young–Mirsky operator-norm bound $\lVert K-K_r\rVert_{\mathrm{op}}=\sigma_{r+1}$ (Theorem 4.4), together with the compact injective example (Example 4.5).

• Semigroup descent (§5). $KU(t)=V(t)K$ for a well-defined $V(t)$ on $\operatorname{Ran}K$ iff $U(t)N\subseteq N$, $N=\ker K$ (Theorem 5.2), with an explicit account of which topology the induced semigroup is strongly continuous in (Proposition 5.3) and the failure corollary (Corollary 5.4).

• Exact projected dynamics (§6). The Mori–Zwanzig-type identity (Theorem 6.2), obtained by variation of constants, exactly, with no ensemble, no random force, no Markov limit, and the closure criterion $PLQ=0$ quantified over all source states (Theorem 6.4).

• Minimal dynamical completion (§7). The Hilbert quotient $E_{\mathrm{dyn}}=\mathcal{H}_{Q}/N_Q$ is the coarsest extension of the resolved state determining every resolved response function (Theorem 7.4), it closes the resolved trajectory through a Volterra equation (Theorem 7.6), and in finite dimensions $N_Q$ is a finite intersection of kernels computable by Cayley–Hamilton (Proposition 7.7), linking the construction to Kalman observability.

• Nonlinear projectability (§8). For a surjective submersion $p:S\to T$, a projected field $Y$ with $Dp\circ X=Y\circ p$ exists iff $Dp_sX_s$ is constant along fibers (Theorem 8.2); this is the nonlinear analogue of $PLQ=0$.

• Elimination (§9–§10). Schur-complement effective operators for bounded quadratic actions with boundedly invertible $C$ (Theorem 9.2), and stationary elimination in a diffeomorphism-invariant action producing a covariant effective stress-energy with total covariant conservation (Proposition 10.2, Theorem 10.5).

What is established mathematics, and what is new here

The component theories on which this paper draws are established mathematics, and we claim none of them. Closed-range theory and the bounded inverse theorem are classical functional analysis [1, 13]; the singular system and the operator-norm optimality of rank-$r$ truncation are due to Schmidt, Eckart–Young, and Mirsky [14, 2, 9]; the theory of $C_{0}$-semigroups and of invariant subspaces under them is standard [3, 11]; the projection identity of Theorem 6.2 is the Mori–Zwanzig identity [20, 10, 21]; the finite rank criterion of Proposition 7.7 is Kalman observability [6, 15]; the Schur complement and block-operator elimination are classical [19]; and Noether identities and the variational definition of stress-energy are textbook general relativity [16]. Where these results appear below they are proved for completeness and stated in the form needed, not asserted as new.

What this paper contributes is their rigorous integration into a single source–readout architecture, and the results that only that integration makes visible. Specifically: the exact descent/closure obstruction, that source dynamics descend to the observable quotient if and only if the readout kernel is invariant (Theorem 5.2), equivalently, in the projected setting, if and only if $PLQ=0$ quantified over all source states (Theorem 6.4), together with the sharp logical relation this obstruction bears to the static phenomena (D1)–(D3), which is neither independence nor implication in the direction usually assumed (1.1); the target-relative minimal dynamical completion $E_{\mathrm{dyn}}=\mathcal{H}_{Q}/N_{Q}$, characterized by a universal property among all sufficient extensions (Theorem 7.4) and shown to close the resolved trajectory (Theorem 7.6), whose minimality is relative to a declared target and changes when the target changes (Remark 7.5); the accompanying separation of algebraic descent from stable readout (Theorem 3.4); and the unified statement of the relation among unresolved initial data, memory, and completion. On that last point the precise claim is the following, and it is deliberately weaker than the slogan it replaces. The vanishing $PLQ=0$ makes the initial-state term, the memory kernel, and the minimal completion all trivial simultaneously, and is equivalent to autonomous closure for every source state (Theorem 6.4). If $PLQ\neq0$, then some unresolved initial data do affect the resolved evolution and $E_{\mathrm{dyn}}$ is nontrivial. The memory kernel $\mathcal{K}(\tau)=PLQ\,e^{\tau QLQ}\,QLP$, however, can vanish identically even when $PLQ\neq0$, most simply when $QLP=0$, so that the resolved sector never feeds back into the unresolved one, but also through cancellation in the composition. Initial-state dependence, memory, and retained completion are therefore three structurally distinct phenomena which need not all be nonzero at once; the exact projection identity is what exhibits them as separate, and the completion $E_{\mathrm{dyn}}$ is the precise price of restoring an initial-value problem when the resolved state alone will not close. The field-theoretic sections then carry the same architecture into elimination: Schur-complement effective operators and a variational, covariant effective stress-energy, with their scope and their failure modes marked.

What is not claimed

We prove no empirical proposition. We do not claim that the physical universe possesses hidden source variables, memory, or any particular effective correction. We supply necessary-and-sufficient criteria and exact reduction identities for models in which a source/readout architecture has been declared. A formal memory kernel, a compact approximation, a Schur complement, or a named effective tensor is, by itself, a piece of mathematics and not a physical discovery; its physical content, if any, lies entirely in the model that supplies $K$, $L$, and the action. We also do not claim that the memory term is stochastic. It is not: at the level of exactness maintained here, both $\eta_{z_0}$ and $\mathcal{K}$ are deterministic linear objects. They acquire a probabilistic reading only after an additional, independent, probabilistic model of $z(0)$ is imposed, which we do not impose.

Relation to the companion papers

This paper is the dynamical layer of a seven-paper foundational sequence. Its companions in that sequence are the preceding companion papers, cited as [22, 23, 24, 25]: descent and equivariant reconstruction obstructions [22]; target-relative necessity of completion [23]; canonical minimal source completion [24]; and geometric realization with variational response [25]. The present paper is self-contained: every definition it needs is restated locally, and every theorem it states is proved here, including the minimal-completion theorem, which we prove directly (Theorem 7.4) rather than importing. Cross-references to [22, 23, 24, 25] are for context only.

Notation and standing conventions

All Hilbert spaces are complex unless stated otherwise, with inner products $\langle \cdot , \cdot \rangle$ linear in the second argument. $\mathcal{B}(\mathcal{H},\mathcal{K})$ denotes bounded linear operators. For $T\in\mathcal{B}(\mathcal{H},\mathcal{K})$ we write $\ker T$, $\operatorname{Ran}T$, $T^{*}$, and $\lVert T\rVert_{\mathrm{op}}$. Orthogonal projections are denoted $P$, $Q=I-P$, with ranges $\mathcal{H}_{P}=P\mathcal{H}$ and $\mathcal{H}_{Q}=Q\mathcal{H}$; both are closed, and $\mathcal{H}=\mathcal{H}_{P}\oplus\mathcal{H}_{Q}$. We freely regard $PLQ$ as an operator $\mathcal{H}_{Q}\to\mathcal{H}_{P}$, $QLQ$ as an operator $\mathcal{H}_{Q}\to\mathcal{H}_{Q}$, and so on, the identification being the obvious restriction/corestriction. Semigroups are one-parameter families $U(t)$, $t\ge0$, of bounded operators with $U(0)=I$, $U(t+s)=U(t)U(s)$, and $t\mapsto U(t)\Psi$ continuous for each $\Psi$ (the $C_0$ property) [3, 11]. In Sections 6–7 the generator $L$ is bounded and $U(t)=e^{tL}$, which makes all exponentials norm-convergent power series and all manipulations below elementary; unbounded generators are addressed only in explicitly flagged remarks, and never silently.