The projected-law dichotomy

The projected-law dichotomy

Descent of dynamics

(16) is the exact formalization of the statement “the present readout determines the future readout”. We stress that it is a deterministic and instantaneous demand: $U_t$ takes as input the readout state at one time, and nothing else , no history, no auxiliary variable, no distribution.

Corollary 5.3 is the single most important negative result of this paper, and the one most easily lost. An observer inside $T_1$ in Case I sees a world with exact deterministic laws, perfectly closed, admitting no anomaly and no residue. That observer’s evidence is entirely consistent with $T_1$ being the whole of physical reality, and entirely consistent with $T_1$ being a proper quotient of a strictly larger source. Dynamical closure is evidence of fiber preservation, not of fiber triviality.

What failed descent actually requires

In Case II the instantaneous readout state is insufficient, and one must say for what. The three targets below are routinely conflated; they are not the same, and only the first is an exact reconstruction.

1. Exact trajectory prediction. To recover $t\mapsto p(\Phi_t(s))$ for the individual realized $s$, one needs strictly more than $p(s)$: either additional state information (a completion variable, i.e. a refinement $e\colon S\to E$ sufficient for the target “future readout”, in the sense of Definition 4.3), or an equivalent history-dependent description that encodes the same information in the past of the trajectory. Which of the two one uses is a matter of representation; the information content required is fixed by the fiber structure.

2. Statistical prediction. One may instead supply a probability distribution over the unresolved states within the fiber $p^{-1}(p(s))$. This yields a distribution over future readouts. But note carefully: this is a prediction only after an additional probabilistic model is supplied , namely the within-fiber distribution, which is not determined by the deterministic data $(S,p,\Phi_t)$. It is an addition to the model, not a consequence of it. And what it delivers is a law over futures, not an exact reconstruction of the individual source trajectory.

3. Memory representation. One may retain a functional of the past readout. This is exact only when the past readout in fact determines the relevant fiber coordinate; in general it does not, and then memory too is only a partial substitute.

We do not assert that any of these is always available. Case II asserts a negative , that the instantaneous readout state does not suffice , and nothing more.

The Hilbert-space instance

Specialize to Example 2.9: $S=\mathcal{H}_{\mathrm{src}}$, $p=K$ bounded with $\ker K$ closed, and let the source evolution be a strongly continuous group $\Phi_t=U(t)$ with generator $L$. The dichotomy takes an operator form.

Now let $P$ be the orthogonal projection of $\mathcal{H}_{\mathrm{src}}$ onto the closed subspace $(\ker K)^{\perp}$, the “resolved” subspace, and $Q^{\!\perp}:=I-P$ the projection onto $\ker K$, the “unresolved” subspace. (We write $Q^{\!\perp}$ rather than $Q$ to avoid collision with the target $Q$ and with the generator of §9; in the Mori–Zwanzig literature this operator is universally called $Q$.) For a bounded generator $L$, invariance (19) is equivalent to the infinitesimal condition $$P\,L\,Q^{\!\perp}=0 ,\tag{20}$$ i.e. the generator does not map unresolved directions into resolved ones. Failure of (20) is Case II. Irrespective of whether (20) holds, the standard Nakajima–Zwanzig–Mori projection [13, 15, 12] yields an exact identity for the resolved component, which for bounded $L$ we derive here by variation of constants, so that the manuscript remains self-contained.

Write $\Psi(t)=U(t)\Psi(0)$, so that $\dot\Psi=L\Psi$, and split it by the two projections: $$p(t):=P\Psi(t),\qquad q(t):=Q^{\!\perp}\Psi(t),\qquad \Psi(t)=p(t)+q(t).$$ (The letter $p$ here denotes the resolved component of the state, following the standard notation of the projection literature; it is not the readout map of §2, and no confusion should arise, as the readout does not appear in this derivation.) Since $P$ and $Q^{\!\perp}$ are constant projections with $P+Q^{\!\perp}=I$, applying each to $\dot\Psi=L\Psi=L(p+q)$ gives the coupled pair $$\begin{aligned} \dot p&=P L P\,p+P L Q^{\!\perp} q, \qquad\text{(21)}\\ \dot q&=Q^{\!\perp} L P\,p+Q^{\!\perp} L Q^{\!\perp} q. \qquad\text{(22)} \end{aligned}$$ Now read (22) as an inhomogeneous linear equation for $q$, with generator $Q^{\!\perp}LQ^{\!\perp}$ and inhomogeneity $Q^{\!\perp}LP\,p(\tau)$ regarded as a given source term. Because $L$ is bounded, so is $Q^{\!\perp}LQ^{\!\perp}$, and $e^{tQ^{\!\perp}LQ^{\!\perp}}$ is a norm-continuous semigroup; variation of constants (equivalently, differentiating $e^{-tQ^{\!\perp}LQ^{\!\perp}}q(t)$ and integrating from $0$ to $t$) gives the unique solution $$q(t)=e^{tQ^{\!\perp} L Q^{\!\perp}}q(0) +\int_0^t e^{(t-\tau)Q^{\!\perp} L Q^{\!\perp}}\,Q^{\!\perp} L P\,p(\tau)\,d\tau .\tag{23}$$ Substituting (23) into the term $PLQ^{\!\perp}q$ of (21), and using $q(0)=Q^{\!\perp}\Psi(0)$ together with $p(\tau)=P\Psi(\tau)$, yields the exact identity $$\frac{d}{dt}\,P\Psi(t) = \underbrace{P L P\,\Psi(t)}_{\text{closed term}} + \underbrace{P L Q^{\!\perp} e^{tQ^{\!\perp} L Q^{\!\perp}} Q^{\!\perp}\Psi(0)}_{\text{unresolved-initial-state term}} + \underbrace{\int_0^t P L Q^{\!\perp} e^{(t-\tau)Q^{\!\perp} L Q^{\!\perp}} Q^{\!\perp} L P\;P\Psi(\tau)\,d\tau}_{\text{memory term}} .\tag{24}$$

Four points must be made about (24), because it is the most frequently over-read equation in this circle of ideas.

First: $P\Psi$ is a source-side coordinate, not the declared readout. The projection formalism evolves $P\Psi$, the component of the source state in $(\ker K)^{\perp}$. The declared readout of our architecture, by contrast, is $K\Psi$. These are different objects living in different spaces, and the identification of the two requires an argument. Under the closed-range hypothesis it is available: the restriction $$K|_{(\ker K)^{\perp}}\colon (\ker K)^{\perp}\longrightarrow \mathop{\mathrm{Ran}}K$$ is a bounded linear bijection between Banach spaces (injective because $(\ker K)^{\perp}\cap\ker K=\{0\}$; surjective onto $\mathop{\mathrm{Ran}}K$ by the orthogonal decomposition $\Psi=P\Psi+Q^{\!\perp}\Psi$ and $K\Psi=KP\Psi$), and $\mathop{\mathrm{Ran}}K$ is closed, hence complete; so the bounded inverse theorem supplies a bounded inverse. Thus $P\Psi\mapsto K\Psi$ is a topological isomorphism, and the two observable-state descriptions are equivalent: convergence, boundedness, and continuity statements transfer between them. Without closed range this stability fails , the inverse need not be bounded , and although the set-theoretic quotient (4) still holds, one may no longer freely trade $P\Psi$ for $K\Psi$ in any topological or approximation-theoretic claim.

Second: (24) is exact and deterministic. No stochasticity, no coarse-graining approximation, and no thermodynamic limit has been invoked. The second and third terms are not noise; they are definite functionals of the initial unresolved data $Q^{\!\perp}\Psi(0)$ and of the resolved history $\{P\Psi(\tau)\}_{\tau\le t}$ respectively.

Third: the exact relation between $PLQ^{\!\perp}=0$ and the vanishing of the two anomalous terms. This must be stated with care, and the naive reading , that the two terms vanish or fail to vanish together , is false. The correct statement is:

• $PLQ^{\!\perp}=0$ is equivalent to autonomous closure for every source state (Proposition 5.4 together with (20));

• when $PLQ^{\!\perp}=0$, both the unresolved-initial-state term and the memory kernel vanish identically, since each carries $PLQ^{\!\perp}$ as a left factor , recovering $\tfrac{d}{dt}P\Psi=PLP\,\Psi$. This is Theorem 5.2 in operator dress;

• conversely, if $PLQ^{\!\perp}\neq0$, then the initial-state response is nonzero for some unresolved initial state , choose $Q^{\!\perp}\Psi(0)$ with $PLQ^{\!\perp}\Psi(0)\neq0$ and evaluate at $t=0$;

• but the memory kernel may nevertheless vanish separately. It carries the additional right factor $Q^{\!\perp}LP$, so it vanishes identically whenever $Q^{\!\perp}LP=0$ , the case where the generator feeds unresolved directions into resolved ones but not conversely , and it may also vanish through cancellation in the intervening propagator $e^{(t-\tau)Q^{\!\perp}LQ^{\!\perp}}$.

We therefore do not claim that memory must be nonzero whenever closure fails. Failure of closure guarantees a nonzero initial-state response for some unresolved datum; it guarantees nothing about the memory term.

Fourth: what the unresolved-initial-state term signifies. It depends on $Q^{\!\perp}\Psi(0)$, which is exactly the fiber coordinate , the information the readout discards. Its presence is the analytic signature of the non-source projection: the projected law is not merely inconvenient but genuinely underdetermined by $P\Psi(t)$ alone. Calling the term “noise” is an additional modelling decision (cf. §5.2(2)), not a theorem; and so, equally, is any replacement of it by a memory kernel.

Examples