Target-Relative Necessity of Completion: When Readout Loss Obstructs, and What a Sufficient Extension Must Retain

Finite-time dynamical descent

Jeremy RodgersSection 13 of 16Original PDF ↗Publication DOI ↗

Finite-time dynamical descent

Our final application of the criterion concerns the question that motivates most practical coarse-graining: can the projected description evolve on its own?

Setting

Let $\Phi_\tau\colon S\to S$ denote the source evolution through elapsed time $\tau$ , assumed to be a well-defined map for the fixed $\tau$ under consideration. No group law, continuity, or reversibility is needed for what follows; $\tau$ is fixed throughout this section and we prove a finite-time statement. Let $p\colon S\to T$ be the readout as before.

Definition 12.1 (Future-readout target).

The future-readout target at time $\tau$ is the target $Q_\tau$ with answer set $A_{Q_\tau}=T$ and correct-answer map $a_\tau:=p\circ\Phi_\tau\colon S\longrightarrow T, \qquad a_\tau(s)=p\bigl(\Phi_\tau(s)\bigr).\tag{57}$ In words: “given the source state now, what will the readout be after time $\tau$ ?”

Definition 12.2 (Autonomous readout propagator).

An autonomous readout propagator at time $\tau$ is a map $U_\tau\colon T\to T$ such that the projection intertwines the dynamics: $p\circ\Phi_\tau=U_\tau\circ p .\tag{58}$

Existence of $U_\tau$ is precisely the statement that the readout description is dynamically closed at time $\tau$ : the future readout value can be computed from the present readout value alone, without consulting the source.

The closure criterion

Theorem 12.3 (Existence criterion for the autonomous propagator).

An autonomous readout propagator $U_\tau$ satisfying (58) exists if and only if $p(s)=p(s') \qquad\Longrightarrow\qquad p\bigl(\Phi_\tau(s)\bigr)=p\bigl(\Phi_\tau(s')\bigr) \qquad\text{for all } s,s'\in S.\tag{59}$ When it exists, $U_\tau$ is unique.

Proof. Equation (58) states exactly that $a_\tau=p\circ\Phi_\tau$ factors through $p$ , i.e. that the target $Q_\tau$ of Definition 12.1 is exactly solvable from the readout with $\bar a_{Q_\tau}=U_\tau$ . By Theorem 3.1 applied to $Q_\tau$ (with $A_{Q_\tau}=T$ and $a_{Q_\tau}=a_\tau$ ), such a factorization exists if and only if $a_\tau$ is constant on every fiber of $p$ , which is condition (3) of that theorem and reads, after substituting $a_\tau=p\circ\Phi_\tau$ , exactly as (59). Uniqueness of $U_\tau$ is the uniqueness clause of Theorem 3.1. ◻

Corollary 12.4 (Dynamical obstruction).

Suppose there exist $s_0,s_1\in S$ with $p(s_0)=p(s_1) \qquad\text{but}\qquad p\bigl(\Phi_\tau(s_0)\bigr)\neq p\bigl(\Phi_\tau(s_1)\bigr).\tag{60}$ Then no map $U_\tau\colon T\to T$ satisfies (58). Concretely: two source states with the same present readout but different future readouts cannot both be reproduced by any deterministic evolution law acting on the present readout state alone.

Proof. Apply Corollary 3.2 to the target $Q_\tau$ : (60) is precisely (6) for $a_\tau$ . For any candidate $U_\tau$ , the value $U_\tau(t)$ at the common readout $t=p(s_0)=p(s_1)$ is a single element of $T$ and hence cannot equal both $p(\Phi_\tau(s_0))$ and $p(\Phi_\tau(s_1))$ . ◻

Prediction at time $\tau$ versus autonomous evolution

The general theory of Section 7, Section 8 applies to $Q_\tau$ verbatim, and it is worth stating exactly what it delivers , and, just as importantly, what it does not.

By Theorem 7.1, any readout extension $e\colon S\to E$ sufficient for $Q_\tau$ must separate the obstructing pair $s_0,s_1$ of (60). By Proposition 8.2, Theorem 8.3, the coarsest such extension is the joint target image $E_{Q_\tau}=\operatorname{Im}\bigl(p,\;p\circ\Phi_\tau\bigr)\subseteq T\times T,\tag{61}$ the present readout paired with the correct future readout at time $\tau$ .

Proposition 12.5 (What $Q_\tau$ -sufficiency provides).

Let $e\colon S\to E$ be a readout extension sufficient for $Q_\tau$ . Then there is a unique map $b_\tau\colon E\longrightarrow T \qquad\text{with}\qquad p\circ\Phi_\tau=b_\tau\circ e ,\tag{62}$ so that the $E$ -value at the present instant determines the readout value at time $\tau$ .

Proof. This is Definition 6.3 and Lemma 6.5 applied to $Q_\tau$ , whose answer set is $A_{Q_\tau}=T$ and whose correct-answer map is $a_\tau=p\circ\Phi_\tau$ . ◻

Proposition 12.5 is a statement about prediction of a $T$ -valued quantity at the one specified time $\tau$ . It says the enriched present state determines the future readout. It does not say that the enriched state evolves , that is, it produces a map $E\to T$ , not a map $E\to E$ , and there is no sense in which (62) can be iterated. Autonomous evolution on $E$ is a strictly stronger requirement, and it is governed by its own descent condition.

Definition 12.6 (Autonomous evolution on an extension).

Let $e\colon S\to E$ be a readout extension. An autonomous evolution on $E$ at time $\tau$ is a map $V_\tau\colon E\to E$ satisfying $e\circ\Phi_\tau=V_\tau\circ e .\tag{63}$

Proposition 12.7 (Criterion for autonomous evolution on $E$ ).

A map $V_\tau$ satisfying (63) exists if and only if $e(s)=e(s') \qquad\Longrightarrow\qquad e\bigl(\Phi_\tau(s)\bigr)=e\bigl(\Phi_\tau(s')\bigr) \qquad\text{for all } s,s'\in S,\tag{64}$ and it is then unique.

Proof. Apply Theorem 3.1 to the surjection $e$ and the map $e\circ\Phi_\tau\colon S\to E$ : a factorization $e\circ\Phi_\tau=V_\tau\circ e$ exists iff $e\circ\Phi_\tau$ is constant on the fibers of $e$ , which is (64), and $V_\tau$ is then unique by the uniqueness clause of that theorem. ◻

Conditions (62) and (63) are different requirements, and the implication between them runs in exactly one direction.

Proposition 12.8 (Autonomous evolution implies $Q_\tau$ -sufficiency).

Let $e\colon S\to E$ be a readout extension, with structure map $u\colon E\to T$ satisfying $p=u\circ e$ . If $e$ admits an autonomous evolution $V_\tau\colon E\to E$ at time $\tau$ , in the sense of (63), then $e$ is sufficient for $Q_\tau$ , and one may take $b_\tau=u\circ V_\tau .\tag{65}$

Proof. Using $p=u\circ e$ and then (63), $p\circ\Phi_\tau =u\circ e\circ\Phi_\tau =u\circ V_\tau\circ e =\bigl(u\circ V_\tau\bigr)\circ e ,$ which is (62) with $b_\tau:=u\circ V_\tau\colon E\to T$ . Hence $a_\tau=p\circ\Phi_\tau$ factors through $e$ , and $e$ is $Q_\tau$ -sufficient by Definition 6.3. ◻

The converse fails in general. Sufficiency for $Q_\tau$ constrains $e$ to determine the future readout , a $T$ -valued quantity , whereas (64) constrains $e$ to determine the future $E$ -value, which is in general a finer quantity, since $e$ refines $p$ (Lemma 6.2 (3)). Knowing where the present $E$ -state sends the time- $\tau$ readout says nothing about where it sends the time- $\tau$ $E$ -state. The following finite example shows that the gap is real and not merely unproved.

Example 12.9 (A $Q_\tau$ -sufficient extension admitting no autonomous evolution).

Let $S=\{a,b,c,d\}, \qquad T=\{*\},$ with $p\colon S\to T$ the constant readout, and let $E=\{x,y,z\}, \qquad e(a)=e(b)=x, \quad e(c)=y, \quad e(d)=z .$ Then $e$ is surjective and $p=u\circ e$ for the constant map $u\colon E\to T$ , so $e$ is a readout extension. Define a source evolution $\Phi_\tau\colon S\to S$ by $\Phi_\tau(a)=c, \qquad \Phi_\tau(b)=d,$ with $\Phi_\tau(c)$ and $\Phi_\tau(d)$ chosen arbitrarily , say $\Phi_\tau(c)=c$ and $\Phi_\tau(d)=d$ .

Since $p$ is constant, the composite $p\circ\Phi_\tau$ is constant, hence constant on every fiber of $e$ ; by Lemma 6.5 it factors through $e$ , so $e$ is $Q_\tau$ -sufficient. (Concretely, $b_\tau\colon E\to T$ is the constant map to $*$ .)

Nevertheless $e$ admits no autonomous evolution. We have $e(a)=e(b)=x$ , so $a$ and $b$ lie in a common fiber of $e$ , while $e\bigl(\Phi_\tau(a)\bigr)=e(c)=y \;\neq\; z=e(d)=e\bigl(\Phi_\tau(b)\bigr).$ Thus $e\circ\Phi_\tau$ is not constant on the fiber $e^{-1}(x)$ , so (64) fails, and by Proposition 12.7 no map $V_\tau\colon E\to E$ satisfies $e\circ\Phi_\tau=V_\tau\circ e$ . Directly: such a $V_\tau$ would have to send the single element $x$ to both $y$ and $z$ .

So $Q_\tau$ -sufficiency does not imply autonomous $E$ -evolution, and autonomy is strictly stronger in general, with $Q_\tau$ -sufficiency the weaker consequence of it established in Proposition 12.8.

Remark 12.10 (What $E_{Q_\tau}$ is, and what it is not).

Accordingly, the joint target image (61) should be described in exactly one way: it is the universally coarsest completion of the readout for predicting the $T$ -valued readout at the fixed time $\tau$ . By Theorem 8.3 every $Q_\tau$ -sufficient extension maps onto it, and by Proposition 8.2 it is itself sufficient for that prediction.

Its $Q_\tau$ -sufficiency does not by itself make it an autonomous state space, supply an iterable dynamics, or furnish a semigroup completion. Nothing proved here supplies a map $E_{Q_\tau}\to E_{Q_\tau}$ satisfying (63), and the results of this section do not assert one. By Proposition 12.8, autonomy would be a strictly stronger property than the $Q_\tau$ -sufficiency that Theorem 8.3 establishes for $E_{Q_\tau}$ , and Example 12.9 shows that the two can come apart; the stronger property is not established here.

This leaves the question open rather than settled against it. In a particular system, $E_{Q_\tau}$ may or may not happen to admit an autonomous $V_\tau$ ; what does not hold is that any such evolution follows from the universal construction of Theorem 8.3. Whether a given extension admits an autonomous $V_\tau$ , whether such extensions can be chosen compatibly across a range of times $\tau$ , and whether the resulting family composes as a semigroup are all questions beyond Theorem 12.3, Proposition 12.7, which are finite-time factorization statements at a single fixed $\tau$ . Those questions belong to [10] and are not addressed here.

What the obstruction does and does not license

Corollary 12.4 rules out one thing precisely: a deterministic autonomous law on $T$ alone. It is silent about what to do instead, and it is worth being explicit about the available modeling responses, none of which is an automatic consequence of the theorem.

State refinement. One may replace $T$ by a $Q_\tau$ -sufficient extension $E$ , on which, by Proposition 12.5, a deterministic prediction $b_\tau\colon E\to T$ of the time- $\tau$ readout does exist. Obtaining a deterministic evolution on $E$ is a further demand, requiring (64), and is not delivered by $Q_\tau$ -sufficiency. Whether an extension meeting either condition is available in practice, and whether its variables are measurable, are separate questions the theorem does not settle. What the theorem does settle is a lower bound on what $E$ must distinguish.
Memory. One may allow the evolution law to depend on the past history of the readout rather than on its instantaneous value. This changes the shape of the modeling problem , the state is now a trajectory segment, not a point of $T$ , and whether the history supplies the missing distinctions is not settled by Theorem 12.3, which concerns the instantaneous readout. Projection-operator methods in nonequilibrium statistical mechanics [12, 5] are the classical setting in which such history-dependent reduced equations arise; the construction of exact memory equations for the present framework belongs to [10] and is not developed here.
Probabilistic modeling. One may posit a probability law and predict a distribution over future readouts. See Remark 12.11.
Domain restriction. One may restrict $S$ to a subset on which (59) does hold, obtaining an exactly closed law on the restricted problem. This is legitimate and often useful, but it is a change of the source domain, and the theorem then applies to the restricted problem rather than the original one.

Remark 12.11 (Stochasticity is not exact trajectory recovery).

It is sometimes suggested that the obstruction of Corollary 12.4 is dissolved by making the reduced law stochastic. It is not, and the reason is the one already given in Remark 5.6. Suppose $p(s_0)=p(s_1)=t$ but the futures differ. A stochastic law on $T$ assigns to $t$ a distribution over $T$ . If that law is intended not to exclude any source-compatible future , and a law that did exclude one would be predicting the impossibility of something that can in fact occur , then its support must contain both $p(\Phi_\tau(s_0))$ and $p(\Phi_\tau(s_1))$ . It therefore does not tell us which of the two actually occurs, and if the system is in fact in state $s_0$ , the stochastic law has not predicted the realized trajectory; it has predicted a spread that contains it. This is a strictly weaker claim, and it answers a different target , “what is the distribution of the future readout, given the present readout and a specified law on the fibers?” , than the one we posed.

Note also the italicized clause. Even the distributional target requires input the readout does not contain: a probability law on the fibers $F_t$ , which is a piece of source-level information. Absent such a law, the most that can be said from the readout alone is the set-valued statement of Theorem 5.4, namely that the future readout lies in $\mathsf{A}_{Q_\tau}(t)=p\bigl(\Phi_\tau(F_t)\bigr)$ , and that this set is the least sound prediction available. Stochastic modeling is a legitimate and powerful response to Corollary 12.4; it is not a refutation of it, and it should not be presented as recovering the realized state.

Finite-time dynamical descent

Setting

The closure criterion

Prediction at time τ\tauτ versus autonomous evolution

What the obstruction does and does not license

Prediction at time $\tau$ versus autonomous evolution