Operator-Theoretic Foundations of Recursive Stability

A Non-Equivalence Framework for Distributionally Robust Agency

Overview

This work develops a new way of understanding stability in recursive systems—systems that repeatedly update both their internal model of the world and the policy that acts on that model.

The central claim is simple but structural:

Stability in recursive systems cannot be fully understood using value functions alone.
It requires reasoning about how probability distributions themselves evolve under feedback.

This leads to a dual perspective on decision-making systems:

A value-space view (what is optimized)
A measure-space view (what actually changes in the world model and data distribution)

The gap between these two views is where instability emerges.

Part I: The Recursive Dynamical System

1. The Recursive Stability Gap

Modern learning systems are not static. They continuously update two coupled objects:

A policy: how the system acts
A model of the environment: how the system believes the world behaves

We can write this as a feedback loop:

The policy evolves:
$\pi_{t+1} = \mathcal{U}(\pi_t, \hat{P}_t)$
The learned model evolves:
$\hat{P}_{t+1} = \Phi(\hat{P}_t, \pi_t)$

Key Insight

This creates a closed-loop system where learning changes the data distribution itself.

This breaks a foundational assumption in classical learning theory:

Standard generalization theory assumes data is independent and identically distributed (i.i.d.).
But recursive systems generate their own data distribution.

As a result, the system is no longer learning from a fixed environment—it is reshaping the environment it learns from.

2. Two Types of Dynamics: Value vs Measure

Recursive systems naturally split into two interacting mathematical objects:

(A) Value Dynamics (Bellman Operator)

This is the standard reinforcement learning view:

T_P^\pi V = r + \gamma \mathbb{E}_P[V]

It describes how expected reward is computed and optimized.

(B) Measure Dynamics (Occupation Evolution)

This describes how state visitation patterns evolve over time:

\mathcal{T}_P^\pi \rho

This governs the distribution of where the system actually goes.

Key Insight

Most classical RL theory focuses only on value dynamics.

But safety, stability, and failure modes depend heavily on measure dynamics—what states are actually visited repeatedly.

3. What Is Epistemic Deviation?

We define a fundamental quantity:

\Delta_t = \|P_{true} - \hat{P}\|_{TV}

This measures how far the learned model is from reality using total variation distance.

Why this matters

Even small errors in the model can produce large shifts in long-run behavior when feedback loops are present.

Stability Redefined

Instead of asking:

“Does the value function stay accurate?”

we ask:

“Does the system remain within the same region of state-space over time?”

This leads to a new definition of stability:

Stability = invariance of the support of the occupation measure

In simple terms:

A system is stable if it keeps visiting the same kinds of states, even under uncertainty.

Part II: Why Failure Is Generic

4. Why Stability Gaps Are Not Rare

A key result of this framework is that instability is not exceptional—it is structurally common.

Even when value-based learning appears stable, the underlying distribution can drift.

Key idea

There exists a large set of environments where:

Expected reward remains stable
But the distribution of visited states shifts into unsafe regions

This is not a special case—it is typical in systems with feedback.

5. The “Demonic Regime” (Phase Transition View)

When small changes in the environment cause large shifts in where the system ends up, we get a phase transition in behavior.

At this point:

The system’s state distribution “jumps”
Mass concentrates in previously rare or unsafe regions
Long-run behavior changes qualitatively

This is what we call:

A structural instability phase transition in the invariant measure

Informally:

The system doesn’t just behave slightly differently—it starts living in a different part of its world.

6. The Wisdom Functional

We define a stability radius:

W(\pi) = \sup \{\epsilon : \rho^\pi_{P'} \subseteq \mathcal{S}_{safe}, \forall P' \in \mathcal{U}_\epsilon\}

Interpretation

This is the largest amount of model error a policy can tolerate while still avoiding unsafe regions.

Key Insight

This is different from standard robustness:

Standard robustness protects expected reward
This protects where the system actually goes

These are not equivalent.

Part III: Why Standard Methods Fail

7. Why This Cannot Be Reduced to Standard RL Objectives

We compare two perspectives:

Value-based robustness (standard RL / CMDPs)
Support-based robustness (this framework)

Main result

These two are structurally different:

You cannot generally convert support constraints into value penalties.

Intuition:

Value functions compress behavior into a single number (expectation)
Support constraints depend on the geometry of the full distribution

Compression loses information that safety depends on.

8. Why CMDPs Are Not Enough

Constrained Markov Decision Processes (CMDPs) optimize:

reward
plus penalty constraints

But this approach assumes:

Safety can be expressed as an expectation.

This framework shows:

Some safety requirements are inherently geometric and cannot be represented as scalar penalties without strong assumptions (like absorbing failure states).

So CMDPs are not wrong—they are:

A restricted projection of a more general measure-space problem.

Part IV: What This Means for Intelligent Systems

9. Stability as a Scaling Problem

We propose a structural law:

R(I) \geq C(I)

Where:

$R(I)$ = representational capacity (what the system can model)
$C(I)$ = causal complexity (what the environment can generate)

Interpretation

A system is only stable if:

Its ability to represent the world grows at least as fast as the complexity of the world it operates in.

Otherwise:

model error accumulates
distribution shifts intensify
the system enters unstable regimes

10. The Central Conclusion

Recursive systems are not unstable because of bad optimization.

They are unstable because:

Standard learning theory tracks the wrong object.

It focuses on:

values (expectations)

But stability depends on:

measures (distribution of experience)

Final Takeaway

The key conceptual shift is:

From “optimizing outcomes”
to “controlling the geometry of experience distributions”

This reframes:

alignment
robustness
safety

as properties of measure invariance under feedback dynamics, rather than properties of reward maximization alone.

Open Questions

Can measure-space control be made computationally tractable?
What approximations preserve support invariance?
Can policies be designed that directly optimize occupation measures?
What are the minimal conditions for avoiding distributional collapse?

Operator-Level Non-Equivalence of Value and Support Dynamics

Full Proof Skeleton (Banach Space Formulation)

0. Preliminaries and Notation

Let:
$\mathcal{S}$ : measurable state space (Polish)
$\mathcal{A}$ : action space (finite or compact)
$\gamma \in (0,1)$ : discount factor

We define two coupled dynamical systems:

Value Space

Let:

\mathcal{B} := (\mathcal{V}, \|\cdot\|_\infty)

where $\mathcal{V}$ is the space of bounded measurable functions $V:\mathcal{S}\to\mathbb{R}$ .

Bellman operator:

T_P^\pi : \mathcal{B} \to \mathcal{B}

Measure Space

Let:

\mathcal{M} := (\mathcal{P}(\mathcal{S}\times\mathcal{A}), \|\cdot\|_{TV})

Occupation measure:

\rho_P^\pi(s,a) = (1-\gamma)\sum_{t=0}^\infty \gamma^t \mathbb{P}_P(s_t=s, a_t=a)

Occupation evolution operator:

\mathcal{T}_P^\pi : \mathcal{M} \to \mathcal{M}

1. Well-Posedness of Both Dynamical Systems

Lemma 1 (Bellman Well-Posedness)

For fixed $P,\pi$ , the operator $T_P^\pi$ is a $\gamma$ -contraction on $(\mathcal{B},\|\cdot\|_\infty)$ .

Result:

\|T_P^\pi V - T_P^\pi W\|_\infty \le \gamma \|V - W\|_\infty

Implication:

Unique fixed point $V_P^\pi$
Standard robust MDP theory applies

Lemma 2 (Occupation Operator Well-Posedness)

For fixed $P,\pi$ , the operator $\mathcal{T}_P^\pi$ induces a well-defined affine map on $\mathcal{M}$ , preserving probability mass.

Key property:

\|\mathcal{T}_P^\pi \rho - \mathcal{T}_P^\pi \rho'\|_{TV} \le \|\rho - \rho'\|_{TV}

Important distinction:

Not a contraction in general
Fixed point corresponds to stationary distribution induced by $\pi$

2. Structural Separation of Spaces

Lemma 3 (Non-Isomorphism of State Spaces)

There exists no norm-preserving isomorphism between:

(\mathcal{V}, \|\cdot\|_\infty) \not\cong (\mathcal{P}(\mathcal{S}\times\mathcal{A}), \|\cdot\|_{TV})

Reason:

$\mathcal{V}$ is linear (Banach space)
$\mathcal{M}$ is convex but not linear
Extreme points correspond to Dirac measures

Consequence:
Any map between them must lose structure (non-bijective or nonlinear).

3. Information Collapse Lemma

Lemma 4 (Expectation Collapse of Measure Structure)

Let $\Lambda:\mathcal{M}\to\mathcal{B}$ be any map of the form:

\Lambda(\rho)(s) = \int f(s,a)\, d\rho(s,a)

Then:

$\Lambda$ is non-injective
Distinct measures $\rho_1 \neq \rho_2$ may satisfy:

\Lambda(\rho_1) = \Lambda(\rho_2)

Key implication:
Support information is lost under any expectation-based embedding.

4. Support Sensitivity Asymmetry

Lemma 5 (Discontinuous Support Functional)

Define support projection:

\mathrm{Supp}(\rho) = \{x : \rho(x) > 0\}

Then the map:

\rho \mapsto \mathrm{Supp}(\rho)

is:

not continuous under total variation topology
sensitive to arbitrarily small perturbations

Key consequence:
Support constraints are topologically non-smooth objects.

5. Failure of Commuting Diagram

We consider the diagram:

\begin{aligned} \mathcal{M} &\xrightarrow{\mathcal{T}_P^\pi} \mathcal{M} \\ \downarrow \Lambda & \quad \downarrow \Lambda \\ \mathcal{B} &\xrightarrow{T_P^\pi} \mathcal{B} \end{aligned}

Lemma 6 (Non-Commutativity of Projection)

For any expectation-type mapping $\Lambda$ ,
there exists $P,\pi,\rho$ such that:

\Lambda(\mathcal{T}_P^\pi \rho) \neq T_P^\pi(\Lambda \rho)

Reason:

$\mathcal{T}_P^\pi$ preserves support evolution
$T_P^\pi$ collapses distributional structure into expectation

Interpretation:

Measure dynamics and value dynamics evolve in incompatible geometries

6. Contraction Preservation Failure

Lemma 7 (Loss of Contraction Structure)

Assume $\Lambda$ exists such that diagram commutes.

Then induced operator:

\tilde{T} := \Lambda \circ \mathcal{T}_P^\pi \circ \Lambda^{-1}

cannot satisfy:

\|\tilde{T}x - \tilde{T}y\|_\infty \le \gamma \|x-y\|_\infty

Reason:

Support perturbations induce discontinuous jumps in pushforward measures
These cannot be bounded in sup-norm after projection

Key technical point:
Contraction in $\mathcal{V}$ is incompatible with TV-induced discontinuities in $\mathcal{M}$ .

7. Main Theorem (Non-Equivalence)

Theorem (Operator-Level Non-Equivalence)

Let:

$\mathcal{B} = (\mathcal{V}, \|\cdot\|_\infty)$
$\mathcal{M} = (\mathcal{P}(\mathcal{S}\times\mathcal{A}), \|\cdot\|_{TV})$

Then there exists no continuous mapping

\Lambda: \mathcal{M} \to \mathcal{B}

such that all of the following hold simultaneously:

Constraint preservation
$\rho^\pi \in \mathcal{S}_{safe} \Rightarrow \Lambda(\rho^\pi) \in \mathcal{K}$
Operator commutativity
$\Lambda \circ \mathcal{T}_P^\pi = T_P^\pi \circ \Lambda$
Contraction preservation
$T_P^\pi \text{ retains modulus } \gamma$

8. Proof (Contradiction Closure)

Assume such a $\Lambda$ exists.

Step 1 (Information Collapse)

From Lemma 4:

\Lambda \text{ is non-injective}

Step 2 (Support Sensitivity)

From Lemma 5:

\mathcal{T}_P^\pi \text{ induces discontinuous support changes}

Step 3 (Commutativity requirement)

Assumption implies:

\Lambda(\mathcal{T}_P^\pi \rho) = T_P^\pi(\Lambda \rho)

forces measure dynamics to be representable in expectation space.

Step 4 (Contradiction)

But:

support evolution is discontinuous in $\mathcal{M}$
value evolution is Lipschitz in $\mathcal{B}$

Therefore:

\text{No continuous }\Lambda \text{ can preserve both geometries}

Thus assumption fails.

\boxed{\text{Q.E.D.}}

$\boxed{ \text{No continuous mapping } \Lambda \text{ exists satisfying all three conditions.} }$

Thursday, June 4, 2026