The View from Within: What Can Embedded Observers (Not) Learn?

Technical Summary: "The View from Within: What Can Embedded Observers (Not) Learn?"

Problem Statement
Classical physics traditionally adopts a third-person perspective, assuming an observer-independent reality where properties are precisely measurable and disturbances are correctable. However, observers are themselves physical systems embedded within the world they observe. This paper investigates the tension between the third-person description of physical laws and the first-person perspective of an embedded observer. Specifically, it asks: What limitations on learning (epistemic horizons) arise for an observer who is a physical system interacting with another system, even within a fully deterministic classical framework? The authors aim to determine whether fundamental limits to knowledge, analogous to quantum uncertainty, can emerge from the physical constraints of the observation process itself, rather than from postulated epistemic restrictions.

Methodology
The authors employ a "nomic toy theory," a deterministic classical model of particles with position and momentum coordinates defined on a symplectic vector space. The framework formalizes four key aspects of learning (Figure 1):

1. Fundamental Physical Theory: The kinematics and dynamics of classical particles (symplectic maps and discarding operations).
2. Manifest Variable: The physical variable of the subject (observer) that constitutes its empirical record. The authors consider three types:
   • Trivial: The record carries no information.
   • Poisson: The record distinguishes only a coarse-graining of the subject's state (e.g., position projection), representing partial self-knowledge.
   • Complete: The record is the subject's full ontic state, representing complete self-knowledge.
3. Ready States: The initial informational resources available to the subject (the "ready system" $R$). These are defined by the variability of the initial state:
   • Trivial: No constraint on the initial state ($V=R$).
   • Poisson: The initial state is constrained to a coisotropic subspace (partial information).
   • Complete: The initial state is fixed ($V=\{0\}$).
4. Type of Learning: The temporal direction and repeatability of the interaction:
   • Retrodiction: Learning about the object's past state.
   • Prediction: Learning about the object's future state.
   • Repeatable Measurements: Interactions where the retrodicted information persists in the object, allowing simultaneous learning of past and future.

Learning is modeled as a physical interaction (measurement) $M: B \oplus R \to B \oplus S$, where $B$ is the object, $R$ is the ready system, and $S$ is the subject. The authors analyze the "retrodictively inaccessible subspace" ($I_{ret}$) and "predictively inaccessible subspace" ($I_{pre}$) to determine which variables of the object can be inferred from the subject's record.

Key Contributions and Results
The paper systematically maps the epistemic horizons for all combinations of manifest variables, ready states, and learning types (summarized in Tables 2–4). The primary findings are:

1. Emergence of Epistemic Horizons in Determinism: Even in a deterministic classical theory, embedded observers face fundamental limitations on what they can learn. These horizons are not postulated but derived from the physical constraints of the interaction and the observer's resources.
2. The Knowledge-Balance Principle: For subjects with Poisson manifest variables and Poisson ready states, the paper recovers Spekkens' knowledge-balance principle: a subject can learn at most half of the object's degrees of freedom. Specifically, only Poisson variables (those with coisotropic kernels) can be learned. This recovers the epistemic features of Spekkens' toy theory and, by extension, the stabilizer subtheory of quantum mechanics for odd-dimensional systems.
3. The Necessity of Ready State Constraints: The authors demonstrate that partial self-knowledge (Poisson manifest variables) alone is insufficient to generate an epistemic horizon.
   • If a subject has a Poisson manifest variable but access to complete ready states, they can retrodict the full ontic state of any object (no epistemic horizon).
   • However, if the subject has a complete manifest variable (full self-knowledge) but only Poisson ready states, a non-trivial epistemic horizon persists if and only if the measurement is required to be repeatable. In this case, only Poisson variables can be repeatably retrodicted.
4. Asymmetry Between Retrodiction and Prediction:
   • Retrodiction is generally less constrained than prediction. For example, a subject with a complete manifest variable and Poisson ready states can retrodict any variable (Corollary 3.4) but faces a knowledge-balance horizon for repeatable retrodiction (Theorem 3.8).
   • Prediction is sensitive to the disturbance of the measurement. With trivial ready states, prediction is impossible (total epistemic horizon) because surjective maps cannot create correlations from unconstrained inputs. With Poisson ready states, prediction is limited to Poisson variables (Theorem 3.11).
5. Repeatability as a Strong Constraint: Repeatable measurements impose stricter limits than either retrodiction or prediction alone. They require the retrodicted information to be preserved in the object, effectively ruling out interactions that merely swap states or disturb the object to encode information. The paper proves that for repeatable measurements, the epistemic horizon is determined entirely by the available ready states, not just the manifest variable.

Significance and Claims
The paper claims to provide a language and formal tools to study how the first-person perspective differs from the third-person perspective beyond simple toy models. Its significance lies in:

• Deriving Uncertainty from Physicalism: It shows that uncertainty-like limitations (epistemic horizons) can arise naturally from the physical constitution of the observer and the nature of physical interactions, without invoking the knowledge-balance principle as a fundamental postulate.
• Clarifying the Role of Resources: It disentangles the roles of self-knowledge (manifest variables) and initial information (ready states). The authors argue that the epistemic horizon derived in their previous work [26] was not solely due to partial self-knowledge but relied critically on the assumption of imperfect initial information (Poisson ready states).
• Connection to Quantum Theory: The results suggest that quantum-like features (specifically the stabilizer subtheory) can emerge in a deterministic ontology when observers are restricted to Poisson ready states. This offers a potential pathway to explaining inherent uncertainty in quantum theory from the standpoint of embedded observers.
• Perspectivalism: The framework supports "moderate physical perspectivalism," where a perspective-neutral description exists, but the system's appearance (epistemic state) depends on the subject's specific physical resources and interactions.

The authors remain modest, noting that extending these results to full quantum theory is non-trivial due to interpretational issues regarding ontic states and the separability of subjects and objects. They do not claim to derive full quantum mechanics but rather to identify the structural conditions under which quantum-like epistemic restrictions emerge in a classical setting.