Platonic Projection Structures: Operator-Induced Observability in Representation Learning

Abstract We introduce Platonic Projection Structures (PPS), an operator-theoretic framework for analyzing observability in representation learning systems. Rather than treating observable outputs as direct reflections of latent representations, PPS models observation as a geometry induced by a self-adjoint positive semidefinite operator acting on a latent Hilbert space. Within PPS, a system is represented as a triple (H, Π, O), where Hdenotes a latent representation space, Π ⪰0 is an observation operator, and O(v) = ⟨v, Πv⟩defines an induced observable quantity. The framework characterizes observability through the quotient geometry H/ ker(Π), which represents equivalence classes of latent states that are indistinguishable under observation. From this perspective, observable behavior is governed not by latent representations themselves, but by the geometry induced through the observation operator. We show that both quantum measurement and deep learning inference can be formulated within this common operator-theoretic structure while differing in the algebraic properties of their observation operators. Within this perspective, quantum measurement serves primarily as a mathematically canonical example of projection-mediated observability. The correspondence developed in PPS is therefore structural rather than physical. Within the same framework, representation transfer and knowledge distillation can be interpreted as approximate preservation of observable geometry through the intertwining condition ΦΠT ≈ΠSΦ. PPS further reveals a structural limitation of output-based interpretability: latent components contained in ker(Π) are fundamentally inaccessible from observables generated through the induced observation process. Accordingly, attribution and explanation methods inherit intrinsic constraints imposed by the observation geometry itself. We provide controlled empirical validations demonstrating kernel-invariant observability, projection-induced attribution gaps, and rank-controlled observable geometry in latent representation spaces. Overall, PPS provides a mathematically explicit operator-theoretic perspective for analyzing observability, interpretability, and representation transfer in projection-mediated learning systems