# Platonic Projection Structures: Operator-Induced Observability in Representation Learning

arXiv: https://arxiv.org/abs/2607.05175 · HTML: https://mlsanity.com/paper/2607.05175/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 31 references that are in the map.
- First posted on arXiv 6 Jul 2026.

## Builds on (most cited first, 16 of 16)

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## Abstract (arXiv)

We characterize observability in representation learning through Platonic Projection Structures (PPS), an operator-theoretic framework for analyzing representation accessibility under partial observation. Rather than treating observable outputs as direct reflections of latent representations, PPS models observation through a self-adjoint positive semidefinite operator acting on a latent representation space. A system is represented as a triple (H, Pi, O), where H is a latent representation space, Pi succeq 0 is an observation operator, and O(v)=langle v,Pi vrangle defines an induced scalar observable. Observability is characterized by the quotient geometry H/ker(Pi), representing equivalence classes of latent states indistinguishable under observation. We show that quantum measurement and representation inference under linear observation models share this operator-theoretic structure while differing in the algebraic properties of their observation operators; the correspondence is structural rather than physical. Representation transfer and knowledge distillation can likewise be interpreted as approximate preservation of observable geometry through Phi PiT ≈ PiS Phi. PPS also reveals a structural limitation of output-based interpretability: latent components in ker(Pi) are inaccessible from induced observables, imposing intrinsic constraints on attribution and explanation methods. Controlled empirical validations demonstrate kernel-invariant observability, projection-induced attribution gaps, and rank-controlled observable geometry in latent representation spaces. PPS thus provides an explicit characterization of observability through operator-induced quotient geometry and a unified perspective on representation accessibility, interpretability, and projection-mediated inference.