The Structure of Observables

Yoshio Ohnuki, S. Kamefuchi

Koji Okino

This work introduces **Structural Differentiation Observation (SDO) v1.0**, a minimal and falsifiable framework in which observation is redefined as a structural process that produces a realized state through interaction. In SDO, reality is not treated as a pre-existing fixed entity. Instead, multiple differentiated states exist as possibilities, and interaction drives the fixation of one realized configuration. The framework is built upon a single structural transition: S_possible → (interaction) → S_fixed A key contribution of SDO is the reinterpretation of observation as a dynamical and generative process rather than passive measurement. The theory introduces: - A structural definition of interaction (I) as coupling between differentiated states- A fixation dynamics law: R_fix = 1 − exp(−αI)- An emergent time relation: t_obs ∝ Σ F_i Importantly, the interaction term I is connected to measurable quantities such as correlation strength, information flow, or interaction energy, providing a direct bridge between abstract structure and experimental observables. The framework is supported by a complete figure set illustrating: - State fixation (branching → convergence)- Network propagation and fixation cascade- Emergence of time through ordered fixation- Structural production of realized states- Interaction-dependent fixation dynamics SDO suggests that observation does not merely reveal reality, but structurally produces it through fixation. This provides a clear, decisive, and directly testable pathway toward experimental validation across physical and informational systems. This work is part of the broader Structural Differentiation Framework (SDC/SDI/SDT/SDO),aiming toward a unified description of structure, information, time, and observation.

Atsushi Okamoto

The Intrinsic Observability Framework (IOF) distinguishes structural decomposition from operational decomposition as complementary levels of organization. Structural decomposition establishes the observable and hidden domains as the architecture of observability, whereas operational decomposition assigns the latent structure to these domains under a given observational interaction. The present work extends this framework by introducing two complementary forms of totality: structural totality and operational totality. Structural totality represents the completeness of the structural architecture established through structural decomposition, consisting of the observable and hidden domains. Operational totality, in contrast, represents the completeness of the latent structure as operationally assigned across these domains through operational decomposition. The distinction therefore concerns different objects of completeness: structural totality refers to the architecture within which assignment is possible, whereas operational totality refers to the content completely assigned within that architecture. Operational totality is consequently neither an incomplete portion nor a degraded representation of structural totality. Different observational interactions may establish different operational totalities within a common structural totality. Such observer dependence does not imply incompleteness or mutually isolated operational worlds. Each operational totality is complete with respect to its own operational assignment, while the admissible relations among independently established operational totalities are constrained by the Global Consistency Rule (GCR). The resulting framework therefore distinguishes structural completeness, operational completeness, and global consistency, providing an organizational interpretation of how distinct observer-dependent totalities can coexist while collectively supporting a shared and coherent operational reality without modifying the mathematical formalism or empirical predictions of quantum mechanics.

Kazuo Ishii, Bishnu Prasad Gautam, Jieling Wu +1 more

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