This preprint introduces the Trans–Integral (TI) framework, a reformulation of classical probability theory that removes the assumption of ownership—the requirement that all observables be defined on a single global probability space. Using a minimal finite example, the paper demonstrates that three locally classical and perfectly consistent probability models cannot be jointly embedded under ownership. The contradiction is expressed entirely at the level of expectations, revealing its structural equivalence to the Bell–CHSH inconsistency. To clarify the mechanism, the paper constructs context-dependent Markov semigroups whose invariant measures encode the relations A = B, B = C, and A ≠ C. The global obstruction arises only when these context-specific attractors are forcibly unified into a single stochastic world. The Trans–Integral framework resolves this obstruction by replacing each observable with a context-indexed family and defining expectations via a two-level integration over outcomes and contexts. The result preserves all local classical structures while eliminating the global contradiction. The work suggests a broader course correction for probability theory, with implications for stochastic processes, Markov semigroups, and the Kolmogorov extension theorem.
This preprint introduces the Trans–Integral (TI) framework, a reformulation of classical probability theory that removes the assumption of ownership—the requirement that all observables be defined on a single global probability space. Using a minimal finite example, the paper demonstrates that three locally classical and perfectly consistent probability models cannot be jointly embedded under ownership. The contradiction is expressed entirely at the level of expectations, revealing its structural equivalence to the Bell–CHSH inconsistency. To clarify the mechanism, the paper constructs context-dependent Markov semigroups whose invariant measures encode the relations A = B, B = C, and A ≠ C. The global obstruction arises only when these context-specific attractors are forcibly unified into a single stochastic world. The Trans–Integral framework resolves this obstruction by replacing each observable with a context-indexed family and defining expectations via a two-level integration over outcomes and contexts. The result preserves all local classical structures while eliminating the global contradiction. The work suggests a broader course correction for probability theory, with implications for stochastic processes, Markov semigroups, and the Kolmogorov extension theorem.
Classical probability theory and standard derivations of the CHSH inequality rely on the assumption of a single global probability space—an implicit form of ownership of all possible values. Under this assumption, Bell’s bound |S| ≤ 2 follows pointwise. The Trans–Integral (TI) framework removes this assumption of ownership. Each measurement context θ generates its own measurable space (Ω_θ, F_θ, P_θ), and expectation is computed as a double integral over context-indexed domains: E_{\mathrm{TI}}[X] = \int_{\theta \in C} \left( \int_{\Omega_\theta} X_\theta \, dP_\theta \right) \, \Pi(d\theta). Within the Trans–Integral program, space itself is not primitive but appears as a trace of generative time. TI01 derives the Tsirelson bound 2√2 as a necessary constraint on relations between such already generated spaces, once global ownership is abandoned. The bound is thus interpreted as a relational–spatial limit rather than a purely probabilistic artifact. Important note. The observer-dependent, generative foundation of Trans–Integral integration and the emergence of Assembler Time are developed in TI03 – Trans–Integral: Observer–Dependent Measure and the Emergence of Time https://doi.org/10.5281/zenodo.17671551 Important note (update). The same bound 2√2 is re-derived dynamically as a curvature limit of generative time (time fusion / Assembler Time) in TI07 – Generative Time and the Reconstruction of the Path Integral https://doi.org/10.5281/zenodo.17844994
Suppose that a family of commutative measurement contexts is given, together with the corresponding classical probability laws. We study the problem under what conditions these contextwise descriptions can be combined into a single classical representation. We distinguish two ways in which this can be done. In {\em global integration}, the prescribed observable algebras are required to fit into a single commutative context and the corresponding probability laws to arise as marginals of one joint law. In {\em protocolwise integration}, by contrast, the context label is retained as part of the description, and the contextwise laws appear as conditional distributions within a single model of the measurement protocol. We show that global integration involves two distinct notions of compatibility: operator-algebraic compatibility of the underlying observables and probabilistic compatibility of their classicalised laws. The former implies the latter, but the converse fails; under the topological hypotheses stated below, probabilistic compatibility is characterised by a Kellerer-type marginal criterion. Protocolwise integration, on the other hand, always exists and is represented operator-algebraically by a direct-sum construction. When global integration exists, the two constructions arise as reductions of a common commutative refinement. We illustrate the distinction in the Bell--CHSH experiment, where Fine's theorem relates global probabilistic integration to Bell locality and the CHSH inequalities. This shows that the obstruction revealed by Bell--CHSH concerns not classical representation as such, but classical representation preserving specified cross-context identifications.
This preprint introduces TI-M0, a minimal algebraic model capturing the core transformation structures appearing in the TI03–TI05 framework of observer-dependent representations. A finite family of contexts is equipped with transformation maps forming a small category. Rigid designation appears as a fixed-point condition of commutative diagrams. Doppler-type transformations are realized as one-dimensional linear deformations, while a two-component lifting provides the smallest extension compatible with observer-dependent time scaling. The model shows that rigidity, Doppler deformations, and two-component observer transformations form three levels of a single algebraic hierarchy. TI-M0 functions as a transparent finite instance of key transformation principles and serves as a conceptual bridge toward more elaborate observer-indexed systems. This short paper is mathematically self-contained and does not rely on any unpublished material. Important note. The structural and measure-theoretic foundations of observer-indexed measurable spaces used here are developed in:Trans–Integral: Observer–Dependent Measure and the Emergence of Time (TI03)https://doi.org/10.5281/zenodo.17671551
This paper develops a measure-theoretic unification of statistical, quantum, and cosmological Doppler phenomena. Although these effects arise in distinct physical contexts—thermal velocity distributions, recoil-induced quantum spectral shifts, and cosmological expansion—they share a common mathematical structure: each is generated by an observer-dependent measurable transformation acting on a spectral measure. The central contribution of this work is the introduction of the Trans-Integral (TI) framework, a minimal generalization of Lebesgue integration that internalizes observer parameters into the measure space itself. TI treats Doppler transformations as families of measurable maps D_\theta : X \to X, \qquad \nu_\theta = (D_\theta)_\# \nu_S, \mathcal{T}_{D,\mu}[\nu_S] = \int_\Theta (D_\theta)_\# \nu_S \, d\mu(\theta). This formulation reveals a structural unity hidden in traditional treatments of Doppler effects. Statistical Doppler broadening, quantum spectral deformation, and cosmological redshift—typically presented by unrelated formalisms—are shown to be instances of the same underlying pushforward geometry. The paper also outlines mathematical and physical open problems suggested by the TI viewpoint, including: geometry of parameterized measure spaces, stability and singularity formation under TI operators, TI flows and semigroup structures, categorical interpretations (Kan extensions and enriched colimits), and potential extensions to time-generation models. Appendices provide the foundational measure-theoretic background, diagrammatic intuition, and a structural analysis of observer-dependent qualitative emergence. This work does not propose new physical laws. Instead, it identifies a measure-theoretic skeleton common to diverse Doppler phenomena and places it within a coherent mathematical framework that may aid future developments in spectral theory, quantum foundations, and cosmological analysis. Important note. The present paper develops an application of the observer-dependent measure framework introduced in [Trans–Integral: Observer–Dependent Measure and the Emergence of Time (TI03)](https://doi.org/10.5281/zenodo.17671551), which provides the mathematical backbone of the Trans-Integral (TI) program. For a minimal algebraic core that isolates the transformation structure used here in a finite and self-contained model, see [TI-M0: A Minimal Transformational Model of Observer-Dependent Representations](https://doi.org/10.5281/zenodo.17791192). This version contains the PDF file only, replacing the previous ZIP+PDF combination for clarity.
Quantum cognition often explains order effects, contextuality, and violations of the law of total probability by replacing classical probability with quantum probability on a fixed event structure. This paper proposes a different interpretation: quantum probability is the fixed-spacetime projection of contextual spacetime formation under finite-state requirements.
The framework begins not with time, space, objects, or probabilities, but with requirements such as finite representational capacity, single-state semantic stability, context-sensitive intervention, avoidance of explicit context labels, coherent world-formation, and intersubjective transformability. When these requirements cannot be realized within a single global Boolean event structure, the mismatch appears, under fixed-spacetime projection, as noncommutativity, interference, and quantum-like probability.
Building on prior single-state approaches to contextuality, we reinterpret classical contextual bookkeeping cost as the fixed-spacetime shadow of contextual spacetime formation. Auxiliary memory or context labels in a classical representation correspond, in this account, to holonomy-like mismatch among locally Boolean logic-worlds. The interference term is the cross term generated when locally classical realization contributions are nontrivially glued and projected back into a fixed classical spacetime form.
The result is a transcendental-operational realist account: objecthood, eventhood, probability, and spacetime are treated as forms of realization under requirements, while objectivity is defined by invariants preserved across observer- and history-dependent spacetime formations.
A set of quantum data can look classical in every local test and still fail to admit a single classical explanation of the whole composite system. We formulate this failure as global contextuality. Here global means global in the physical sense of the whole multipartite system, not the local/global terminology of sheaf theory. Each party's local statistics are noncontextual and each measured multipartite context admits a generalized local hidden-variable description, but the GLHV block descriptions cannot be promoted to a single noncontextual hidden-variable model for the whole system. Three bipartite constructions exhibit this separation. A polarization-path construction gives a direct global obstruction. A qubit-qutrit KCBS construction gives an algebraic scenario-level example, with explicit formulas for the unconditional KCBS operator, the correlation-polytope constraints, and the postselected violation. A flagged qutrit Werner-local state gives a state-level example: the state is entangled and local for all projective measurements, its local qutrit marginals do not violate KCBS, yet postselection rules out a single GNCHV model. We also spell out the classical composition lemma: classical conditional hidden variables can be absorbed into a larger hidden variable, whereas quantum contextual data need not allow such a factorization. Within the general Bell-type framework considered here, with arbitrary parties and arbitrary local compatible contexts, but no cross-party joint measurements, the absence of local contextuality and GLHV-type generalized Bell nonlocality does not imply the existence of a global noncontextual hidden-variable model. Global contextuality is thus a compositional obstruction to classical explanation.