Ontological Synthesis: Modeling Quantum Foundations

• Ontological synthesis is the systematic process of constructing, integrating, and optimizing realism-based models that merge abstract structures with empirical data.
• It employs techniques like nonnegative matrix factorization and convex decompositions to transition from indeterministic to deterministic, context-sensitive representations.
• The approach enhances classical simulation in quantum physics by minimizing ontic state spaces and clarifying the role of measurement contextuality in hidden variable frameworks.

Ontological synthesis is the process of systematically constructing, integrating, and optimizing ontological models that provide explicit, realism-based representations for complex domains—most notably in foundational physics, information systems, formal semantics, epistemology, and the engineering of conceptual frameworks. Central to ontological synthesis is the unification of abstract structures (types, states, events, relations) with empirical data and the identification of the constraints and affordances that such representations introduce. Contemporary research delineates several key dimensions where ontological synthesis provides both conceptual clarification and practical methodologies for analyzing and simulating complex systems.

1. Ontological Models and Factorizations of Probabilistic Data

In the context of quantum theory and related probabilistic frameworks, ontological synthesis is exemplified by the construction of ontological models that factor empirical data tables into components reflecting a putative underlying reality (“ontic” states). Each preparation procedure is associated with a probability distribution over a finite set of ontic states, while measurements are modeled by sets of positive indicator functions. Mathematically, this is formalized as a data table factorization $D = M P$, where $P$ encodes preparation probabilities over ontic states and $M$ encodes measurement statistics.

Key properties of this framework include:

• Column-stochasticity for both preparation distributions and measurement indicator matrices, ensuring that all modeled probabilities are positive and normalized.
• Contextuality, which is generically enforced by the constraints of the factorization and appears, for example, in scenarios covered by the Kochen–Specker theorem. Even physically identical projectors must, in general, be represented by different indicator functions depending on their measurement context in a deterministic ontological model.
• Conversion from indeterministic to deterministic models via expansion of the ontic state space, typically leading to a large (potentially exponential) number of ontic states but allowing representation of all data with binary (0/1) indicator functions.
• Ontological compression, or the search for minimal representations—that is, the minimal number $\Omega$ of ontic states consistent with the empirical data and model constraints. This is closely related to nonnegative factorization rank (cp–rank) and is critical for understanding the limits of classical simulation.

Typical key expressions:

• $D^{(x)} = M^{(x)} P$ for each measurement $x$
• $$P_{(j_1,\ldots,j_m), k} = \prod_{x=1}^m D_{j_x, k}^{(x)}$$ (deterministic construction)
• $M_{i,(j_1,\ldots,j_m)}^{(x)} = \delta_{i,j_x}$

This synthesis provides an explicit, positive-probability and contextual alternative to quasi-probability and $r$–$p$ formalism representations, making explicit the structure of hidden variable models (0709.1149).

2. Measurement Contextuality and Realism Constraints

A defining insight of ontological synthesis in the quantum setting is the non-trivial manifestation of contextuality at the level of representation. Even when two measurement events correspond to the same projector in the Hilbert space, the stricter constraints of the ontological model (notably, column-stochasticity and assignment of definite outcomes per ontic state) require that the corresponding rows of the measurement matrix $M$ may be distinct. Concrete cases, such as those constructed with the Kernaghan Kochen–Specker set, demonstrate that identical projectors arising in different measurement contexts must be assigned differing binary values. This upholds the necessity of measurement-based contextuality for deterministic ontological models and underscores the departure from non-contextual classical representations.

3. Transforming Indeterministic to Deterministic Models

One of the principal systematic procedures in ontological synthesis is the transition from indeterministic to deterministic models. This is accomplished through:

• Trivial indeterministic model: $M$ identified directly with the data table; $P$ is the identity (each preparation mapped to its own ontic state), yielding no internal structure but retaining indeterminism in the measurement.
• Deterministic expansion: Ontic state space is expanded to tuples of measurement outcomes, with indicator functions set by Kronecker deltas. All randomness is transferred to the probabilistic choice of initial ontic state, with measurement functions returning deterministic outcomes.
• Spreading procedures: Indeterminism is simulated via “spreading” weight over additional ontic states, as in Bell-type transformations and rational approximations.

This process is not mere mathematical reshuffling, but structurally exposes and enforces hidden contextuality and reflects the depth to which underlying classical structures can be forced to match quantum statistics, provided one allows sufficient ontic state complexity.

4. Model Optimization and Minimal State Spaces

Ontological synthesis encompasses not only the existence of factorizations but their optimization, particularly minimization of the ontic state space. Strategies include:

• Recursive “peeling” of minimal probability support, decomposing probability vectors as convex sums of binary vectors to strip down the ontic carrier set.
• Block permutations and probability “shifting” in tensor-product–structured models, merging equivalent columns of measurement matrices wherever supports are disjoint.
• Achieving reductions from naive upper bounds (e.g., $s$ or $d^m$ states) to constructions at or near $s(dm - 1)$ ontic states in concrete settings.

The practical significance lies in the classical simulability and epistemic/ontic interpretations allowed by more compressed models: minimal overlap in supports points to $\psi$-ontic behavior, while substantial overlap connects to $\psi$-epistemic interpretations.

5. Mathematical Characterization and Formalism

The mathematical structure of ontological synthesis is captured by:

• Nonnegative matrix factorization: $D = M P$ with $M$ and $P$ column-stochastic.
• Kronecker delta assignment: $\delta_{i, j_x}$ for deterministic indicator functions.
• Convex decompositions: $\sum_{i=1}^{\Omega} q_i^{(k)} \lambda_i = p^{(k)}$ expressing preparations as mixtures over binary ontic state labels.
• Conservation constraints: Each column of $M$ sums to 1, ensuring deterministic measurement yields.
• Compression operations: Equivalence classes of columns in $M$ merged if their supports in $P$ do not overlap.

This formalism facilitates quantitative analysis of contextuality, simulation complexity, and resource requirements for ontological models.

6. Implications for Quantum Foundations and Computational Simulation

The ontological synthesis framework provides:

• A direct bridge between empirical data (measurement statistics), realist hidden variable models, and explicit constraints derived from foundational results (e.g., Kochen–Specker, Bell-type no-go theorems).
• Explicit characterization of resource requirements (number of ontic states), critical for both foundational arguments and for classical simulation or emulation of quantum behavior (e.g., in variational algorithms, quantum-inspired computation).
• Clarification of the distinction between non-contextual and contextual representations, making manifest the structural reasons for quantum–classical simulation gaps.
• Foundational rigor: By insisting on explicit positivity and contextuality, ontological synthesis sharpens the debate surrounding the epistemic vs. ontic status of the quantum state, and elucidates the limits of classical “explanation” of quantum statistics.

In summary, ontological synthesis—through systematic construction, transformation, and optimization of ontological models—integrates empirical data with underlying structural postulates, sharply distinguishes classical and quantum representational paradigms, and provides a mathematically explicit framework for both foundational inquiry and pragmatic simulation strategies in quantum theory (0709.1149).