We propose a discrete ontological reconstruction of quantum mechanics based on a graph-theoretic model with deterministic cyclic dynamics. The model is founded on the hypothesis that quantum mechanics is an effective description of reality from the perspective of an internal observer, that is, an observer who is part of the finite system (world) being observed. A central element of the model is the principle of subjective incompleteness: in general, the set of states of the system (ontological states) exceeds the set of observable states and cannot be fully distinguished by the internal observer. This leads to taking the quotient of the set of ontological states with respect to indistinguishability and naturally induces a probabilistic structure in which probabilities are determined by the cardinalities of the corresponding equivalence classes. It is shown that the cyclic structure of the dynamics, combined with this quotient construction, leads to the emergence of complex amplitudes and the Born rule. Thus, the formalism of quantum mechanics arises as an effective description of a quotient-induced deterministic dynamics. It is further shown that the geometry of the edge space of a complete graph realizes a discrete phase space with an action of the Heisenberg–Weyl group, and that in the continuum limit the model leads to the Klein–Gordon equation. Within this approach, the structure of quantum mechanics appears as a representational level of description of the quotient cyclic dynamics.
We propose a discrete ontological reconstruction of quantum mechanics based on a graph-theoretic model with deterministic cyclic dynamics. The model is founded on the hypothesis that quantum mechanics is an effective description of reality from the perspective of an internal observer, that is, an observer who is part of the finite system (world) being observed. A central element of the model is the principle of subjective incompleteness: in general, the set of states of the system (ontological states) exceeds the set of observable states and cannot be fully distinguished by the internal observer. This leads to taking the quotient of the set of ontological states with respect to indistinguishability and naturally induces a probabilistic structure in which probabilities are determined by the cardinalities of the corresponding equivalence classes. It is shown that the cyclic structure of the dynamics, combined with this quotient construction, leads to the emergence of complex amplitudes and the Born rule. Thus, the formalism of quantum mechanics arises as an effective description of a quotient-induced deterministic dynamics. It is further shown that the geometry of the edge space of a complete graph realizes a discrete phase space with an action of the Heisenberg–Weyl group, and that in the continuum limit the model leads to the Klein–Gordon equation. Within this approach, the structure of quantum mechanics appears as a representational level of description of the quotient cyclic dynamics.
Abstract We propose that quantum mechanics emerges as the effective description of a deterministic physical reality from the perspective of an internal observer that constitutes a subsystem of a finite closed world. Because such an observer cannot access the complete ontological state of the world to which it belongs, its description is intrinsically incomplete. We refer to this fundamental limitation as subjective incompleteness . It naturally induces a factorization of the ontological state space with respect to gauge degrees of freedom that are indistinguishable to the observer. Within the proposed framework, the projective geometry of quantum state space is understood as a direct consequence of subjective incompleteness. This provides a common geometric origin for characteristic quantum phenomena, including nonlocality, contextuality, and intrinsic randomness.
We propose a discrete, node-based framework in which quantum mechanics emerges as a continuum approximation of underlying update dynamics. Fundamental entities are nodes related by causal update relations, with no probabilistic postulate assumed at the microscopic level. Transition amplitudes are assigned as minimal two-dimensional internal degrees of freedom, while observations correspond to coarse-grained node densities. We show that the Born rule is not a fundamental axiom but an inevitable quadratic statistical form arising from phase elimination, σ-additivity, and non-contextuality in the continuum limit. The Schrödinger equation appears as an effective generator of near-identity node updates, and probability measures are uniquely fixed by information‑geometric consistency. This framework reinterprets quantum theory as a shadow of discrete node dynamics rather than a primary description of reality.
Quantum mechanics is widely regarded as a fundamental theory of physical reality. In this work, we demonstrate that this status is not supported at the structural level. The quantum formalism is shown to be neither closed nor generative, relying on externally imposed elements such as probabilistic rules and measurement prescriptions. We introduce a closed variational framework based on a quartic functional, from which admissible configurations emerge through explicit variation. The second variation defines a Hessian operator whose spectral structure exhibits intrinsic discreteness and degeneracy, without the introduction of quantization postulates. We prove that apparent probabilistic behavior arises from structural degeneracy and indistinguishability of admissible configurations, rather than from intrinsic indeterminism. The role of wavefunction collapse is replaced by a global deterministic selection functional applied to the set of stationary solutions. Within this framework, quantum states, probabilities, and observables are reinterpreted as effective representations of an underlying deterministic structure. Consequently, quantum mechanics does not constitute a fundamental description of physical reality, but a statistical projection of unresolved structural configurations.
We propose an interpretation of quantum mechanics in which the wavefunction represents unresolved potential relative to an observer, measurement is deterministic resonance stabilisation, and Born-rule probability quantifies the observer’s incomplete access to the full entropic and phase-geometric state of the observer–system interaction. An observer is defined operationally as any bounded system capable of converting external entropy into internal coherence. On this view,superposition is not a set of co-existing realities but an unresolved field of possible coherence resolutions; collapse is not a stochastic discontinuity but a deterministic phase-lock between observer and system; and the classical world is the high-coherence limit in which environmental observation is dense enough to continuously stabilise macroscopic facts.The interpretation preserves the full predictive machinery of quantum mechanics while providing a unified account of the measurement problem, the Born rule, decoherence, entanglement, delayed choice, Wigner’s friend, and the quantum-to-classical transition. We formalise the framework through a resonance-selection rule, an observer equation of collapse, and a prime-modal state-space structure, and identify five classes of empirically accessible predictions.
This paper derives the complete mathematical structure of quantum mechanics—complex Hilbert spaces, unitary evolution, and Born-rule probabilities—from five information-theoretic axioms, without assuming quantum mechanics at the outset. Starting from an ontology of microhistories (fully specified microscopic trajectories), we define branches as equivalence classes of operationally indistinguishable microhistories. We then show that five physically motivated constraints on how information is stored, mixed, and transformed uniquely force the effective state space to be a complex projective space with unitary dynamics.
This work (O3 — Part II) completes the derivation of the quantum mechanical formalism within the Emergent Pre-Quantizable (EPPQ) framework by establishing the necessary emergence of the complex Hilbert space structure from purely relational dynamics. Building upon O3 — Part I, where the Schrödinger equation and the Born rule were derived conditionally from coarse-grained Markov dynamics on metastable basins, this paper addresses the remaining foundational gap: the origin of complex amplitudes and quantum phases. We demonstrate that a discrete U(1) gauge structure naturally arises from the contextual layer of the EPPQ vacuum, formalized as a principal bundle over the vacuum graph. The associated holonomies (Wilson lines) induce geometric phases along microscopic trajectories. By analyzing the interference of these trajectories, we prove that: • Transition amplitudes must take values in the complex field ℂ • The Born rule P = |ψ|² is the unique probability measure compatible with superposition and gauge invariance • The effective Hamiltonian emerges from the skew-symmetric component of the coarse-grained Markov generator and is directly determined by contextual holonomies Furthermore, we show that the same structure yields, in a specific spectral sector, the Yukawa matrices of the Standard Model, including nontrivial complex phases responsible for CP violation. The results establish that the complex structure of quantum mechanics is not an independent postulate, but a necessary consequence of relational holonomy and trajectory interference. This provides the missing link between the geometric constructions of O1/O4 and the emergent quantum dynamics of O3. This work is part of a broader program (EPPQ) aimed at deriving spacetime, quantum theory, and particle physics from a minimal relational ontology.