A pregeometric framework in which correlations and global compatibility constraints are taken as primitive, while spacetime geometry, matter content, and quantum behaviour arise only as emergent, projected observables. The core construction specifies a constraint manifold equipped with a correlation kernel and a family of projection maps into an effective manifold, from which metric structure, causal cones, and relativistic propagation can be reconstructed from induced two-point correlators. Quantum statistics are treated measure-theoretically, with outcome weights determined by the compatible preimage structure of the projection, providing a route to Born-type probabilities. The program is designed to reproduce established low-energy limits when calibrated, while enabling falsifiable deviation claims and direct numerical exploration on discrete realizations.
A pregeometric framework in which correlations and global compatibility constraints are taken as primitive, while spacetime geometry, matter content, and quantum behaviour arise only as emergent, projected observables. The core construction specifies a constraint manifold equipped with a correlation kernel and a family of projection maps into an effective manifold, from which metric structure, causal cones, and relativistic propagation can be reconstructed from induced two-point correlators. Quantum statistics are treated measure-theoretically, with outcome weights determined by the compatible preimage structure of the projection, providing a route to Born-type probabilities. The program is designed to reproduce established low-energy limits when calibrated, while enabling falsifiable deviation claims and direct numerical exploration on discrete realizations.
We propose a non-naive realist pregeometric model in which spacetime, matter, and quantum states emerge fromconstrained projections of fundamental correlations. In this approach, no spacetime or material degrees of freedomare assumed as primitive; instead, physically meaningful observables arise from compatibility conditions on theunderlying correlation structure. This framework provides a unified perspective on the emergence of geometry,matter content, and quantum statistical structure from a common pregeometric foundation.
We develop a pregeometric reconstruction framework in which quantum theory, entanglement structure, gauge symmetry, locality, Lorentzian spacetime geometry, gravitation, and effective physical constants emerge from recursively stabilized coherent relational organization. The framework begins from a primitive relational substrate consisting of compositional accessibility processes endowed with complex coherent transport amplitudes. No background spacetime manifold, metric geometry, particle ontology, or external classical time parameter is assumed fundamentally. The central proposal is that coherent compositional structure is more fundamental than spacetime geometry. Hilbert organization emerges from consistency requirements on coherent relational composition. Observable physical sectors arise through projection stabilization: a dynamical stability-selection process acting on coherent relational transport. Decoherence partitions coherent accessibility structure into approximately factorizable sectors, producing emergent locality and classical persistence. Gauge symmetry emerges naturally as redundancy in coherent transport representation. Lorentzian geometry arises statistically from stabilized accessibility organization, while gravitation appears as thermodynamic relaxation of projected coherent geometry. The framework is formulated as a reconstruction program rather than a completed unified theory. Its aim is not to replace existing physics at the phenomenological level, but to explain why physical reality organizes itself through Hilbert structure, gauge redundancy, locality, relativistic spacetime, and thermodynamic gravitational dynamics. Particular emphasis is placed on the possibility that quantum nonseparability is more fundamental than spacetime locality.
We develop a pregeometric reconstruction framework in which quantum mechanics, gauge symmetry, probabilistic structure, matter, Lorentzian spacetime geometry, gravitation, and physical constants emerge from recursively stabilized coherent relational transport organization. The framework begins from a primitive causal-relational substrate consisting only of fluctuating accessibility relations and compositional transport structure. No background spacetime manifold, metric geometry, particle ontology, or external temporal parameter is assumed fundamentally. The theory further proposes that the preconditions underlying Einsteinian physics must already exist at the pre-geometric level as deeper relational consistency principles. Lorentz invariance, causal structure, covariance, conservation laws, and gravitational equivalence are interpreted as emergent macroscopic expressions of more primitive coherent accessibility constraints. A generalized spectral transport operator governs coherent relational evolution and generates stable eigensectors corresponding to persistent physical structure. Quantum mechanics emerges from coherent compositional consistency and norm-preserving recursive transport flow. Gauge symmetry emerges as local redundancy of coherent transport representation. Classical mechanics emerges through stationary coherent transport stabilization. A reconstruction of the Born rule is proposed in which quadratic probability amplitudes arise from multiplicative persistence consistency under independent coherent composition. Lorentzian spacetime geometry emerges statistically from stabilized causal accessibility organization, while gravitation emerges thermodynamically as equilibrium relaxation of coherent accessibility geometry. The framework further proposes that physical constants correspond to frozen stability invariants selected during primordial coherent transport evolution. In particular, the fine-structure constant is interpreted as a stable surviving coherent phase-accessibility ratio associated with recursive phase closure. The theory therefore attempts not merely to reproduce known equations, but to explain why physical reality universally organizes itself through Hilbert structure, gauge redundancy, localization, Lorentzian geometry, and thermodynamic gravitational dynamics. In particular, the framework proposes that Einsteinian spacetime physics emerges from deeper pre-Einsteinian coherent relational conditions governing recursive transport accessibility before spacetime itself exists.
This publication presents an extended axiomatic revision of the Probability-Space Axiomatic Foundation. The original minimal logical core (K1–K4), governing realization, temporal asymmetry, and constrained selection among unrealized possibilities, is preserved verbatim and unchanged. This revision introduces an Extended Core (v2.0) consisting of three additional axioms (K5–K7) that are additive rather than revisionary. These axioms are required to support observer-dependent projection of observables, non-signaling correlation via shared collapse ancestry (entanglement), and the emergence of persistent relational deformation interpreted as effective curvature. The extended axioms do not modify the original logical foundation; instead, they constrain how realized structure is accessed, correlated, and stabilized when instantiated as a physical system. The work is structured as a strict in-place extension of the original manuscript. All original text, numbering, and logical development are preserved word-for-word. New material is inserted only where logically required, and is explicitly labeled as part of the Extended Core. No experimental claims are made. No force-based or field-first primitives are introduced. Physical quantities such as mass, energy, or spacetime curvature are not assumed at the axiomatic level. This axiomatic framework is intended to function as a constraint layer for physical theories built on Probability-Space concepts, ensuring compatibility with observer locality, non-signaling correlation, and emergent large-scale structure without altering the minimal logical basis. The framework is falsifiable in principle: any physical realization must recover standard quantum statistics and empirically tested gravitational behavior in appropriate limits. This record supersedes and extends the author’s prior Zenodo publication by providing a formally defined Extended Core (v2.0), while maintaining full continuity with the original axiomatic foundation.
This paper constitutes the fourth installment of the Mapping the Time research program and provides explicit toy models that concretely instantiate the abstract framework developed in the preceding papers. Building on the kinematic emergence of time established in Paper 1, the tri-level projective architecture introduced in Paper 2, and the dynamical constraints derived in Paper 3, this work constructs low-dimensional, fully explicit examples that realize all structural requirements simultaneously. The goal is not to introduce new theoretical claims, but to demonstrate that the framework is operationally consistent and mathematically realizable. The paper presents two hierarchical toy models that explicitly implement: A tri-level structure consisting of an Extended Volumetric Substrate (EVS), Observed Spacetime (OS), and a Reduced Informational Manifold (RIM); Two strictly non-invertible projections linking these levels; A volumetric spatial coordinate in the EVS that projects to the temporal coordinate in observed spacetime; Projection-compatible dynamics defined on the extended level; Emergent irreversibility at the observed and informational levels despite reversibility at the extended level; A clear distinction between the abstract evolution parameter and observed time. The central results demonstrate that: All constraints derived in Papers 1–3 can be satisfied simultaneously within simple, explicit models; Structural irreversibility arises purely from projective constraints, without invoking thermodynamics, entropy, or fundamental time asymmetry; The induced dynamics at the observed and informational levels naturally form semigroups, even when the underlying dynamics remain reversible; The mechanism is robust under changes in coordinate choices and fiber geometry. Importantly, the analysis remains strictly structural and pedagogical: No physical interactions or forces are modeled; No empirical claims are made; No new postulates are introduced beyond those already established in the trilogy; The models are illustrative rather than physically realistic. By providing explicit realizations, this paper closes the foundational phase of the Mapping the Time program. It validates the internal coherence of the framework and clarifies how the abstract results of the previous papers can be instantiated concretely, preparing the ground for future work on physically motivated models or applications. Keywords: emergent time, toy models, projective time, non-invertible projections, tri-level structure, projection-compatible dynamics, structural irreversibility, pre-dynamical physics, spacetime emergence, information loss, mathematical physics, differential geometry, foundations of time, theoretical physics.
This preprint is the second installment in a research program that develops a geometric ontic framework for finite-dimensional quantum theory. Paper I (“A Geometric Ontic Framework for Quantum States Based on SU(2) Symmetry and Non-Factorizable Projections”) introduced the kinematic layer of the theory: quantum states were represented as points on an underlying ontic manifold equipped with SU(2) symmetry, and standard single-particle quantum structure emerged from geometric projection rules. That first step reconstructed the full Hilbert-space picture for a single degree of freedom and identified how non-factorizable geometry underlies entanglement. The present paper extends this framework by introducing a dynamical layer. We postulate that the ontic metric G depends on the global configuration of the system, and we study its evolution under a general metric flow. Two principles guide this construction. The Quantum Equivalence Principle (QEP) requires reversible operational dynamics to preserve local distinguishability, recovering unitary evolution in symmetric limits. The Quantum Mach Principle (QMP) assigns dynamical content to the metric by allowing mild back-reaction from the ontic distribution. Together with a coarse-graining map RN(λ), these ingredients generate operational states and lead to an emergent Schrödinger-type dynamics. Standard quantum mechanics appears as the symmetric fixed point of the metric flow, while small departures from symmetry lead to controlled deviations from unitary evolution and the Born rule. This provides a geometric explanation for quantum kinematics and suggests testable signatures beyond the standard theory. This work lays the foundation for Paper III, where composite systems and curvature-induced entanglement dynamics are developed.