Abstract: We present a model-theoretic framework that derives the mathematical structures of physics from operational coherence constraints. Uniquely, we employ a “nominalist air-gap,” extracting these structures from a first-order language containing no mathematical objects in its ontology. Our construction has three main components: (1) We show how any minimal coherence model M |= T_min generates a canonical effect algebra E(M ) via a constructive free completion. (2) We demonstrate that the apparent conflict between Quantum Mechanics and General Relativity can be expressed as a difference in the scope of the Riesz Decomposition Property (RDP). Global RDP (T_rich) forces the algebra to embed into an ordered real vector space, while Local RDP (T_grav) gives rise to a presheaf of local algebras over a derived context topology. (3) We analyze a Path Independence property: global RDP is naturally associated with path-independent transport in context space, whereas non-trivial holonomy obstructs any such global linear structure. Motivated by this obstruction, we formulate a Logical Preservation Principle according to which physical evolution must restrict to locally coherent sectors when global RDP would lead to inconsistency, and we interpret this restriction as objective collapse.
Abstract: We present a model-theoretic framework that derives the mathematical structures of physics from operational coherence constraints. Uniquely, we employ a “nominalist air-gap,” extracting these structures from a first-order language containing no mathematical objects in its ontology. Our construction has three main components: (1) We show how any minimal coherence model M |= T_min generates a canonical effect algebra E(M ) via a constructive free completion. (2) We demonstrate that the apparent conflict between Quantum Mechanics and General Relativity can be expressed as a difference in the scope of the Riesz Decomposition Property (RDP). Global RDP (T_rich) forces the algebra to embed into an ordered real vector space, while Local RDP (T_grav) gives rise to a presheaf of local algebras over a derived context topology. (3) We analyze a Path Independence property: global RDP is naturally associated with path-independent transport in context space, whereas non-trivial holonomy obstructs any such global linear structure. Motivated by this obstruction, we formulate a Logical Preservation Principle according to which physical evolution must restrict to locally coherent sectors when global RDP would lead to inconsistency, and we interpret this restriction as objective collapse.
We present a rigorously formulated, internally self‑consistent foundational framework for physics based on a minimal algorithmic substrate and the mathematics of category theory and presheaf topoi. The construction begins with a universal prefix Turing machine and the algorithmic probability distribution as the sole primitive, from which arithmetic, discrete topology, and gauge symmetries emerge as the most compressible relational structures. Quantum mechanics is not postulated but derived: the non‑Boolean, contextual nature of physical propositions is captured by the Isham–Butterfield presheaf topos SetsV(H)opSetsV(H)op, whose internal logic is intuitionistic and whose spectral presheaf resolves the Kochen–Specker obstruction. Within this setting, we propose an Algorithmic Information Bound that provides a novel, computational explanation for the Yang–Mills mass gap: a strictly positive lower bound on the algorithmic complexity per lattice site forces a finite correlation length, guaranteeing a non‑zero mass gap. The observer is formalised as an Independent Ontological Simulation Algorithm (IOSA)—a finite, algorithmic subsystem that executes decoherence and context selection, reducing the global intuitionistic logic to a classical Boolean description. We embed the entire ontology in a symmetric monoidal category, exhibiting a precise Rosetta Stone mapping between physics, computation, and logic. Crucially, the framework explicitly acknowledges its own limits via Gödel and Turing, thereby avoiding the paradoxes of absolute self‑description. The resulting theory is internally contradiction‑free, mathematically explicit, and generates falsifiable predictions for lattice gauge simulations, cosmological initial conditions, and quantum‑contextual experiments.
Quantum Mechanics as Reversible Closure Representation and the Emergence of Relativistic Structure (Paper 2) Abstract This work develops a representation-theoretic extension of the closure-first discrete gravity framework established in Reality is What Misspoken Language Cannot Deliver (Rev17). Starting from finite primitive state spaces and reversible closure algebra, we construct a unitary representation of the closure group on a complex Hilbert space. Quantum evolution arises naturally as the continuous representation of discrete reversible transitions, with phase structure determined by restoration periods. Defect transport within the closure algebra yields dispersion relations that recover relativistic propagation in the macroscopic limit. Local flatness emerges as the tangent limit of the discrete curvature substrate, giving Special Relativity as a local geometric approximation of the underlying simplicial structure. General Relativity appears as the coarse-grained curvature density of defect-labelled simplicial geometry, preserving the Einstein field equations within their established effective domain of validity. Quantum mechanics is therefore not postulated independently, but arises as the representation layer of reversible finite closure, while relativistic structure emerges from geometric coarse-graining of the same primitive algebra. No new macroscopic gravitational equations are introduced; the present work concerns the structural origin of existing relativistic frameworks rather than their modification. Introduction In the preceding work (Rev17), finite entropy bounds were shown to exclude primitive continuum ontology and to imply a finite state space for bounded physical regions. Under assumptions of deterministic closure, no primitive erasure, and inverse realisability, the reversible sector of primitive transitions forms a group. Defect labels associated with group elements define discrete curvature quanta, which embed naturally into a Regge-type geometric framework. A minimal continuum-limit argument yields the Einstein–Hilbert action under refinement. Importantly, this construction does not alter the Einstein field equations at macroscopic scales. General Relativity remains valid within its established regime as an effective continuum description. The prior work addresses the structural substrate from which that continuum description emerges. The present work extends this foundation by addressing two remaining structural questions: How does quantum mechanics arise from finite reversible closure? How does relativistic kinematics emerge from the same primitive structure? Rather than introducing Hilbert space or Lorentz symmetry axiomatically, we derive them as representation and coarse-graining layers of the previously established closure algebra. The guiding principle is minimal structural extension: no new primitive entities are introduced. All additional structure follows from representation consistency and macroscopic limit coherence. Quantum theory appears as the unitary representation of reversible closure; relativistic structure emerges as the geometric compression of defect-labelled algebra under scale separation. References[1] J. Bloggs, Aligning GR with Reality – Fully and Finally, Zenodo (2026). DOI: 10.5281/zenodo.18673556.[2] T. Regge, General relativity without coordinates, Nuovo Cimento 19 (1961) 558–571. DOI:10.1007/BF02733251.[3] J. W. Barrett, D. Oriti, and R. M. Williams, Tullio Regge’s legacy: Regge calculus and discrete gravity, arXiv:1812.06193 (2018). DOI: 10.48550/arXiv.1812.06193.[4] R. Loll, Discrete approaches to quantum gravity in four dimensions, Living Rev. Relativ. 1 (1998) 13. DOI: 10.12942/lrr-1998-13.[5] B. Dittrich and S. Steinhaus, Path integral measure and triangulation independence in discrete gravity, Phys. Rev. D 85 (2012) 044032. DOI: 10.1103/PhysRevD.85.044032.
We develop a mathematical framework in which physical description is organized through predictively sufficient closure of richer realization spaces. A closure map identifies realizations that are equivalent relative to specified observables, while dynamic sufficiency requires the induced reduced state to support autonomous evolution. This yields a precise formulation of the principle that law can close before realization does: a law-level state may be fully predictive within a declared domain even when multiple underlying realizations remain possible. We distinguish quotient closure, realization fibers, idempotent closure maps, refinement, partial closure, closure memory, obstruction, and saturation of multiple closure constraints. These structures are then tested against explicit physical examples involving compact phase winding, topological defects, gauge compensation, exchange statistics, spinorial closure, quantum measurement channels, and effective propagation geometry. The analysis exposes several limitations of a motivating one-scalar coherence construction. In particular, a smooth exact phase connection has vanishing local curvature; a relative phase does not generate orthogonality or Pauli exclusion; winding alone does not supply spin- or the observed electric-charge spectrum; and a single scalar gradient cannot define a generic nondegenerate four-dimensional metric. Rather than abandoning the coherence hypothesis, these obstructions motivate a minimally enriched effective architecture involving amplitude/phase, gauge connection, spinor, and geometric frame sectors. We distinguish this effective reconstruction from the stronger hypothesis that all such structures descend from a common parent coherence object. The resulting framework does not claim a completed derivation of the Standard Model, general relativity, or quantum measurement. Instead, it identifies the mathematical and physical gates such derivations must satisfy and isolates noncommuting saturation, closure obstruction, representation reconstruction, stable particle sectors, and universal emergent geometry as principal theorem and physics programs. Keywords: closure mathematics; predictive sufficiency; quotient dynamics; gauge theory; topological defects; particle identity; quantum foundations; emergent geometry; noncommuting closure; unification.
This series presents a structured and logically sequential formulation of a discrete informational framework governed by global coherence constraints. Although it originates from the same research program developed in the earlier exploratory preprint series (P-series), the works presented here constitute an autonomous and self-contained series, organized in a progressively axiomatic form and focused on the explicit derivation of the framework’s effective structures. The goal of the construction is to investigate whether structures commonly associated with quantum mechanics, thermodynamics, and geometry may emerge from a discrete relational substrate without being introduced as fundamental postulates. Each article addresses a specific level of the theoretical derivation and forms part of a progressive logical structure based on results established in the preceding works of the series. This article develops the axiomatic foundation of the framework and introduces the discrete informational system underlying the construction. The paper defines the admissible configuration space, the local update structure, the global coherence constraint, and the dynamical selection principles leading to the emergence of a coherent sector. The analysis establishes the existence, invariance, and structural uniqueness of the coherent sector, together with the effective restrictions that prepare the minimal two-component structure developed in Paper 02 and the emergent quantum framework constructed in Paper 03. No quantum-mechanical, geometric, or probabilistic structure is assumed at the foundational level. The purpose of the work is to establish the minimal structural and dynamical layer from which the later effective constructions emerge.
This document provides a technical review and supplemental material for the poster "A Quantum-Cosmological Picture with Operator-Algebraic Relational Frames", which details an operational framework where algebra is taken as a fundamental physical structure, while spacetime, curvature, particles, interactions, propagation, and gauge-like symmetries are derived from mathematical relationships that can be proven to occur by algebraic necessity. The framework is developed from the bottom-up, using unitary quantum-mechanical first principles, and provides a coherent way to incorporate particles into quantum gravity, yielding an elegant structure where all the pieces fit together.The poster proposes an algebraic origin for matter and interaction, which is based on quantum relational frames. Particle-like objects arise as local non-closure defects, while curvature arises from the related failure of local boundary operations to commute. Fermionic sectors, bosonic generators, particle and anti-particle orientation, and propagator behavior are then placed in the same algebraic setting, rather than being added as independent field-theoretic inputs. The poster's framework treats a physical observer or system as a local frame, defined as finite algebraic state with its own internal history. Other possible histories, future alternatives, and non-local systems are represented through the algebra available to that local frame. In this setting, spacetime is not introduced as a pre-existing manifold. Instead, event coordinates are obtained from observables that are shown to map algebraic histories into a local event representation. This gives a route from quantum-histories to special-relativistic distances and then to general-relativistic behavior through the comparison of local event descriptions. Moreover, the quantum measurement problem (QMP) is resolved at the framework's fundamental level of construction by treating the algebra's quantum frames as ensemble of density matrices constructed on recursive branching channels of quantum information, where a sequence of positive-operator-valued measurements (POVM) updates each local state's history.The first purpose of this review is to validate the poster's equations and clarify their significance. The poster is condensed, so several of its central claims require expansion in order to elucidate how the pieces fit together. In particular, the validation focuses on: how the poster's overall equations are constructed and whether the framework is consistent and coherent (Section 1.2); how the poster's local-frame construction supports its particle/signature claims about SU(3) x SU(2) x U(1) gauge symmetry (Section 4.1); and, how its event observables recover the metric distance structure of special and general relativity (Sections 4.2.3 to 4.2.10).The second purpose is to show that the framework's equations lead to several other significant results that form a complete picture. In particular, a generalized form of loop quantum gravity (LQG) is derived (Section 5.9) by passing from algebraic histories to event geometry and then to a dual graph structure of event simplexes. This is significant because the same algebraic framework also supplies matter and propagator structures in that setting, giving a possible first-principles route to fermions, bosonic generators, and propagation within the LQG correspondence. Another important result is the derivation of quadratic quantum gravity (QQG) from the algebra (Sections 3 and 14.9). Because the local curvature operator is represented by a finite commutator, its trace vanishes. As a result, the usual linear curvature scalar is absent at the fundamental algebraic level, and the first nonzero invariant is quadratic in curvature. This gives a curvature-square gravitational functional as a consequence of the operator algebra rather than as an assumed action. Moreover, the review refines the poster's marginal curvature equation (defined within local configuration space) into a reduced curvature (defined as a push-forward into event space) (Section 6.5). This clarifies how the poster's curvature equation should be interpreted and leads to an operator algebraic form for general relativity (Section 12.2). Discussion about black holes (Section 7.1) and their event-space gravitational description (Section 7.2) is provided.The other sections use the operator-algebra framework to study interactions (Sections 2.2 and 9.1); particle-count observables (Section 8); black-hole-like trapping (Section 13.1); anisotropic minimal black-hole support (Section 10.5); and the Green's functions of radiation sectors (Section 13.4). These topics show that the poster's equations are not isolated formulas. They form a unified algebraic framework that can be used to organize several areas of quantum gravity, particle structure, and cosmology from the same starting point. A discussion about how quantum fields (Section 15.1) and particle masses (Section 16.4) may be obtained from the operator algebra is also provided
The relationship between quantum mechanics and general relativity is typically pursued via quantizing the spacetime metric or modifying the Standard Model. This paper proposes an alternative structural approach: treating both frameworks as local differential theories whose admissible global solutions are restricted by the geometry of their domains. The Schrödinger equation, expressed in Madelung form, makes this explicit: probability is conserved, and the resulting current must route around regions where the wavefunction is zero. This same structure of current routing around obstructions explains double-slit interference, Bell correlations, gravitational lensing, and the geometric role of phase in coherent evolution. Here, "phase" is used in its geometric sense as a connection on the space of states, while "density" is the conserved quantity whose zeros define the topological obstructions. The incompatibility therefore dissolves not by quantizing gravity or modifying quantum mechanics, but by recognizing their common structure. This paper initiates a series (Unitary Scale Theory, UST) examining scale-independent consequences of covering-space constraints on conserved currents. Version history v1 (January 2026): Initial publication. v2c (February 2026): Title refined to clearly reflect scope. Holonomy language updated throughout for consistency with covering-space / lift-projection framework developed in subsequent papers in the series. No changes to mathematical content or results.