A Set-Based Alternative to the Hilbert-Space Formalism: Two-Slit Interference in the CEGS Framework This work presents a deterministic reformulation of the two-slit experiment within the Calculus of Element Generated Sets (CEGS), a set-based framework for modeling physical systems without assuming Hilbert space as a fundamental structure. Instead of representing quantum states as vectors in a complex Hilbert space, physical states are described as structured sets of phase-generating elements subject to intrinsic constraint operations prior to aggregation. Within this framework, observable quantities arise only after a coherent aggregation map acts on constrained generative sets. The standard interference pattern of the two-slit experiment is recovered exactly at the level of observable predictions, preserving empirical equivalence with conventional quantum mechanics. A central conceptual shift is that complex amplitudes are not treated as fundamental entities. Instead, they emerge as effective compressed representations of underlying distributed phase structures under observer-dependent projection. In this view, Hilbert space is interpreted as an emergent effective description rather than a primary ontological structure. The formulation provides a deterministic and structural alternative perspective on quantum interference, contributing to ongoing discussions in quantum foundations, interpretation of quantum mechanics, and set-based or non-vectorial reformulations of physical theory. The analysis is restricted to the two-slit experiment as a minimal test case, while maintaining consistency with standard quantum predictions.
A Set-Based Alternative to the Hilbert-Space Formalism: Two-Slit Interference in the CEGS Framework This work presents a deterministic reformulation of the two-slit experiment within the Calculus of Element Generated Sets (CEGS), a set-based framework for modeling physical systems without assuming Hilbert space as a fundamental structure. Instead of representing quantum states as vectors in a complex Hilbert space, physical states are described as structured sets of phase-generating elements subject to intrinsic constraint operations prior to aggregation. Within this framework, observable quantities arise only after a coherent aggregation map acts on constrained generative sets. The standard interference pattern of the two-slit experiment is recovered exactly at the level of observable predictions, preserving empirical equivalence with conventional quantum mechanics. A central conceptual shift is that complex amplitudes are not treated as fundamental entities. Instead, they emerge as effective compressed representations of underlying distributed phase structures under observer-dependent projection. In this view, Hilbert space is interpreted as an emergent effective description rather than a primary ontological structure. The formulation provides a deterministic and structural alternative perspective on quantum interference, contributing to ongoing discussions in quantum foundations, interpretation of quantum mechanics, and set-based or non-vectorial reformulations of physical theory. The analysis is restricted to the two-slit experiment as a minimal test case, while maintaining consistency with standard quantum predictions.
This work addresses the long-standing claim that interference phenomena—most notably in double-slit–type experiments—cannot be physically modeled without invoking wave propagation or particles following multiple trajectories. We show that this claim rests on an implicit restriction to local, dynamical descriptions acting between emission and detection. We introduce a minimal framework in which detection events are treated as discrete realizations occurring at an absorbing boundary, while the observed interference pattern arises from global structural admissibility constraints of the full experimental configuration. The model does not attempt to generate or predict individual detection events. Instead, it characterizes the statistical distribution of outcomes as a property of realization closure rather than of local evolution in transit. Within this approach, interference patterns are understood as emergent statistical signatures of global realization structure, not as evidence of propagating waves, hidden variables, or oscillatory dynamics in space. The framework preserves all empirical features of standard interference experiments while avoiding assumptions about intermediate physical processes that are neither observed nor required by theory. This work is consistent with relativistic and quantum no-go results concerning photon trajectories, localization, and in-flight observables. It complements earlier structural interpretations of light and realization by providing a concrete, physically explicit model demonstrating how stable interference statistics arise without propagation. The contribution clarifies the precise sense in which interference is modelable, and the sense in which it is not: individual events remain intrinsically unpredictable, while their global organization is physically intelligible at the level of realization structure. The model introduces no new forces, fields, or modifications of established equations, and is intended as a refinement of conceptual foundations rather than a replacement of existing formalisms.
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 emergence of Hilbert-space structure and Schrödinger-type effective dynamics from the coherent two-field sector established in Paper 02. The work demonstrates that the global coherence constraints imposed on the discrete informational substrate naturally generate an effective complex representation endowed with linear superposition properties and unitary-preserving evolution. The analysis shows that the two-component coherent sector admits a mathematically stable reformulation equivalent to a Hilbert-space structure, and that the effective evolution of the coherent amplitudes leads to a Schrödinger-type dynamical equation without introducing quantum mechanics as an initial assumption. The article further derives the probabilistic interpretation associated with the effective amplitudes and clarifies how quantum-like statistical behavior emerges from the internal coherence conditions of the underlying discrete system. No quantum postulates are assumed at the foundational level. The purpose of the article is to demonstrate that the formal structures associated with quantum mechanics can emerge as effective consequences of coherence-preserving informational dynamics. The article does not assume quantum mechanics as a foundational theory and does not postulate the Hilbert-space formalism a priori. Instead, the work demonstrates that, within the internal hypotheses and coherence constraints of the proposed discrete informational framework, structures mathematically equivalent to Hilbert-space representations and Schrödinger-type evolution can arise as effective emergent descriptions. The purpose of the derivation is therefore not to reproduce standard quantum mechanics by assumption, but to show that quantum-like formal structures may consistently emerge from deeper coherence-preserving informational dynamics.
The standard Copenhagen interpretation of quantum mechanics relies on a set of historically established, ad hoc postulates to describe particle-wave duality, measurement, and non-locality. In this paper, we propose a single foundational axiom that fully reproduces the standard mathematical formalism of quantum mechanics while eliminating the need for supplementary assumptions. From this singular postulate, we rigorously derive the fundamental origin of the wave function and the mathematical generator of the Schrödinger equation. Furthermore, we demonstrate via isomorfism that all seemingly inexplicable quantum phenomena—including wave-function collapse and quantum entanglement—are mathematically and conceptually reducible to the underlying dynamics of the double-slit experiment. By establishing a direct fundamental correlation between the double-slit interference mechanism and the Monty Hall phenomenon, we prove that these effects are distinct manifestations of the exact same principle. We framed the double-slit experiment as an interference of the space of possibilities affected by information and simetrys. Ultimately, by anchoring the existing quantum mathematical framework to this unique axiom, this work provides a coherent, paradox-free ontological foundation that resolves the interpretational crisis of quantum mechanics without altering its empirical predictive power.
This paper develops the structural modeling core of a global realization approach to quantum interference. Building on prior work that established the possibility of non-propagational descriptions, the present study addresses a more precise question: how quantum interference can be explicitly modeled once local propagation, trajectories, and in-transit dynamics are excluded. The model introduces realization admissibility functionals defined over the complete experimental configuration, including source, apertures, and absorbing boundaries. These functionals are non-dynamical and non-ontic; they do not represent physical fields or evolving states, but encode global compatibility conditions that govern where detection events can coherently close. Detection probabilities emerge through statistical closure at the absorbing boundary, reproducing standard interference patterns without invoking wave superposition in space. A minimal geometric formulation is presented to demonstrate how slit configurations generate admissibility structures that yield conventional interference statistics. Within this framework, interference minima arise naturally as forbidden realizations rather than as products of destructive interaction between propagating entities. The model preserves full empirical equivalence with standard quantum mechanics while clarifying the physical status of interference as an organizational, rather than dynamical, phenomenon. This work does not modify quantum dynamics or introduce additional degrees of freedom. Its contribution is structural: it sharpens the distinction between quantum dynamical states and realization, delineates the limits of local modeling, and provides a concrete, explicit framework for modeling interference at the level where empirical outcomes occur. The paper serves as the modeling core of a broader global realization program, situated between conceptual foundations and empirical analysis.
This paper develops a phase-first reconstruction of quantum interference within standard quantum mechanics and wave optics. Its central idea is that interference patterns become conceptually clearer when the complex phase is treated as the organizing structure and probability or intensity density is treated as the visible statistical expression of that structure. The guiding principle is: the phase structures the pattern; the density renders it visible. The paper shows how this principle applies to single-slit diffraction, double-slit interference, Airy patterns, Fourier optics, Fresnel propagation, probability current, the Madelung representation, which-way marking, reduced coherence, and quantum eraser experiments. In each case, the observed density pattern is interpreted as the visible result of coherent phase relations among amplitudes. The reconstruction is strictly conservative. It does not modify quantum mechanics, the Born rule, unitary dynamics, or standard wave optics. It introduces no new postulates and makes no new empirical prediction. Its purpose is conceptual and pedagogical: to clarify the architecture of quantum interference by placing relative phase at the center of the analysis. This work is Part I of the series The Geometry of Phase. It establishes the phase-geometric foundation for later parts of the series, where the same perspective is extended to de Broglie matter waves, proper-time phase, relativistic action, and possible curvature-coupled phase extensions.
Quantum interference is conventionally formulated using complex amplitudes associated with quantum alternatives or through Feynman’s path integral over possible trajectories. This article investigates a different ontology, the Single-History Record Framework (SHRF), in which a physical system has one realized history together with a record space of phase-carrying contributions associated with that history. Observable amplitudes are obtained through History-Record Summation (HRS). SHRF deliberately retains established quantum ingredients wherever possible, including complex-amplitude addition, the Born probability rule, propagation phases, and marker-state overlaps. These mathematical devices are used as predictive rules and do not imply that multiple physical histories, paths, or marker outcomes coexist.A preliminary model is developed in which rapidly varying regions of record space largely cancel while coherent record bands provide the dominant detector amplitude. For double-slit geometry, two aperture-selected bands give A(x)≈H_A (x)+H_B (x).With the established propagation phases H_A=Ce^(ikL_A ),H_B=Ce^(ikL_B ),SHRF recovers the textbook far-field fringe spacing Δx=λD/d.The same formal structure is applied to single-slit diffraction, multiple slits, which-path marking, partial distinguishability, and quantum-eraser subensembles. In particular, marker records modify the interference term through the standard overlap factor ⟨M_A∣M_B⟩,without requiring a physical superposition of marker records. The present model assumes rather than derives its phase law, record-space measure, and Born rule. It is therefore a recovery-oriented consistency construction and ontological reinterpretation, not an independent replacement for standard quantum mechanics.