Quantum mechanics is based on fundamental notions such as the wave function,superposition, probability, measurement, entanglement, and nonlocality. These no-tions are remarkably successful from both mathematical and experimental points ofview, yet they raise deep conceptual questions. Is the wave function a real physicalobject or only a computational tool? Does superposition mean that an object reallyexists in several states at once? Does entanglement imply an instantaneous actionat a distance? Is probability fundamental, or does it arise from partial information?In this article, we propose an exploratory relational interpretation of these no-tions within a discrete framework based on the additive group (nZ, +). In thisframework, the fundamental entities are not isolated objects, nor points of a pre-existing spacetime, but indexed relational degrees of freedom. A physical confgura-tion is described by a global relational scene, and admissible scenes are selected bya relational master equation derived from a coherence functional.The aim is not to deny the empirical success of quantum mechanics, but topropose a conceptual dictionary in which the wave function is interpreted as anefective representation of a deeper relational structure. In this perspective, su-perposition becomes a composition of compatible relations, entanglement becomesrelational non-separability, measurement becomes relational selection, and probabil-ity becomes an efective description associated with the observer’s partial access tothe global structure.
Quantum mechanics is based on fundamental notions such as the wave function,superposition, probability, measurement, entanglement, and nonlocality. These no-tions are remarkably successful from both mathematical and experimental points ofview, yet they raise deep conceptual questions. Is the wave function a real physicalobject or only a computational tool? Does superposition mean that an object reallyexists in several states at once? Does entanglement imply an instantaneous actionat a distance? Is probability fundamental, or does it arise from partial information?In this article, we propose an exploratory relational interpretation of these no-tions within a discrete framework based on the additive group (nZ, +). In thisframework, the fundamental entities are not isolated objects, nor points of a pre-existing spacetime, but indexed relational degrees of freedom. A physical confgura-tion is described by a global relational scene, and admissible scenes are selected bya relational master equation derived from a coherence functional.The aim is not to deny the empirical success of quantum mechanics, but topropose a conceptual dictionary in which the wave function is interpreted as anefective representation of a deeper relational structure. In this perspective, su-perposition becomes a composition of compatible relations, entanglement becomesrelational non-separability, measurement becomes relational selection, and probabil-ity becomes an efective description associated with the observer’s partial access tothe global structure.
This preprint proposes a discrete relational framework indexed by the additive group (nZ,+), in which neither fundamental time, nor primitive dynamics, nor ontological randomness, nor a physical wave ontology is assumed. The global structure is described by a relational configuration (Ψ,R)(\Psi,R)(Ψ,R) constrained by a master equation of coherence, interpreted as an atemporal global constraint (no derivatives, no evolution law). The work introduces the notion of a “relational book”: events are assumed to be fully pre-inscribed in a negative branch of the index set, while the positive branch contains virtual events that become apparent only through a relational connection to their pre-inscribed symmetric counterparts. Within this setting, quantum measurement is reformulated as constrained relational selection/activation driven by the measurement device, with no wave collapse and no underlying wave propagation. The Born rule is reinterpreted as a relational compatibility density whose observed statistical regularities emerge from partial relational access to the global structure. The preprint also discusses interference “without waves” (e.g., double-slit scenarios) as a signature of global relational constraints, and includes a structural comparison with Everett (Many-Worlds), Bohmian mechanics, Relational Quantum Mechanics, and QBism.
For centuries, science has attempted to understand reality by identifying its most fundamen-tal objects. Atoms, particles, molecules, felds, neurons, symbols and data have progressivelybecome the elementary building blocks through which natural phenomena are described.This object-centered view has produced remarkable scientifc achievements. Nevertheless,many phenomena observed in physics, mathematics, language, biology and artifcial intelligencesuggest that the behavior of a system cannot always be understood by studying its componentsindependently.The central idea developed throughout this book is that relations may be more fundamentalthan the objects themselves.Objects acquire meaning through the network of relations in which they participate. Observ-able behavior emerges from coherent relational organizations rather than from isolated entities.This book proposes a discrete relational framework based on the additive group(nZ, +)as a possible mathematical foundation for describing relational organization, emergence,measurement, artifcial intelligence and several conceptual questions arising in modern quantumphysics.The objective is not to replace existing theories but to explore a complementary perspectivein which relations become the primary elements of scientifc description.
In this text, we propose a conceptual study of the relational framework througha detailed analogy with language. The central idea is that physical objects, events,measurements, and some notions of quantum physics may be interpreted as elementswhose meaning depends on their position within a global relational structure.In language, letters form words, words form sentences, sentences form paragraphs,paragraphs form chapters, and chapters form books. An isolated letter does not yetpossess the meaning of a word. An isolated word may have several possible meanings.A sentence selects a more precise meaning, while the book gives global coherence to thewhole structure.By analogy, the elementary bricks of physics may be viewed as letters, objects orparticles as words, events as sentences, physical scenes as paragraphs, coherent domainsas chapters, and the universe as an indexed relational library.This analogy also allows us to propose a relational reading of several quantum no-tions. The corpuscular aspect of a particle would correspond to the stable existenceof a word in a book. Measurement would correspond to the reading of a word in asentence, which selects one precise meaning among several possibilities. Wave-packetreduction would then be interpreted as a selection of relational meaning. Entanglementwould correspond to a correlation between two words belonging to the same structure ofmeaning, as in the example of the word director and the word school. Finally, entropymay be understood as a measure of the organization or dispersion of words, events, andpossible readings.The aim of this study is not to reduce physics to linguistics, but to use language asa structuring analogy in order to clarify the relational framework, especially the ideaof a universe after zero already inscribed in the form of indexed books, symmetric to astructure before zero.
Quantum mechanics represents physical states through complex amplitudes, state vectors, and density operators, yet the ontological status of these objects remains interpretation-dependent. This paper develops a closure-based formulation in which a quantum state is interpreted as a representation of contextually partial relational closure: the state possesses definite mathematical and relational structure while remaining unresolved relative to one or more observational contexts. Within this language, decoherence differentiates closure channels and redistributes locally accessible coherence into relational structure; environmental redundancy stabilizes records; and classical facts emerge as approximately fixed, effectively commuting, redundantly recorded structures. Measurement is separated into three problems—context, weight, and outcome—so that decoherence, Born weighting, and unique outcome selection are not conflated. The paper then distinguishes two stronger research programs from the exact reformulation. First, it asks whether Hilbert-space quantum mechanics can be reconstructed from primitive closure, incompatibility, composition, and consistency principles. Second, it formulates a falsifiable closure-history extension in which physically relevant information might exist beyond the present density operator. Quantum Partial Closure Theory is therefore presented as a relational interpretation and information calculus, a reconstruction program, and a prospective experimental framework rather than as an established replacement for quantum mechanics. Keywords: quantum foundations; closure; wavefunction ontology; coherence; entanglement; decoherence; measurement; classical emergence; quantum information; Hilbert-space reconstruction The wavefunction question Quantum theory is operationally clear about what a wavefunction does. For a normalized pure-state vector ψ∈H and a projective measurement with alternatives {Π_i}, the Born weights are p_i=Tr(ρ_ψ Π_i ), ρ_ψ=ψψ^†. For a rank-one outcome Π_i=e_i e_i^†, this reduces to p_i=|e_i^† ψ|^2. Yet the mathematical clarity of the formalism does not by itself settle what the state vector is ontologically. Interpretations of quantum theory differ over whether the state should be regarded as physical, epistemic, relational, branch-relative, dispositional, or otherwise. The present paper proposes a specific interpretive and reconstructive language: partial relational closure. The central claim is not that the wavefunction is literally a new material substance called closure. Rather, ▭(ψ "is interpreted as a Hilbert-space representation of coherent partial relational closure" .) The word partial is crucial. It does not mean that a state is a scalar fraction of being real or complete. It means that information and constraint are distributed across local, coherent, and relational sectors, and that closure is always specified relative to a context. A Bell state, for example, can be locally indefinite while being relationally exact. A decohered state can be closed with respect to one effective macroscopic context while retaining a larger quantum embedding. A selective outcome can be locally definite without implying that every larger-scale relation in the universe is globally closed. This formulation is motivated by three observations. First, projective dephasing is naturally closure-like because it is idempotent. Second, modern quantum information already supplies exact measures for local coherence, correlations, and record redundancy. Third, the quantum-to-classical transition is naturally relational: local coherence can disappear from a subsystem while information migrates into system-environment correlations. The paper has three epistemic layers. Layer A uses established quantum mathematics. Layer B derives exact information identities and interprets them in closure language. Layer C states reconstruction conjectures and an explicitly speculative beyond-QM closure-history hypothesis. These layers are kept separate throughout. A quantum system is not structureless before measurement. Its amplitudes, phases, conservation relations, and correlations are tightly organized. What remains unresolved is how that structure will become locally definite relative to a chosen measurement context. The partial-closure view therefore distinguishes three broad regimes: Coherent partial closure. Alternatives remain phase-related and can interfere. Contextual differentiation. Interaction or dephasing suppresses coherence between alternatives relative to a context without necessarily choosing one. Selective local closure. A particular outcome is conditioned or realized relative to the measurement relation. In this language, wave-like behavior corresponds to spatially or otherwise contextually unresolved coherent closure, while particle-like detection corresponds to localized closure. Entanglement is closure stored in relations rather than in isolated subsystems. Decoherence is not simply destruction: in a globally pure description it can be a transfer of local coherence into relational entanglement. Classical objectivity appears when relevant records become stable and redundantly accessible. The framework does not assign consciousness a privileged dynamical role. An observer is a partial closure process for constraining possibilities. Explainer video Watch on YouTube (Roots of Reality): https://www.youtube.com/watch?v=UYx_omk3h3A The video, its thumbnail and description are included in this record.
We propose a relational reinterpretation of the concept of wave within a discreteframework based on the additive group (nZ, +). Instead of postulating the waveas a fundamental entity, we explore the possibility that a wave may be understoodas an efective, coherent manifestation of a distributed relational quantity. Twocomplementary representations are introduced. In the additive representation, a waveis modeled as a succession or superposition of local quantities. In the multiplicativerepresentation, the global wave-like manifestation is modeled as a product of localrelational factors. In this second case, the natural logarithm transforms the globalproduct into a sum of local contributions, thereby revealing the internal structure ofthe manifestation. This approach connects the wave concept with relational scenes,graph Laplacians, coherence functionals, efective localization, and emergent physicaldescriptors. The work is presented as a mathematical and conceptual exploration,not as a replacement of standard wave theory
We propose a conceptual analysis of probability within a discrete relational frameworkbased on the additive group (nZ, +). The central claim is that, under the simultaneousassumptions of exact laws, global symmetries, and strict conservation principles, probabilityis more naturally interpreted as epistemic than as the manifestation of ontological chance.In the framework considered here, events are treated as elements of a globally constrainedrelational structure, without introducing fundamental time or a physically postulated wave.We discuss the possible relevance of this perspective for the conceptual relation betweengeneral relativity and quantum mechanics.