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 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
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.
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.
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.
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.
This work presents the complete mathematical and conceptual formulation of the Relational Self-Coherence Framework — a unified physical model in which relationships, rather than spacetime or fields, are taken as the fundamental building blocks of reality. In this framework, all physical laws emerge from the minimisation of a single global quantity: the Total Relational Instability Functional, \mathcal{I} = \sum_{i,j} f(\mathbf{H}_{ij}, \Delta \ell_{ij}, \Delta \sigma_{ij}, S_{ij}, W_{ij}), The model proposes a probability rule for the actualisation of relations, P(R_{ij}=1) \propto \left(-\frac{\delta \mathcal{I}}{\delta R_{ij}}\right)\exp(-S_{ij}), Through coarse-graining and variation of the instability functional, the framework naturally reproduces: Einstein-like geometric structure, quantum path-sum amplitudes, conservation laws, interference and phase behaviour, and emergent constants such as , , and . This release includes the full extended formulation, derivations, tensor definitions, and appendices, providing a complete resource for theoretical exploration, simulation, or further development.It is intended as a stable, citable reference for the emerging relational approach to fundamental physics.
This work proposes a systematic development of an analogy between the discrete rela-tional framework and language, in order to shed light on the emergence of usual physicalquantities from a more fundamental structure of relations.Within this perspective, letters, words, sentences, paragraphs, chapters, and the bookitself are interpreted as hierarchical levels of organization, comparable to relational scenesof increasing richness.The aim is not to reduce physics to linguistics, nor to transform the universe into amere textual metaphor. Rather, language is used as an exemplary structure of discretecomposition, non-commutativity, local and non-local correlations, and emergence of globalcoherence.Such an analogy allows space to be reformulated as relational proximity, time asan order of reading or appearance between scenes, velocity as a rate of variation of aconfguration, energy as a measure of tension or coherence, and entropy as an index ofdispersion of confgurations or admissible interpretations.The book is organized as a conceptual and mathematical foundation intended to beprogressively developed. It begins by recalling the foundations of the discrete relationalframework, then introduces language as a hierarchical model of organization. It thenshows how emergent quantities can be interpreted through this analogy, and fnally opensthe way toward a more rigorous mathematical formalization and future theoretical devel-opments.