This work is Part II of a two-part series on the Modular Superselection Framework. Part I (DOI: 10.5281/zenodo.17834017) established the theoretical foundation; Part II provides computational validation through AlexMax-Q discrete quantum dynamics. Building upon the Modular Superselection Framework established in Part I (Lazzari, 2025; DOI: 10.5281/zenodo.17834017), we present a concrete computational implementation showing that discrete ℤ_M-graded state spaces can support robust quantum interference. Using AlexMax fractal-arithmetic encoding to construct natural-number states organised into modular sectors, we systematically explore the emergence of quantum behaviour in a fully discrete setting. We identify a sharp distinction between purely geometric models and dynamically driven ones. In Framework v1.x, where overlap is determined solely by a fractal metric and discrete phase assignments, the resulting quantum–classical deviations never exceed ≈3%, placing the model firmly in a quasi-classical regime. In Framework v2.0, by contrast, we introduce explicit unitary dynamics generated by fractal-weighted Hamiltonians. Numerical simulations reveal strong interference, with quantum–classical deviations ranging from 18% to 43% across a broad parameter range of fractal decay rates, path asymmetries, and coupling geometries (fractal, modular, hybrid). These results provide the first explicit computational realisation of the postulates formulated in Part I: modular-arithmetic structures, when equipped with sector-preserving unitary evolution, can generate genuinely non-classical correlation patterns while maintaining a deterministic, ℤ_M-graded description of measurement. The AlexMax-Q framework thus connects number-theoretic fractal geometry with discrete quantum dynamics and supports an epistemic interpretation of quantum measurement within ℤ_M-graded structures.
This work is Part II of a two-part series on the Modular Superselection Framework. Part I (DOI: 10.5281/zenodo.17834017) established the theoretical foundation; Part II provides computational validation through AlexMax-Q discrete quantum dynamics. Building upon the Modular Superselection Framework established in Part I (Lazzari, 2025; DOI: 10.5281/zenodo.17834017), we present a concrete computational implementation showing that discrete ℤ_M-graded state spaces can support robust quantum interference. Using AlexMax fractal-arithmetic encoding to construct natural-number states organised into modular sectors, we systematically explore the emergence of quantum behaviour in a fully discrete setting. We identify a sharp distinction between purely geometric models and dynamically driven ones. In Framework v1.x, where overlap is determined solely by a fractal metric and discrete phase assignments, the resulting quantum–classical deviations never exceed ≈3%, placing the model firmly in a quasi-classical regime. In Framework v2.0, by contrast, we introduce explicit unitary dynamics generated by fractal-weighted Hamiltonians. Numerical simulations reveal strong interference, with quantum–classical deviations ranging from 18% to 43% across a broad parameter range of fractal decay rates, path asymmetries, and coupling geometries (fractal, modular, hybrid). These results provide the first explicit computational realisation of the postulates formulated in Part I: modular-arithmetic structures, when equipped with sector-preserving unitary evolution, can generate genuinely non-classical correlation patterns while maintaining a deterministic, ℤ_M-graded description of measurement. The AlexMax-Q framework thus connects number-theoretic fractal geometry with discrete quantum dynamics and supports an epistemic interpretation of quantum measurement within ℤ_M-graded structures.
This fourth volume in the series explores the deep structural connection between the modular spectrum of the type-III fibres arising in the direct-integral decomposition of the emergent algebra M and the fractal, self-similar branching architecture of the avalanche process itself. Building on the classification of the centre Z(M) (Volume II) and the experimental signatures of sectorisation (Volume III), we demonstrate that the Connes invariant S(Mω ) = R+ of each fibre is not an abstract feature but is directly encoded in the binary-tree structure of the avalanche generations. The recursive definition of the entangled states |Zk ⟩ generates a self-similar hierarchy whose scaling exponents determine the continuous modular spectrum. We derive an explicit relation between the branching ratio, the overlap decay parameter p r = 1 − |η|2 , and the flow of weights on the fibres. This fractal-modular correspondence furnishes a concrete physical realisation of Tomita–Takesaki theory in a measurement context and suggests new invariants that characterise the algebraic Heisenberg cut. All constructions remain faithful to the avalanche-photodiode model of Van Den Bossche and Grangier (2025) while revealing a previously unnoticed fractal geometry underlying the type-III structure. K 1 S C Also available on Academia.edu: https://www.academia.edu/attachments/133072812/download_file?s=portfolio
Preprint: This work is a preprint, has not undergone peer review, and is made publicly available to establish a public scientific record. This work reports an experimental two-dimensional validation of the Dynamical Information Geometry (DIG) Twin-Sign protocol under fixed audit and regime constraints. Dynamical Information Geometry assigns an effective, state-dependent information geometry to quantum many-body dynamics via the curvature of a coarse-grained mutual-information profile. Earlier work introduced the theoretical framework (Part I), provided numerical evidence for sign-sensitive refraction effects (Part II), validated a minimal operational consequence on quantum hardware (Part III), established regime structure and operational necessity (Part IV), and formulated the intrinsic tensorial nature of information curvature in dimensions greater than one (Part V). The present work occupies a strictly operational and consolidating role within the DIG series. It does not introduce new theoretical postulates, derivations, observables, or interpretive claims. Instead, it tests whether the multidimensional, projection-based structure articulated in DIG Part V remains internally consistent when subjected to the mandatory audit logic and regime language fixed in DIG Part IV. Twin-Sign tests are performed in two spatial dimensions along two non-collinear directions, without privileging any spatial axis. All results are derived exclusively from raw measurement counts under fixed audit invariants. No readout mitigation, post-selection, fitting procedures, or cross-platform normalization is applied. In addition to positive validation runs, the dataset includes a structurally defined negative control (One-Sided Twin), which is shown to fail deterministically. Across sampled values of the lens-strength protocol parameter, the observed outcomes recover the finite onset–window–breakdown regime structure introduced in DIG Part IV. This behavior is reported descriptively, without invoking universality, sharp thresholds, or new scaling claims. The results demonstrate that the operational and structural content consolidated in DIG Part IV persists under the multidimensional extension of the framework. Directional asymmetries and null projections are observed and are consistent with the tensorial formulation of information geometry in dimensions greater than one. All results reported in this work are fully reproducible from the accompanying public dataset without additional assumptions. Related DIG materials DIG Part I (theoretical framework):https://doi.org/10.5281/zenodo.17983399 DIG Part II (numerical validation):https://doi.org/10.5281/zenodo.17982058 DIG Part III (hardware validation):https://doi.org/10.5281/zenodo.18100057 DIG Part IV (regime structure and operational necessity):https://doi.org/10.5281/zenodo.18202656 DIG Part V (multidimensional formulation):https://doi.org/10.5281/zenodo.18210315 DIG Part VI (dataset):https://doi.org/10.5281/zenodo.18226272 DIG Part VII (dataset):Dataset for Experimental Validation of Dynamical Information Geometry in Two Dimensionshttps://doi.org/10.5281/zenodo.18309526 DIG — N1 (definitions, scope, and non-equivalences):https://doi.org/10.5281/zenodo.18001179 DIG — N2 (definitions/interpretation/non-equivalences) https://doi.org/10.5281/zenodo.18380514 - Correspondence regarding this work may be directed to:kaya@cab-film.com
This work introduces the Modular Superselection Framework, a mathematical and conceptual proposal based on the assumption that a quantum system may be structured into modular sectors labeled by a conserved quantity defined modulo an integer MMM. In this framework, observable states must remain within a single sector, and transitions between sectors are restricted by the modular selection rule ΔQ≡0(modM)\Delta Q \equiv 0 \pmod{M}ΔQ≡0(modM). The work develops this idea axiomatically, presenting formal definitions, structural propositions, and examples illustrating its internal coherence. Unlike standard superselection rules derived from symmetry groups or environmental decoherence, the modular architecture is introduced as a primitive structural feature of Hilbert space. Its goal is not to describe existing physics, but to provide a mathematically consistent, falsifiable, and expandable theoretical scaffold for future investigations in quantum foundations, mathematical physics, and the study of constrained quantum systems.
This record contains the preprint version of the article “PhaseGeometry Z₂: Decoherence, Quantum Darwinism and Branching in a Binary Phase Medium”, submitted to a peer-reviewed journal. The work develops a unified quantum–foundational layer of the PhaseGeometry Z₂ programme, in which a single real scalar field with a Z₂-symmetric double-well potential plays a dual role:– as a cosmological substrate, whose homogeneous broken phase sources dark-energy–like expansion and whose inhomogeneities generate dark-matter–like halo profiles and static black holes with binary phase hair;– as a quantum phase medium, within which systems, apparatuses, observers and classical branches are all realised as patterns of the same underlying field. On the physical side, the article introduces a finite Z₂ chain evolving under a transverse-field Ising Hamiltonian as a fully explicit “toy Universe” of the phase medium. Within this model it demonstrates, under strictly unitary dynamics:– rapid decoherence of macroscopically distinct binary events;– saturation of the subsystem entropy at classical values;– the emergence of a clear Quantum Darwinism plateau in the mutual information between the system and environment fragments, signalling redundant objective records. On the conceptual side, the work formulates an ontology in which:– observers are stable, information-rich patterns in the phase medium;– branches are identified with disjoint amplitude humps in Z₂ configuration space;– delayed-choice and quantum-eraser scenarios are reinterpreted in a purely unitary, medium-based language. Taken together, these results show that the PhaseGeometry Z₂ phase medium can simultaneously underpin dark-sector phenomenology and the emergence of classical observers and histories, providing a concrete Everett–Zurek–style measurement framework in a closed Universe. A related two-part foundations set (“PhaseGeometry Z₂ Foundations IV–V: Phase medium, observers, decoherence and Quantum Darwinism”, DOI: 10.5281/zenodo.17772072) presents a more modular split between ontology (IV) and numerical implementation (V). This preprint is part of the PhaseGeometry Z₂ Core Package v1.0: “Binary Phase, Dark Energy and Dark Matter” (series DOI: 10.5281/zenodo.17652957). All materials are released under the Creative Commons Attribution 4.0 International (CC BY 4.0) licence.
Etudes calculatoires de l'intrication et de la contextualité quantiques dans la perspective de leur vérification formelle Bien que les ordinateurs quantiques actuels se limitent à l'utilisation de quelques qubits, les fondations de la programmation quantique se sont développées au cours des 20 dernières années. Ces fondations ont été théorisées dès les années 80 mais la complexité de leur mise en œuvre a rendu ces pistes inaccessibles jusqu'à très récemment. Dans ce contexte, l'objectif de cette thèse est de contribuer à l'adaptation des méthodes de spécification formelle et de vérification déductive des programmes classiques aux programmes quantiques. Je présente donc mes contributions à l'étude des propriétés quantiques dans le but de les formaliser. J'étudie en particulier l'intrication quantique et la contextualité quantique. Ces propriétés permettent de classifier les états et les protocoles quantiques, et en particulier de les différencier des états et protocoles classiques. Mon étude de l'intrication est fondée plus spécifiquement sur l'évolution de l'intrication au cours de deux algorithmes quantiques : l'algorithme de Grover et la transformée de Fourier quantique. Pour quantifier l'intrication le long de ces algorithmes, j'utilise des polynômes de Mermin, qui ont l'avantage de pouvoir être implémentables dans des ordinateurs quantiques réels. Mon étude de la contextualité s'appuie, elle, sur des géométries finies représentant des expériences, qui sont dites contextuelles quand aucune théorie classique non-contextuelle ne peut en prédire les résultats. Ces géométries sont associées à des espaces polaires symplectiques binaires dont nous explorons la structure, avec comme objectif d'utiliser cette structure pour obtenir des intuitions sur les protocoles quantiques en utilisant la contextualité. L'étude de ces propriétés a conduit à des conjectures intéressantes que nous avons commencé à formaliser dans des environnements de preuve tels que Coq et Why3, mais qui sont laissées en perspective car ces travaux n'ont pas encore abouti.
Preprint: This work is a preprint, has not undergone peer review, and is made publicly available to establish a public scientific record. This work formulates the Dynamical Information Geometry (DIG) framework in spatial dimension d > 1. Dynamical Information Geometry assigns an effective, state-dependent geometry to quantum many-body systems. Earlier work focused on the one-dimensional setting, where the structure of the framework can be developed in its simplest and most transparent form. This manuscript states the multidimensional formulation of DIG explicitly. In dimensions higher than one, information-induced geometry is inherently multidirectional and cannot be reduced to a single scalar quantity associated with a preferred coordinate. Directionality and projection therefore enter as intrinsic structural aspects of the framework. The presentation is theoretical and structural in nature. It fixes the multidimensional formulation of Dynamical Information Geometry at the level of the framework itself and is not tied to particular realizations, implementations, or test scenarios. For reference and context, related DIG manuscripts include: DIG Part I (theoretical framework):https://doi.org/10.5281/zenodo.17983399 DIG Part II (numerical validation):https://doi.org/10.5281/zenodo.17982058 DIG Part III (hardware validation):https://doi.org/10.5281/zenodo.18100057 DIG Part IV (regime structure and operational necessity):https://doi.org/10.5281/zenodo.18202656 DIG Part VI (scaling 1D hardware validation): https://doi.org/10.5281/zenodo.18226369 DIG Part VII Dynamical Information Geometry:Two-Dimensional Twin-Sign Validation under Fixed Audit and Regime Constraintshttps://doi.org/10.5281/zenodo.18313460 DIG — N1 (definitions, scope, and non-equivalences):https://doi.org/10.5281/zenodo.18001179 DIG — N2 (definitions/interpretation/non-equivalences) https://doi.org/10.5281/zenodo.18380514 This manuscript introduces no new theoretical framework beyond the existing DIG construction. Its role is to make explicit the multidimensional form implied by that construction and to fix its structural consequences at the level of the framework itself. Correspondence regarding this work may be directed to:kaya@cab-film.com