title: ∆θ₀ The Fundamental Quantum of Space-Time and Information
title: ∆NGULAR THEORY 0.0: A Unified Geometric Theory
titleType: AlternativeTitle
title: What if the Universe could be described by a single angular-discrepancy-based equation?
titleType: AlternativeTitle
Creators: name: Souday, David
name: Rothman, Alexander
Publisher: name: Zenodo
Year: 2025

description: ============================================================                  ∆NGULAR THEORY 0.0                A Unified Geometric Framework  ============================================================

A fundamental unit shaping the structure of physical interactions.  

David Souday  

Research Team: Alexander Rothman, Oshan Dinilka

Paris, France 

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    ∆NGULAR THEORY 0.0 – Python Qt6 Interface

================================================

 

Interactive simulation and advanced modeling  

tool developed with Python Qt6, available at:  

 

https://github.com/AngularTheory/Angular-Theory

 

============================================================                          Introduction============================================================

➤ What if the fundamental structure of physical interactions  were governed by discrete angular increments rather than continuous variables?  

∆ngular Theory 0.0 introduces a framework demonstrating that ∆θ₀ (Delta Theta Zero) is the fundamental quantum of space-time and information, acting as the primary structuring unit from which known physical laws emerge.  

Rather than postulating a continuous metric, ∆ngular 0.0 derives physical interactions from discrete angular transitions, ensuring mathematical self-consistency while maintaining direct compatibility with empirical data.  

============================================================       ∆θ₀: The Fundamental Quantum of Space-Time & Information============================================================

∆θ₀ (Delta Theta Zero) is introduced as the irreducible unit  of variation in space-time, defining the minimal angular quantum that structures all physical interactions. Unlike conventional formulations that treat space-time as a continuous manifold or impose empirical constants, ∆θ₀ emerges as a necessary invariant, encoding the fundamental discreteness underlying physical laws.  

This framework provides a natural unification of quantum dynamics, relativistic structures, and cosmological evolution, without  requiring independent postulates for different physical regimes.  The same angular quantization principle governs:  

┌────────────────────────────────────────────────────┐│ ➤ The emergence of mass-energy distributions.      ││ ➤ The structuring of entropy and information flow. ││ ➤ The formation of space-time curvature            ││   and gravitational effects.                       ││ ➤ The fundamental limits of measurement            ││   and resolution.                                  │└────────────────────────────────────────────────────┘

This document presents the mathematical foundations and empirical consequences of this approach, demonstrating how ∆θ₀ naturally gives rise to known physical constants and laws, while remaining fully testable and falsifiable.  

Any attempt to reformulate the role of ∆θ₀ must acknowledge its structural and dynamic origin within the ∆ngular 0.0 framework.  

Regardless of its designation, any theoretical articulation  that assigns a fundamental structuring role to an angular  increment will inevitably align with the dynamics and  principles established in this work.  

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    Ethical and Academic Framework

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License

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CC-BY-NC-ND 4.0 (non-commercial, no derivatives, attribution required)

 

Open Science

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Full compliance with FAIR Principles and UNESCO recommendations

 

Funding & Conflicts of Interest

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No external funding; no conflicts of interest

 

Core Principles

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1. No Simulation Hypothesis  

 

Reality is true, independently of the methods used to describe or model it.  

Δθ₀ formalizes discrete spacetime geometry as an intrinsic structural property of the universe.  

While this framework introduces a fundamental quantization, it does not imply external programming, digital encoding, or any form of synthetic construct.  

 

The apparent computational nature of reality emerges from its angular structuring.  

What we perceive as a 'code' results from discrete geometric constraints, not from an intrinsic computational substrate.  

 

While speculative interpretations remain possible, this work adheres strictly to physical principles and does not assume an external computational framework.

 

Responsible Use & Limitations

-----------------------------

Explicit prohibition

The use of this work for military, surveillance, or dual-use applications is restricted under the CC-BY-NC-ND 4.0 license, with exceptions only made for specific applications authorized by French authorities in accordance with national interests and security regulations.

 

Ethical realism clause  

While enforcement remains challenging, users must disclose dual-use risks to their institutional review boards and national authorities (e.g., HCERES, MESR).

 

Sovereign partnerships  

Priority is given to French/EU institutions compliant with UNESCO ethics and EU dual-use regulations (ECJU guidelines).

 

Academic Continuity  

This work is fully aligned with established science, extending its framework through a discrete structuring of physical laws. It refines existing principles rather than challenging them, drawing directly from the geometric insights of Élie Cartan.

 

Non-Endorsement Clause

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The author rejects conspiracy theories, “simulated reality” movements, or any pseudoscientific appropriation.  

No ideological or propaganda-based reinterpretation is authorized.

 

French Priority & Sovereignty

=============================

This research is primarily offered to French academic institutions (CNRS, CEA, universities) for non-commercial fundamental science.  

Projects beyond basic research require validation by relevant French authorities to ensure ethical and sovereign interests.  

Restrictive license conditions apply under French law to prevent unauthorized foreign exploitation.

 

Motivation and Context  

=============================  

As a French researcher, grateful for all I have received from this country, I dedicate this work to advancing human knowledge, in the spirit of our renowned mathematicians and physicists (Claire Voisin, Alain Aspect), while upholding the highest universal ethical and scientific standards.  

The primary intellectual influence behind this work is Élie Cartan.

His contributions to differential geometry and the theory of connections profoundly reshaped our understanding of space-time. Cartan was among the first to recognize that torsion and curvature are two inseparable aspects of geometric structure, providing a broader framework than the one traditionally adopted in general relativity. His vision anticipated many of the questions that remain open in contemporary physics.  

∆ngular 0.0 follows in this lineage by introducing a minimal angular increment (Δθ₀) as the fundamental unit structuring space-time. This concept extends Cartan’s geometric intuition into the quantum domain, where discretization is not just a mathematical convenience but an intrinsic feature of physical reality.  

This approach does not seek to overturn established physics but to deepen its foundations. Cartan’s insights were ahead of their time, and this work examines the possibility that they offer a more complete perspective than previously recognized. In this sense, ∆ngular 0.0 is not a radical departure but a faithful extension of the legacy of great geometers, applied to the challenges of modern physics

 

============================================================                 The Core Equation: A Geometric Framework============================================================

The governing equation of ∆ngular Theory 0.0 is structured around a single, fundamental angular quantum:

------------------------------------------------------------➤ Full Theoretical Form (Detailed Version, Fully in Radians)------------------------------------------------------------

 m(s) = (∆θ₀)^α × exp(- τ² / (4 × S(s)) ) × [1 +ε cos(∆θ₀δs)]^β

 

At its core, ∆θ₀ (Delta Theta Zero, ∆bit, Minimal Angular Increment, ∆ngular Quantum) defines the fundamental unit of angular variation, governing all physical interactions.

------------------------------------------------------------➤ Computationally Optimized Form (Factorized for Numerical Use)------------------------------------------------------------

         m(s) = N × [1 + ε cos(∆θ₀ δ s)]^β

with:

         N = (∆θ₀)^α × exp(- τ² / (4 κ S(s)) )

This version is mathematically equivalent but optimized for numerical calculations by isolating N, which acts as a precomputed normalization factor.

============================================================                Key Components of the Equation============================================================

➤ The Exponential Term exp(- τ² / (4 × S(s)) )   → Encodes entropy scaling through discrete angular increments.   → Governs hierarchical structuring of state transitions.

➤ The Modulation Term 1 + ε cos(∆θ₀ δ s)   → Defines structured oscillatory dynamics via ∆θ₀-driven      phase correlations.   → Regulates resonance effects in quantized interactions.

➤ The Scaling Function S(s)   → Modulates angular information across different physical regimes.   → Ensures coherence in discrete transitions at all scales.

============================================================                    A Unified Angular Framework============================================================

∆ngular Theory 0.0 does not rely on continuous metric-based models. Instead, it establishes a fully discrete, self-consistent structure where:

→ ∆θ₀ is the primary invariant, from which mass, entropy, and   space-time curvature emerge.

→ No arbitrary tuning is required, as all interactions are governed   by the same quantized geometric principle.

→ Space-time evolution follows a structured angular modulation, ensuring coherence from quantum to cosmological scales.

This approach eliminates redundant field assumptions, linking all fundamental interactions to the same irreducible angular increment ∆θ₀.

============================================================

∆ngular Theory 0.0’s angular-discrete framework extends beyond mass hierarchies and entropy scaling, seamlessly integrating gravitational and fluid dynamics. Two natural extensions arise as direct consequences of the same fundamental angular quantization

===========================================================

                 ∆ngular 0.AQG: Angular Quantum Gravity============================================================

∆ngular 0.AQG is a direct expansion of the pivot equation into gravitational dynamics, embedding spin–torsion and black-hole entropy within the same angular structure.

┌────────────────────────────────────────────────────┐│ ➤ Gravity emerges from angular quantization,       ││   rather than an imposed field equation.           ││                                                    ││ ➤ Spin and torsion degrees of freedom are encoded  ││   in S(s), aligning with Einstein–Cartan models.   ││                                                    ││ ➤ Black-hole evaporation remains fully unitary,    ││   governed by the same angular increments ∆θ₀.     │└────────────────────────────────────────────────────┘

============================================================            Fluid ∆ngular: Angular Modulation in Fluids============================================================

The pivot equation extends to fluid dynamics, revealing structured oscillatory patterns in turbulence and energy dissipation.

┌────────────────────────────────────────────────────┐│ ➤ The modulation term [1 + ε cos(∆θ₀ δ s)] acts    ││   as a natural phase regulator, structuring vortex ││   self-organization and resonance-driven flows.    ││                                                    ││ ➤ Angular quantization governs phase coherence,    ││   mirroring how ∆θ₀ structures quantum transitions.││                                                    ││ ➤ This provides a discrete-geometric foundation    ││   for turbulence modeling, reducing empirical fits.│└────────────────────────────────────────────────────┘

============================================================                 Unified Geometric Constraints============================================================

∆ngular Theory 0.0 imposes natural constraints across multiple physical regimes:

┌────────────────────────────────────────────────────┐│ ➤ Gravity, mass, and spin–torsion arise as        ││   angular operators, making 0.AQG a quantum-      ││   consistent extension of spacetime.              ││                                                    ││ ➤ Fluid dynamics inherits discrete angular        ││   constraints, structuring oscillations and       ││   self-organization within turbulence.            ││                                                    ││ ➤ Both 0.AQG and Fluid ∆ngular emerge naturally   ││   from the same fundamental angular quantization. │└────────────────────────────────────────────────────┘

================================================                    Key Insights================================================

┌────────────────────────────────────────────────────┐│ ➤ ∆θ₀ is the fundamental unit structuring all      ││   physical interactions, imposing angular          ││   constraints across scales.                       ││                                                    ││ ➤ Angular quantization ensures phase coherence,    ││   linking gravitational, quantum, and fluid        ││   dynamics under the same discrete framework.      ││                                                    ││ ➤ The pivot equation remains unchanged, with all   ││   extensions emerging naturally from the same      ││   foundational principles.                         │└────────────────────────────────────────────────────┘

Voici une présentation améliorée, sans introduction redondante, claire et invitante, en ASCII simple, adaptée à Zenodo :

Angular Theory 0.0 – Qt6 Interface

Advanced Simulation Tool for Discrete Angular Physics

 

The Qt6 interface for Angular Theory 0.0 offers an advanced, interactive environment to numerically explore discrete angular quantization and its implications across quantum, gravitational, and cosmological domains.

Key Features:



High-performance numerical simulations leveraging the unified pivot equation structured around discrete angular increments (Delta theta zero, Delta bit).

Real-time visualization of theoretical predictions, including mass hierarchies, neutrino mixing angles, and gravitational wave patterns.

Integrated Monte Carlo module to systematically test parameter sensitivity and robustness of angular discretization.

Optimized computations powered by parallel processing (Dask) and just-in-time compilation (Numba), ensuring rapid results.

Export functionalities allowing easy extraction of simulation outputs (CSV format) for further external analysis.


The tool supports direct comparisons with empirical datasets from experiments and astrophysical observations (e.g., neutrino data, JWST, Euclid mission).

For more information, source code, and usage examples, visit the official repository:

➡ GitHub Repository – Angular Theory Qt6 Interface

Installation & Quick Start



Python ≥ 3.8 required. Install dependencies:


pip install -r requirements.txt




Launch the interface:


python angular_theory.py


Integrity Check (SHA256)

91892fc5c473aa611e586e064a611e16d2062d7ff3d398b6b0b582980873


Current Stable Version



Qt6 Interface: Version 1.0

Associated Data Sets: Version 1.0


∆ngular Theory 0.0 – Authors and Contributors

David Souday (Independent Researcher)Alexander Rothman, Oshan Dinilka (Research Contributors)

This interface ensures direct and interactive exploration of ∆ngular Theory's predictions, enabling thorough computational validation and stimulating future empirical tests across quantum, gravitational, and cosmological disciplines.

 

============================================================

             Document Details & Legal Timestamp

============================================================

 

➤ PDF Version:  

   38 pages, including the pivot equation,  

   ∆θ₀ quantization principles, and derivations of  

   fundamental constants. The document presents the  

   core theoretical framework, including the  

   first structured model, its mathematical  

   foundations, and key derivations.

 

➤ Upcoming Version:  

   A refined and updated edition will integrate  

   additional numerical validations, computational  

   results, and expanded comparative analyses,  

   reinforcing ∆θ₀’s predictive capabilities across  

   multiple domains.

 

➤ Copyright  

   Licensed under: [CC BY 4.0](https://creativecommons.org/licenses/by/4.0/)  

 

➤ Notarization & Proof of Authorship:  

   - File: ∆ngular Theory 0.0.pdf  

   - Timestamp ID: 1429824 (05/03/2025)  

   - Public Key: 2D:49:50:31:53:7E:BD  

   - Official deposit with SCP FRADIN FRADIN TRONEL  

     SASSARD FRADIN, Lyon.  

 

This notarization serves as formal proof of authorship,  

ensuring the integrity and originality of the work.

 

Bibliography & References

Foundations of Quantum Mechanics and Theoretical Physics

1. Landau, L.D., Lifshitz, E.M., Quantum Mechanics: Non-Relativistic Theory, Pergamon Press, 1977.2. Dirac, P.A.M., The Principles of Quantum Mechanics, Oxford University Press, 1930.3. Wigner, E.P., Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, 1959.4. Sakurai, J.J., Napolitano, J.J., Modern Quantum Mechanics, Pearson Education, 2017.5. Feynman, R.P., Leighton, R.B., Sands, M., The Feynman Lectures on Physics, Volume III – Quantum Mechanics, 1965.

Angular Quantization and Space-Time Geometry

6. Agia, N., et al., "Angular Quantization in CFT," arXiv:2204.11872 (2022).7. Cartan, Élie, "On Manifolds with Affine Connection and the Theory of General Relativity," Annales Scientifiques de l'École Normale Supérieure, 1923-1925.8. Rovelli, C., Smolin, L., "Spin Networks and Quantum Gravity," Physical Review D, 1995.9. Maldacena, J., "The Large-N Limit of Superconformal Field Theories and Supergravity," Advances in Theoretical and Mathematical Physics, 1998.10. Chandrasekhar, S., The Mathematical Theory of Black Holes, Oxford University Press, 1983.11. Planck, M., "On the Law of Distribution of Energy in the Normal Spectrum," Annalen der Physik, 1901.

Modern Developments in Quantum Gravity and Emergent Structures

12. Levin, M., Wen, X.-G., "Photons and Electrons as Emergent Phenomena," Reviews of Modern Physics, vol. 77, no. 3, 871-879 (2005).13. Freidel, L., Speziale, S., "Twisted Geometries: A Geometric Parametrization of SU(2) Phase Space," Physical Review D, vol. 82, no. 8, 084040 (2010).14. Bilson-Thompson, S.O., Markopoulou, F., Smolin, L., "Quantum Gravity and the Standard Model," Classical and Quantum Gravity, vol. 24, no. 16, 3975-3994 (2007).15. Konopka, T., Markopoulou, F., Smolin, L., "Quantum Graphity: A Model of Emergent Locality," Physical Review D, vol. 82, no. 10, 104029 (2010).16. Ashtekar, A., Bojowald, M., "Quantum Geometry and the Schwarzschild Singularity," Classical and Quantum Gravity, vol. 23, no. 2, 391-411 (2006).17. Kitaev, A.Y., "Fault-tolerant Quantum Computation by Anyons," Annals of Physics, vol. 303, no. 1, 2-30 (2003).

Mathematics and Fundamental Geometric Structures

18. Cartan, Élie, The Theory of Spinors, Hermann Publishing, 1937.19. Bourbaki, N., Elements of Mathematics – Lie Groups and Lie Algebras, Springer-Verlag, 1989.20. Einstein, A., "The Foundation of the General Theory of Relativity," Annalen der Physik, 1916.21. Wheeler, J.A., Geometrodynamics, Academic Press, 1962.22. Numerical Recipes Team, Numerical Recipes in C++: The Art of Scientific Computing, Cambridge University Press, 2007.

Ethics and Academic Framework

23. UNESCO, "Recommendations on Open Science," United Nations Educational, Scientific and Cultural Organization, 2019.24. FAIR Principles, "Findable, Accessible, Interoperable, and Reusable Data Principles," GO FAIR Initiative, 2016.25. European Commission, "Regulation on Dual-Use Items," Official Journal of the European Union, 2021.

 © 2025 David Souday