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 Theory

 

A fundamental unit shaping the structure of physical interactions.

 

By David Souday & Alexander Rothman

 Paris, France 

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

▪ This document introduces ∆ngular Theory 0.0, a framework demonstrating that Δθ₀ (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.

────────────────────────────────────────────────────────────  Δθ₀: The Fundamental Quantum of Space-Time and Information────────────────────────────────────────────────────────────

Δθ₀ (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, Δθ₀ 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 Δθ₀ naturally gives rise to known physical constants and laws, while remaining fully testable and falsifiable

Any attempt to reformulate the role of Δθ₀ 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  

Military, surveillance, or dual-use applications are banned under CC-BY-NC-ND 4.0.

 

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

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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  

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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

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      Beyond Empirical Inputs: A Structured Approach  

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Modern physics has reached an exceptional level of precision,  

but it remains constrained by its foundational assumptions.  

 

Current models rely on 27+ free parameters and treat fundamental constants  

(G, c, ħ, Λ) as empirical inputs, lacking intrinsic derivation.  

 

These constants are measured, not explained.  

While their numerical values allow predictive accuracy,  

they leave open a fundamental question:  

 

➜ Are these values truly fundamental, or do they emerge from a deeper structure?  

 

Rather than assuming these constants as fixed inputs,  

∆ngular Theory 0.0 derives them from a single geometric principle:  

 

➜ Discrete angular increments encoded in a pivot equation,  

structuring mass hierarchies, entropy scaling, and cosmic evolution.  

 

The Core Equation: A Geometric Framework

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

                    m(s) = (Δθ₀)^α × exp[ −τ² / (4 × S(s)) ] × [1 + ε cos(2π δ s)]^β        At its core, Δθ₀ (Delta Theta Zero, Δbit, Minimal Angular Increment, ∆ngular Quantum)defines the fundamental unit of angular variation, governing all physical interactions.

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Key Components of the Equation

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▶ 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(2π δ s)   □ Defines structured oscillatory dynamics via Δθ₀-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.

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A Unified Angular Framework

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

✔ Δθ₀ 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 Δθ₀.

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∆ngular Theory 0.0 redefines fundamental physics through angular quantization, offering a structured alternative to traditional formalisms.

 

Pivot Equation and Its Natural Extensions

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For comparative analysis, the pivot equation can also be expressed in its π-based form:

            m(s) = (Δθ₀)^α × exp[ −π² / (4 × S(s)) ] × [1 + ε cos(2π δ s)]^β

Both formulations share the same underlying structure; however:

✔ The τ-based representation provides a cycle-normalized perspective, particularly suited for oscillatory and rotational dynamics.✔ The π-based representation aligns with classical periodic structures, offering alternative insights into discrete angular transitions.

This dual formulation enhances the adaptability of ∆ngular Theory 0.0, bridging classical periodicity with the intrinsic discreteness of angular quantization.

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Natural Extensions of the Pivot Equation

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∆ngular Theory 0.0’s angular-discrete framework extends beyond mass hierarchies and entropy scaling, naturally incorporating gravitational and fluid dynamics. Two primary extensions emerge:

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∆ngular 0.AQG: Angular Quantum Gravity

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∆ngular 0.AQG is not an independent theory but 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 intrinsically encoded in S(s), aligning with Einstein–Cartan approaches without additional assumptions.✔ Black-hole evaporation remains fully unitary, governed by the same angular increments Δθ₀ structuring mass transitions at quantum scales.

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Fluid ∆ngular: Angular Modulation in Fluid Dynamics

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A second direct extension applies the pivot equation to fluid dynamics, revealing structured oscillatory patterns in turbulence and energy dissipation.

✔ The modulation term [1 + ε cos(2π δ s)] serves as a natural phase regulator, capturing vortex self-organization and resonance-driven energy transfers.✔ Angular quantization governs phase coherence in turbulent structures, mirroring how Δθ₀ structures mass transitions in quantum physics.✔ This framework provides a discrete-geometric foundation for turbulence modeling, reducing reliance on empirical parameters.

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Key Insight: Unified Geometric Constraints

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∆ngular Theory 0.0’s angular-discrete formalism imposes natural constraints on multiple physical regimes, ensuring:

✔ Gravity, mass, and spin–torsion arise as angular operators, positioning 0.AQG as a quantum-consistent extension of spacetime dynamics.✔ Fluid dynamics inherits discrete angular constraints, structuring oscillations and self-organization within turbulence and wave propagation.

Both 0.AQG and Fluid ∆ngular are not independent formulations but emergent consequences of the same fundamental angular quantization—reinforcing Δθ₀ as the universal structuring invariant.

Δθ₀ is not introduced as an auxiliary hypothesis but as the absolute structuring unit from which observed physical properties emerge. If nature is fundamentally discrete, then Δθ₀ is the quantum of this discreteness, imposing angular constraints on all scales.

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         Consolidation & Future Developments  

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This document provides a structured theoretical foundation,  

offering a self-consistent framework based on angular discreteness.  

 

As research advances, the continued integration of Δθ₀  

across multiple domains will shape future developments,  

validating its role as a fundamental structuring principle.  

 

Beyond its direct applications, this work aims to foster  

new independent investigations, encouraging researchers  

to explore Δθ₀ within their own theoretical landscapes.  

 

The next phase lies in rigorous computational validations,  

falsifiable predictions, and interdisciplinary crossovers.  

 

Independent contributions will be crucial in refining  

and expanding this approach, bridging established paradigms  

with the emergent structure of angular quantization.  

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         Document Details & Legal Timestamp  

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✔ PDF Version: 38 pages, including the pivot equation,  

   Δθ₀; ∆bit interpretations, and the derivation of fundamental constants.  

✔ Copyright: © 2025 David Souday.  

   Licensed under: 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 provides formal proof of authorship,  

ensuring the integrity and originality of the work.