Recursive Superpositional Ontology (RSO) Laboratory

🎉 PUBLISHED RESEARCH 🎉

🔊 Listen to the RSO Deep Dive | 📄 Read the Full Paper on Academia.edu

"Recursive Superpositional Ontology: A Computational Framework for Contradiction-Preserving Logic and Reality Modeling" by Gregory Betti is now available for the global research community.

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Welcome to the RSO Laboratory, an open repository for exploring the metaphysical and symbolic logic framework introduced in Recursive Superpositional Ontology: A Computational Framework for Contradiction-Preserving Logic and Reality Modeling.

This repository accompanies the scientific paper written by Gregory Betti (Betti Labs) and contains reproducible code, visualisations, and documentation needed to investigate the Recursive Superpositional Ontology (RSO). The lab is organised around the idea that reality is not composed of resolved states but of recursive contradictions captured by the operator ⊕ and its associated contradiction field Ξ.

Philosophy

RSO begins from a radical ontological axiom: every property (x) coexists with its negation (¬x). Rather than treating the contradiction (x ∧ ¬x) as an impossibility, RSO elevates it to a fundamental ingredient of being. The universe is described as a superposition of every predicate and its opposite, generating a network of Ξ‑nodes that constantly loop between states. This recursion is not a failure of logic; it is the engine of existence.

Key Principles

• Contradiction‑preserving superposition (⊕): A binary operator that combines a predicate with its negation without resolving their opposition. Iterating ⊕ generates a Ξ attractor, a symbolic structure containing all variations of (x) and (¬x).
• Ξ fields: The self‑referential sets resulting from the ⊕ operation. Each Ξ contains the predicate, its negation, their conjunction, and further nested superpositions. Ξ‑graphs visualise these structures as loops with tension arrows.
• Recursion as reality: Time, space, matter, and consciousness emerge from the velocities and orientations of Ξ cycles. Nothing ultimately collapses—every collapse is a local projection of a deeper recursion.

For philosophical context, RSO resonates with ideas from Heraclitus (unity of opposites and flux), Daoist yin–yang (interdependence of contrary forces), and modern paraconsistent logics that reject explosion in the presence of contradictions【800088766790706†L41-L54】. It also echoes the many‑worlds interpretation of quantum mechanics, which removes wave function collapse and treats the universe as an ever‑branching superposition【751751222457522†L457-L464】. These influences underscore the plausibility of a reality built on recursive contradiction.

Contents

RSO-Lab/
├── figures/            # PNG and PDF diagrams generated from code
├── notebooks/          # Jupyter notebooks for interactive exploration
├── src/                # Python modules implementing RSO concepts
├── README.md           # This file
├── requirements.txt    # Python package requirements
└── …

figures/

The figures directory contains reproductions of the key diagrams used in the paper. They are generated programmatically by src/make_figures.py and include:

• xi_one_X.(png|pdf) – one‑predicate Ξ graph showing the loop between (x), (¬x), and the contradictory state (x∧¬x).
• xi_two_X_Y.(png|pdf) – two‑predicate Ξ² graph illustrating the richer structure of combining predicates (x) and (y).
• contradiction_lattice.(png|pdf) – lattice of the four truth values (True, False, Both, Neither) highlighting the legitimacy of the “Both” state in paraconsistent reasoning.
• recursive_orbit.(png|pdf) – continuous oscillation representing a recursive orbit.
• hot_cold_simulation.(png|pdf) – discrete toy model toggling between hot and cold states.

src/

• xi.py – Implements the XiOscillator class for discrete oscillations, the XiSymbolic class for symbolic predicates using SymPy, and the xi_operator function to build finite approximations to Ξ fields.
• make_figures.py – Generates all diagrams in the figures directory. It uses only matplotlib primitives so that no external graph library is required.

notebooks/

• xi_simulation.ipynb – A short Jupyter notebook acting as a Ξ playground. It demonstrates how to create an oscillator, iterate its states, and construct symbolic contradiction fields.

Testable Predictions

The RSO framework makes several qualitative predictions that can be investigated computationally:

1. Stability thresholds: A Ξ attractor remains stable only if the recursion is continued indefinitely. If the oscillation is interrupted or one state is favoured, a “collapse” occurs. Simple models, such as the hot/cold simulation in this repository, can explore how long it takes for such interruptions to occur under perturbations.
2. Emergent time: The period of oscillation in a Ξ cycle defines an emergent temporal scale. Simulations of coupled oscillators may exhibit phase synchronisation that resembles thermodynamic or causal arrows of time.
3. Composite attractors: Combining predicates via Ξ² should yield richer dynamics, potentially displaying quasi‑periodic or chaotic behaviour. One can numerically explore these multi‑predicate systems by extending XiOscillator to more than one dimension.

Getting Started

Installation

1. Clone or download this repository:

   git clone https://github.com/Betti-Labs/rso-framework.git
   cd rso-framework

2. Install the required dependencies:

   pip install -r requirements.txt

3. (Optional) Install in development mode:

   pip install -e .

Quick Start

Command Line Interface

The RSO framework includes a comprehensive CLI for easy interaction:

# Run interactive demo
python src/cli.py demo

# Generate oscillation sequence
python src/cli.py oscillate --steps 20 --initial true

# Create symbolic Xi attractor
python src/cli.py symbolic --predicate X --depth 3 --validate --verbose

# Run formal verification
python src/cli.py verify

# Generate performance benchmarks
python src/cli.py benchmark

# Generate all figures
python src/cli.py figures

Python API

from src.xi import XiOscillator, XiSymbolic, xi_operator

# Create and run oscillator
oscillator = XiOscillator(True)
history = oscillator.iterate(10)
print(f"Oscillation: {history}")

# Create symbolic predicate and Xi attractor
predicate = XiSymbolic('Consciousness')
attractor = xi_operator(predicate, depth=2)
print(f"Xi attractor has {len(attractor)} expressions")

# Validate the attractor
from src.xi import validate_xi_attractor
validation = validate_xi_attractor(attractor, predicate)
print(f"Validation passed: {validation['validation_passed']}")

Jupyter Notebooks

Launch the interactive notebook for experimentation:

jupyter notebook notebooks/xi_simulation.ipynb

Advanced Usage

Formal Verification

from src.formal_proofs import run_formal_verification

results = run_formal_verification()
print("Verification results:", results)

Quantum Bridge

from src.quantum_bridge import QuantumXiState

# Create quantum superposition state
quantum_state = QuantumXiState(alpha=0.6, beta=0.8)
print(f"P(x) = {quantum_state.probability_x():.3f}")

# Time evolution
evolved = quantum_state.evolve(time=3.14, frequency=1.0)
print(f"Evolved P(x) = {evolved.probability_x():.3f}")

Performance Benchmarking

from benchmarks.performance_suite import RSOBenchmarkSuite

suite = RSOBenchmarkSuite()
summary = suite.run_comprehensive_benchmark()
print(f"Average execution time: {summary['avg_execution_time']:.6f}s")

We hope these tools help you delve deeper into the paradoxical yet structured world of Recursive Superpositional Ontology.

📚 How to Cite

If you use the RSO framework in your research, please cite the published paper:

APA Style:

Betti, G. (2025). Recursive Superpositional Ontology: A Computational Framework for Contradiction-Preserving Logic and Reality Modeling. Academia.edu. https://www.academia.edu/143089984/

BibTeX:

@article{betti2025rso,
  title={Recursive Superpositional Ontology: A Computational Framework for Contradiction-Preserving Logic and Reality Modeling},
  author={Betti, Gregory},
  journal={Academia.edu},
  year={2025},
  url={https://www.academia.edu/143089984/Recursive_Superpositional_Ontology_A_Computational_Framework_for_Contradiction_Preserving_Logic_and_Reality_Modeling}
}

Software Citation:

Betti, G. (2025). RSO Framework: Recursive Superpositional Ontology (Version 1.0.1) [Computer software]. GitHub. https://github.com/Betti-Labs/rso-framework