Zero–Infinity Algebra — Demonstrations

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What Is This?

These notebooks demonstrate Zero–Infinity Algebra (ZIA), a conservative extension of ℝ that preserves distinguishability across singularities where standard arithmetic collapses distinct values into identical representations.

The core problem: stabilisation techniques (clipping, flooring, capping) are necessary for numerical stability, but they destroy information. Two different computational histories can become indistinguishable after stabilisation, even when the underlying difference remains meaningful.

ZIA provides a principled way to preserve that information.

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Notebooks

Notebook	Domain	What It Shows
zia-ml_1_4.ipynb	Machine Learning	LLR clipping collapses distinct evidence histories; a shadow channel restores identifiability (AUC 0.5 → 0.81)
zia-qm-Pipeline_v7_2.ipynb	Inference Pipelines	Near-zero seam + clipping destroys comparative ratios; ZIA preserves them
zia-cfd_1_0.ipynb	CFD / Reactive Transport	Positivity flooring makes different runs look identical; ZIA-shadow exposes hidden differences

Direct Binder Links

• Launch ML notebook
• Launch QM Pipeline notebook
• Launch CFD notebook

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The Algebra (One-Minute Version)

Extended numbers: ∞ₐ ⊕ r where a, r ∈ ℝ

a / 0    =  ∞ₐ         (expansion)
r / ∞ₙ   =  0          (collapse)  
0 / 0    =  0          (convention)

∞ₐ / ∞ₙ  =  a / b      ← ratios survive

Division is total. Commutative ring. Irreversibility is structural.

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Running Locally

git clone https://github.com/MitchellQuinn/zia-examples.git
cd zia-examples
pip install -r requirements.txt
jupyter notebook

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Citation

These notebooks accompany:

Quinn, M. (2026). Zero–Infinity Algebra: Preserving Distinguishability Across Singularities. Preprint v0.5.13.
DOI: 10.5281/zenodo.18280702

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License

• Code (*.py): MIT License
• Documentation & Notebooks (*.ipynb, *.md): CC-BY-4.0

See LICENSE for details.