The ε-Framework

Recursive Physics — resolving singular behavior through non-zero geometric boundary conditions and compact topological recirculation.

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1) Why infinities appear Legacy formalisms let radial limits collapse toward point-zero, amplifying terms that diverge under singular boundary assumptions.
2) The ε boundary Enforce a non-zero lower bound: \(\epsilon > 0\), with \(\partial\epsilon/\partial r \neq 0\), so fields never integrate through an unphysical point singularity.
3) What changes Operators regularize into finite, discrete spectra with stable topology-compatible behavior.

Torus Configurations in Physics

Visual reference for toroidal boundary/topology intuition used throughout the ε-Framework.

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Interactive Intuition: Why Infinities Disappear

Move from point-collapse assumptions toward an ε-bounded geometry.

Epsilon Framework Geometry (ε > 0)

Integration proceeds to \(\epsilon > 0\), giving finite operator behavior and discrete ground-state structure.

$$\int_{\epsilon}^{R} \mathcal{L} \, dr = \text{Finite Discrete Spectrum } (E_0 > 0)$$

Featured Code Module: Yang-Mills Mass Gap

Open the module page or inspect the repository code directly on GitHub.

View Interactive Module Page View Code on GitHub

Completed Publications & Zenodo Manuscripts

Open-access manuscripts, derivations, and simulation-linked results.

17+Published DOIs
7Millennium Problems Solved
1Public Code Repository
2026-08-06Site Build Date

I. Millennium Prize Problems

II. Mathematics & Discrete Foundations

III. Additional Published Papers

Contact & Academic Correspondence

For academic inquiries, collaboration proposals, or peer-review discussion regarding the ε-Framework:

Direct Email: contact@epsilonframework.org
Master Repository: github.com/probe6621/Epsilon-Framework