Delta Core & Non-Singular Topologies

Finite-Radius Core Regularizations, Bounded Invariants & Topological Defect Stabilization

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Research Root · FQFT Frontier · Delta Core Regularization

Relation to FQFT: This node is a downstream branch of Fractal Quantum Field Theory (FQFT), proposing a specific field-structure formalization route for regularizing point-like interactions on discrete multiscale lattices.

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1. Domain Problem

In standard continuum field formulations, point-like interactions and infinitesimal source terms introduce ultraviolet divergences and mathematical singularities ($1/r$ gravitational potentials, self-energy divergences). The domain problem addressed by the Delta Core framework is to replace point-like singularities with a topologically bounded, scale-invariant core geometry within discrete metric-measure spaces $(X_s,d_s,{\mu }_s)$.

2. Position in the Scientific Spine

3. Theory Derivation

The Delta Core is derived by applying scale-dependent cutoff operators and discrete Dirichlet forms ${\mathcal{E}}_s$ to localized source terms. Rather than allowing metric distances $d_s(x,y)\to 0$ unboundedly, the core topology enforces a minimum finite-fiber radius $r_0>0$, bounding energy densities and field gradients.

4. Formal Kernel

The Delta Core mapping within FQFT is defined by the relational operator:

$${\mathcal{Q}}_{\text{FQFT}}:(\mathcal{F},\mathcal{S},\mathcal{O},\mathcal{I})⟶{\mathcal{R}}_{\text{field}}$$

where:

• $\mathcal{F}$: Field structure centered around the localized Delta Core manifold.
• $\mathcal{S}$: Spectral and fractal constraints bounding the core dimensions and modal eigenvalues.
• $\mathcal{O}$: Observer-bound measurement mapping core interactions and energy distributions.
• $\mathcal{I}$: Invariant preservation enforcing topological charge conservation within the core boundary.
• ${\mathcal{R}}_{\text{field}}$: Output field configuration satisfying bounded gradient conditions $\parallel \mathrm{\nabla }\mathrm{\Phi }{\parallel }_{L^{\mathrm{\infty }}}<\mathrm{\infty }$.

5. Established Scientific Grounding & Lineage

This theoretical construct relates conceptually to established mathematical and physical paradigms:

• Quantum Field Regularization: Pauli-Villars, dimensional regularization, and lattice cutoff schemes.
• Topological Solitons & Defects: Skyrmions, vortices, and non-singular finite-energy solutions.
• Metric-Measure Dirichlet Spaces: Bounded Sobolev spaces $H^1(X,d,\mu )$ on metric graphs.

6. Proposed Fabric Model

Within the Science of Fabric Reality, the Delta Core functions as a stable, localized topological knot. It provides a structural foundation for particle-like field excitations without generating singular infinities, serving as a formal hypothesis for discrete-to-continuum field limits.

7. Physics Research Boundary

Epistemic Demarcation

This page presents an authorial theoretical extension and research program. It relates to established domains for comparison and terminology, but it is not presented as standard academic physics, peer-reviewed consensus, or structurally validated physical law. Independent mathematical review, formal proof verification in Lean 4, simulation testbeds, and external critique remain required.

8. Canonical Continuations

Domain	Resource	Focus
FQFT Master Node	Fractal Quantum Field Theory	Parent framework and scale-recursive field grammar
Field Equations	Fabric Field Equation (FFE)	Constrained variational stationarity and saddle-point system
Topological Charges	Monopole Theory	Isolated topological charges and homotopy stability
Microscopic Architecture	Microscopic Sub-Index	Local/global split compiler and finite fiber convergence
Formal Mathematics	Mathematics Gateway	Machine-verified Lean 4 proofs and theorem catalog

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