UNNS Field Explorer
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Topological Field Theory Visualization
UNNS Field Explorer
UNNS as a substrate: meaning, potential applications, and topological impact — with interactive visuals and explanatory micro-animations.
Overview
The UNNS project treats Unbounded Nested Number Sequences as a computational and number-theoretic substrate for discrete fields. This page reflects on the project’s trajectory, its connections to discrete exterior calculus, finite element exterior calculus, and an emerging topological field theory built from recursion.
Core idea
We frame UNNS as a discrete medium: recurrence coefficients act as connection data, echo residues are curvature quanta, and nested lattices implement algebraic embeddings (e.g. $\mathbb{Z}\subset\mathbb{Z}[i]\subset\mathbb{Z}[\omega]$). UNNS inletting — the formal rule by which external data seed the nest — is the canonical interface to physics and number theory.
Inletting is the morphism that maps finite external samples into the UNNS substrate while preserving recurrence compatibility and stability thresholds.
General reflection
UNNS as an emergent substrate: a place where arithmetic structure, discrete geometry, and quantum-like field observables intersect. Rather than a single theorem, UNNS acts as an experimental stage — a pattern gallery that stimulates rigorous follow-up work combining FEEC/DEC, algebraic number theory, and computational experiments.
In this spirit, UNNS serves three roles:
- On-ramp: presents striking numeric and geometric patterns that invite scrutiny.
- Substrate: provides a discrete scaffold to express curvature, holonomy, and topological charges.
- Bridge: links arithmetic invariants (Gauss/Jacobi sums, cyclotomic constants) to discrete PDE discretizations.
UNNS is not a finished theory; it is an attractive, fertile substrate. The project’s strength is in creating a shared visual & computational language so mathematicians, numerical analysts, and physicists can collaborate.