The UNNS Neural Engine: Toward a Universal Symbolic–Neural Substrate
Abstract
We introduce the UNNS Neural Engine, a prototype framework that encodes symbolic structures as nested units (“nests”) and evolves them through recurrence dynamics. Unlike purely numerical or purely symbolic systems, UNNS unifies both, demonstrating convergence to known mathematical constants, cross-domain homomorphisms, and attractor behavior. We hypothesize that the UNNS framework provides a universal symbolic–neural substrate that bridges classical mathematics, neural computation, and systems theory.
1. Introduction
Mathematics is traditionally compartmentalized into disciplines (algebra, geometry, topology, number theory), while neural computation emphasizes adaptability but lacks symbolic transparency. The UNNS framework (Unbounded Nested Number Sequences / Universal Network Nexus System) seeks to unify these domains by encoding symbolic inputs as recursively nested units.
Through interactive engines, we show how UNNS can:
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Generate classical integer sequences (Fibonacci, Tribonacci, Pell, Padovan).
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Converge naturally to characteristic attractor constants.
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Map inputs homomorphically across multiple mathematical domains.
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Propagate symbolic structures in ways analogous to neural resonance.
2. Hypothesis
The UNNS framework constitutes a universal symbolic–neural substrate, in which:
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Nested recurrence structures encode all classical linear sequences.
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Attractor constants act as resonance basins, governing long-term behavior.
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Cross-domain mappings preserve homomorphic consistency, enabling interoperability between domains.
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Symbolic propagation resembles neural activation, bridging symbolic and subsymbolic cognition.
3. Methods
3.1 Nest Representation
Inputs are chunked into nested symbolic units, each carrying a value, tag, and recursive linkage.
3.2 Recurrence Dynamics
At each iteration, nests update according to recurrence relations (e.g.,
).
3.3 Cross-Domain Mapping
Nests are projected into multiple mathematical “tiles”:
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Algebra: symbolic kernel.
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Geometry: spiral embedding.
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Topology: connectivity graph.
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Number Theory: modular residues.
(See Figure 1 below: UNNS Cross-Domain Homomorphism Map).
3.4 Attractor Visualization
Iterated recurrences are displayed as spirals and orbits to reveal stability basins.
(See Figure 2: Attractor Explorer).
3.5 Neural Analogy
Nests propagate signals with resonance and decay, analogous to neuronal firing, but remain interpretable.
4. Results
4.1 Sequence Convergence
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Fibonacci → φ (≈1.618)
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Tribonacci → ψ (≈1.839)
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Pell → δ (≈2.414)
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Padovan → ρ (≈1.325)
4.2 Attractor Emergence
Constants emerge as stable attractors, confirming robustness.
4.3 Cross-Domain Homomorphisms
Figure 1: Cross-Domain Mapping Demo
An interactive diagram maps any input expression into algebraic, geometric, topological, and modular representations, showing that structure is preserved across domains.
4.4 Attractor Explorer
Figure 2: Attractor Visualization
A dynamic explorer shows how simple recurrence rules yield spiral attractors, limit cycles, and stable orbits, grounding abstract constants in vivid visual dynamics.
4.5 Neural-Like Behavior
Nests synchronize and stabilize like neurons, but retain symbolic traceability, unlike black-box neural networks.
5. Discussion
The UNNS Neural Engine demonstrates:
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Universality: Classical sequences and constants emerge from one recursive substrate.
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Cross-Domain Bridges: Homomorphisms provide interoperability across algebra, geometry, topology, and number theory.
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AI Potential: UNNS may serve as a transparent symbolic–neural hybrid architecture, balancing adaptability with interpretability.
Applications could include:
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Detecting recurrence patterns in finance, biology, or climate.
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Designing fault-tolerant protocols via topological invariants.
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Educational interactive explorations of mathematics.
6. Conclusion
The UNNS Neural Engine provides evidence for a universal symbolic–neural substrate. It unifies recurrence, convergence, cross-domain homomorphisms, and attractor dynamics. Unlike traditional AI systems, it offers interpretable symbolic processing with neural-like adaptability, suggesting a foundation for both theoretical mathematics and future AI architectures.
Figures
Figure 1. UNNS Cross-Domain Homomorphism Demo link
(Embed your homomorphism HTML file here in Blogger — interactive expression mapping.)
Figure 2. UNNS Attractor Explorer link
(Embed your attractor engine HTML — visualizing spiral attractors and stable basins.)
Figure 3. UNNS Neural Engine link
(Embed your neural propagation demo — showing recursive symbolic resonance.)