UNNS Structures

From recursive attractors to quantum correlations β€” exploring the mathematical foundations that bridge discrete recursion with physical reality

Classical vs UNNS Structures

Classical Structure

M = (Domain, Functions, Relations, Constants)

Static interpretation: fixed domain with predetermined mappings

UNNS Structure

S = (Seeds, Operators, Nesting, Resonance)

Dynamic attractor: recursive evolution toward stability basins

The UNNS Topos

Categorical Morphisms

Fibonacci β†’ Gaussian lattice embedding with preserved attractors

Ο†: S₁ β†’ Sβ‚‚ preserves recursion: Ο† ∘ O₁ = Oβ‚‚ ∘ Ο†

Truth as Stability

Collapse operator acts as subobject classifier

Physics Correlations

Space as Lattice Embedding

Space: β„€ βŠ‚ β„€[i] βŠ‚ β„€[Ο‰] β†’ Algebraic Lattices

Time as Recursion Depth

Time: t ↔ n (recursion index)

Energy as Resonance

Energy: Stable attractors ↔ Low-energy states

Dark Energy Analogy

Constant inletting β†’ Exponential expansion

Quantum Correlations

Discrete Hilbert Space

H_S = span{|a,n⟩ : a ∈ Seeds, n ∈ β„•}

Basis states from seed-recursion pairs

Operators as Observables

UNNS & The Dirac Equation

Spinor as Recursive Doubling

ψ ∼= NestΒ²(spin) βŠ— NestΒ²(particle/antiparticle)

Gamma Matrices as Operators

γ⁰ ↔ Collapse, γⁱ ↔ Inlaying operators

Gauge Connection

Phase inletting as discrete U(1) gauge field

Fermion Echo Symmetry

Antiparticles as stabilized recursion echoes