The UNNS thermodynamic framework treats recursive nests as microstates in a statistical ensemble. Each nest N ∈ N_k is characterized by coefficients, echo residues {e_n}, and spectral constants σ(N).
An energy functional E: N_k → ℝ≥0 measures instability or complexity:
Entropy satisfies: ∂S/∂⟨E⟩ = 1/T where T = 1/β is the UNNS temperature
Recursive operators act as stochastic maps N → N'. The system evolves under master equations with transition rates W(N → N'), producing entropy through non-equilibrium processes.
For operator evolution with transition rates W:
where J_N→N' is the probability flux between states.
For any operator evolution: ΔS_sys + ΔS_env ≥ 0
The Mandelbrot set M ⊂ ℂ is reinterpreted through UNNS as having structure along a recursion axis orthogonal to the complex plane. The iteration z_{n+1} = z_n² + c generates a recursive grammar.
For recursive process z_{n+1} = f(z_n, c), the recursion axis is an abstract coordinate measuring iteration depth n, orthogonal to the embedding space ℂ.
Let τ(c) be the escape time for parameter c:
The boundary ∂M corresponds to divergence of S(c), forming an echo surface.
UNNS recurrence relations generate algebraic structures that map to spectral operators on lattices. The Fibonacci sequence exemplifies this through its connection to quasicrystals.
Given UNNS coefficients {w_n}, construct discrete Laplacian:
The spectrum reflects UNNS arithmetic structure.
For F_{n+1} = F_n + F_{n-1}, eigenvalues are φ = (1+√5)/2 and φ' = (1-√5)/2.
These define the spectral lattice underlying Penrose tilings and quasicrystal diffraction.
Phase transitions occur when control parameters (inletting rate T₀, repair strength r, noise η) cross critical curves where susceptibilities diverge.
There exist curves in parameter space (T₀, r, η) where:
Let W be operator work and ΔF free energy difference:
Repair operators reduce system entropy but require work:
where d is a metric on nests and R(N) is the repaired state.