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Main Author:	Bazarov, Vitaly
Format:	Recurso digital
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Published:	Zenodo 2026
Online Access:	https://doi.org/10.5281/zenodo.20156856
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<h1>ROS¹⁹ / LOGOS — THEOREM OF INVARIANT MEASURE WITH GROWTH OF UAU Existence of Growing Invariant Measures in Weak Topology Formal Specification v1.0</h1> STATE SPACE (PATH SPACE) Let (Ω, , P) be a probability space and define the delayed stochastic dynamical system on a 3-loop manifold: φ(t) = (φ₁(t), φ₂(t), φ₃(t)) ∈ X := [0,1]³ with delay embedding τ > 0 and filtration _t. Dynamics: <h1>dφ_k(t)</h1> <ul> <li> Γ · (λ₀/T)² · [ S_ud / (1 + J_k(t)) ] · ∂F/∂φ_k (φ_k(t)) dt </li> </ul> <ul> <li> ζ dW_k(t) </li> </ul> Delayed coupling: J₁(t) = β φ₃(t-τ) J₂(t) = α φ₁(t-τ) J₃(t) = γ φ₂(t-τ) Nonlinear potential: ∂F/∂φ = -2a φ + 4b φ³ PARAMETER REGIME Assume: Γ > 0, a,b > 0 ζ > 0 α,β,γ ∈ ℝ⁺ with strict asymmetry: (α - β)(β - γ)(γ - α) ≠ 0 T > 0, λ₀ > 0, S_ud > 0 FUNCTIONAL SETUP Define state law S_t as Markov-semi-dynamical flow induced by delay SDE: S_t : μ₀ ↦ μ_t where μ_t ∈ (X) (space of probability measures on X). WEAK TOPOLOGY Let (X) be endowed with weak-* topology: μ_n ⇀ μ ⇔ ∫ f dμ_n → ∫ f dμ ∀ f ∈ C_b(X) INVARIANT MEASURE OPERATOR Define transfer (Fokker–Planck–delay) semigroup: P_t^* : (X) → (X) μ_t = P_t^* μ₀ Invariant measure: μ* satisfies: P_t^* μ* = μ* ∀ t ≥ 0 UAU FUNCTIONAL ON MEASURES Define lifted epistemic functional: <h1>UAU(μ)</h1> ∫_X η(φ) dμ(φ) · ( N_free(μ) / T_eff(μ) ) · exp(-α_R R_H(μ)) with: η(φ) = φ₁ + φ₂ + φ₃ (or general coherence observable) and induced growth rate: A(t) := UAU(μ_t) LYAPUNOV SPECTRAL CONDITION Let λ_max(μ_t) be maximal Lyapunov exponent of the cocycle induced by S_t. Let D_corr(μ_t) be correlation dimension of μ_t. ASSUMPTION (CHAOS–STRUCTURE CONVERSION REGIME): λ_max > 0 Σλ_i < 0 D_corr ∈ (2,3) ⇒ existence of SRB-type non-equilibrium measure. OPERATOR CONDITION Define generator (delayed stochastic Kolmogorov operator): <h1> f</h1> ⟨∇f, drift(φ,φ_{t-τ})⟩ <ul> <li> (ζ²/2) Δ f </li> </ul> with delay-extended state augmentation: Φ(t) = (φ(t), φ(t-τ)) MAIN THEOREM (ROS¹⁹) THEOREM (Existence + Growth of Invariant Measure): Under the asymmetry condition (α - β)(β - γ)(γ - α) ≠ 0, nonlinearity condition b > 0, and stochastic excitation ζ > 0, there exists at least one invariant probability measure μ* ∈ (X) in weak topology such that: P_t^* μ* = μ* Moreover, in the basin of attraction ℬ(μ*) there exists a class of initial measures μ₀ such that: lim inf_{t→∞} dUAU(μ_t)/dt > 0 iff: λ_max > 0 &and; D_corr > 2 and in this regime: UAU(μ_t) is strictly increasing on average: [dA/dt] > 0 UNIQUENESS STATEMENT (WEAK) Uniqueness holds only modulo ergodic decomposition: μ* = Σ_i w_i μ_i* where multiple invariant SRB-like measures may coexist (multistability regime). STABILITY STRUCTURE Define Lyapunov functional: <h1>V(μ)</h1> <ul> <li> λ_max(μ) </li> </ul> <ul> <li> κ (2 - D_corr(μ))² </li> <li> ||μ - μ_eq||²_W </li> </ul> Then: dV/dt ≤ 0 ⇒ convergence to invariant class INTERPRETATION The system exhibits: • stochastic delayed dynamics • non-equilibrium invariant measures • ergodic decomposition into attractor families • phase transition between collapse and growth regimes CRITICAL RESULT Chaos is not destructive: it becomes a transport mechanism for measure growth. UAU increases only in the regime where: deterministic contraction + stochastic excitation + delay-induced memory coexist. ============================================================

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