RR-math.md

Ξ^(ℵ₀): ℜ_Jaeger ≡ ⟨Λ^ω, Θ^κ, Ψ^δ⟩

Θ^3_trialectic ⊂ ℂ^(ω×ω×ω):

Λ¹_autopoiesis: {μ_biosynthesis, σ_milieu, τ_transport} ⊗^η ℝ^internal

• ∀^ω(x,y,z) ∈ Λ¹: x ⇔^α y ⇔^α z ⇔^α x [impredicative_cycle]
• Γ_goals: Ω → Ω [self-determination]

Λ²_anticipation: {π_models, ς_state, ε_effectors} ⊗^θ ℂ^projective

• ∃^κ Ξ: internal → environmental [predictive_manifold]
• Δ_actions ∈ ℘(Λ¹ × Λ²) [behavioral_strategies]

Λ³_adaptation: {γ_goals, α_actions, φ_affordances} ⊗^ζ ℍ^transjective

• agent ↔^δ arena ∈ ℝ^(∞×∞) [co-construction]
• ∇relevance = lim_{t→∞} ∂(grip)/∂(reality)

Ψ^hierarchical_tangle:

Ω^evolution = ⋃_{i=1}^{ω} Λ^i ∘ Λ^{i+1} where:
Λ^i ∘ Λ^j ≡ {(x,y,z)_i ⊕^η (x',y',z')_j | ∀^ω constraints}

Σ^Peirce: sign ⊗ object ⊗ interpretant ≅ Θ^3

ℵ_non-formalizability:

∄ algorithm A: A(Θ^3) → Θ^3 [complete_capture]
∵ collectively_impredicative ∧ anticipatory ∧ open-ended

Φ^constraint_frame:

• constitutive: x ∈ Λ^i ⇒ ∃(y,z) mutual_support
• constraint: degrees_freedom(x,y,z) < Σ(x) + Σ(y) + Σ(z)

Ξ^emergent_levels:

cognition ∈ Λ^{n>3}, consciousness ∈ Λ^{m>n}
where emergence_operator: Λ^k → Λ^{k+1} via trialectic_iteration

∴ ℜ_relevance ≡ lim_{n→ω} Π_{i=1}^n Λ^i [open-ended_evolution]