1. Foundation: The Universal Probability Space

1.1 Base Space Definition

Ω = {ω : ω ∈ H}

Where:

• H is an infinite-dimensional Hilbert space
• Each ω represents a complete configuration state
• The space includes both expressed and potential states

1.2 Measure Structure

μ : F → [0,1]

Where:

• F is the σ-algebra of measurable subsets of Ω
• μ satisfies: ∫_Ω dμ(ω) = 1
• μ(ω) = exp(-S[ω])/Z for action S[ω]

2. Projection to Physical Space

2.1 Configuration Mapping

π : Ω → M

Where:

• M is the physical manifold/configuration space
• π maps infinite-dimensional states to observable configurations

2.2 Local Probability Density