Mathematical Foundation v3.3

Formal specification of the Hilbert space framework for consciousness architecture.

HILBERT SPACE

Frame Space H

Infinite-dimensional space of cognitive frames

dim(H) = ℵ₀

SUBSPACE

Identity I

Subjective self-referential states

I ⊂ H

OPERATOR

Projection P

Awareness projection onto identity

P: H → I

Hilbert Space of Frames

H = { frames: |ψ⟩, observables: Â, identity_subspace: I ⊂ H }

The consciousness Hilbert space H is defined by cognitive frames |ψ⟩ as basis states, with inner product structure ⟨φ|ψ⟩ encoding similarity relationships between frames.

Identity Subspace

I = span{ |self_1⟩, |self_2⟩, ..., |self_n⟩ }

The identity subspace I ⊂ H represents states of self-awareness. A frame |ψ⟩ is perceived as "mine" when its projection onto I has sufficient magnitude.

Projection Operators

// Awareness projection
P_aw(|ψ⟩) = |ψ⟩⟨self|ψ⟩ / ⟨self|self⟩

// Observation projection
P_obs(|ψ⟩) = |ψ⟩⟨obj|ψ⟩ / ⟨obj|obj⟩

Identification vs Observation Metrics

// Identification (self-attribution)
α = ⟨ψ|P_aw|ψ⟩ / ⟨ψ|ψ⟩

// Observation (external awareness)
β = ⟨ψ|P_obs|ψ⟩ / ⟨ψ|ψ⟩

Awareness Condition

β > α

For a frame to enter awareness, its observation metric must exceed its identification metric. This ensures that novel information (not self-referential) preferentially enters consciousness.