v3.3 — Mathematical Framework
The Architecture
of Awareness
A mathematically grounded architecture of awareness.
A formal framework describing consciousness through three interconnected layers: Frame System, Attentional Canvas, and Consciousness Core — with rigorous mathematical foundations in Hilbert space theory and projection operators.
Core Architecture
Three-Layer Framework
Layer 3 — Consciousness Core
CC
Identity subspace S_id ⊂ H_CC
Layer 2 — Attentional Canvas
AC
Projection operator Π_AC
Layer 1 — Frame System
FS
Hilbert frame bundle H_F = ⊕ᵢ Hᵢ
Significance
Why This Matters
Neuroscience
- • Formal bridge between neural measures and phenomenological reports
- • Testable predictions for EEG/fMRI paradigms
- • Framework for understanding disorders of consciousness
Artificial Intelligence
- • Design principles for aware AI systems
- • Metrics for evaluating machine consciousness
- • Integration with existing architectures
Psychology
- • Unified account of attention, awareness, and self
- • Tools for studying metacognition
- • Clinical applications for therapy and assessment
Broader Impact
Implications
Beyond the core model, this framework offers insights into consciousness research across disciplines — from clinical applications to AI safety.
Core Awareness Condition
Awareness emerges when:
β = rank(Π_CC) > α = dim(S_id)
The rank of the consciousness projection must exceed the dimension of the identity subspace for awareness to occur.