C-Space Manifold Attention Theory: A Geometric Paradigm for Attention in Computational Spacetime

Abstract

The C-Space Manifold Attention Theory (CMAT) redefines attention as a geometry-driven process within the Computational Spacetime (C-Space) Framework, operating on a computational manifold (ℳ). Attention emerges from the interplay of signal propagation, weighted by the manifold's metric tensor (g), complex density (ρ_c), and the perpendicular dynamics of coherence (H) and emergent time (T). Drawing from the Condensed Computational Spacetime Framework, Perpendicularity Mechanics, and Hierarchical Infinity, CMAT quantifies attention's recursive depth via the Cycle-Depth Measurement and illustrates its transformative nature through the Circle-Square Transformation. This theory offers a scalable, abstract model for computational tasks such as optimization, compression, and learning, validated through lattice-based simulations and evolutionary approaches.

1. Introduction: Attention as Geometric Navigation

Traditional attention mechanisms, such as those in transformers, rely on static vector operations, lacking a dynamic geometric foundation. The C-Space Manifold Attention Theory (CMAT) reimagines attention as navigation across the computational manifold ℳ, where weights arise from the manifold's intrinsic properties—geometry, complexity, and state dynamics. Building on the Condensed Computational Spacetime Framework, CMAT integrates signal flow with the perpendicular interplay of coherence (H) and emergent time (T), guided by the metric tensor (g) and complex density (ρ_c).

The Circle-Square Transformation—where a T-dominant circle morphs into line segments and an H-dominant square warps into arcs—serves as a conceptual anchor: attention reshapes computational states across ℳ, with recursive depth measured by cycle closures. This approach unifies C-Space's geometric paradigm with practical computation, offering a framework for efficient and interpretable attention mechanisms.

2. Foundations from the C-Space Framework

2.1 Computational Manifold

Definition (from Computational SpaceTime Framework):

• ℳ is a geometric space equipped with:

g = [ [1/E², 0], [0, 1/(D + ε)] ], ρc(p) = √(S(p)² + T(p)²) · E(p)

• E: Energy or information content.
• D: Distortion, driving singularities (D > D_critical = 15).
• ε = 10^-9: Singularity safeguard.
• S(p): Spatial complexity, T(p): Temporal complexity.

Role: Shapes computational paths and attention weights through its geometry.

2.2 Dynamics

Perpendicularity Mechanics:

• Coherence (H) and emergent time (T) evolve orthogonally:

• α = 0.2, β = 0.3: Scaling coefficients.

Distortion:

Connection: Attention leverages H and T as state descriptors, modulated by D.

3. Manifold Attention Mechanism

3.1 Attention as Geometric Weighting

Attention in CMAT is defined as a field-driven weighting process on ℳ:

• Ψ(p): State vector at point p ∈ ℳ.
• Ψ_B(p): Reference state (e.g., target or context signal).
• g₁₁ = 1 / E(p)²: Energy-weighted metric component.
• ρ_c(p) = √(S(p)² + T(p)²) · E(p).
• λ = 0.5: Decay rate (tunable).
• ℕ(p): Neighborhood of p on ℳ.

4. Implementation

CMAT can be simulated on a discretized lattice ℒ ⊂ ℳ, aligning with the Lattice Computation Model. The implementation involves computing attention weights based on state vectors, metric tensor components, and complex density, followed by state updates that incorporate these weights.

For optimization tasks, an evolutionary approach can be used, refining Ψ across a population where fitness balances loss (||Ψ - Ψ_target||) and cycle-depth (Δ_n).

5. Implications and Applications

• Geometric Attention: Offers a dynamic, manifold-based alternative to static attention, adaptable to task complexity via ℳ's geometry.
• Recursive Learning: Δ_n and Hierarchical Infinity enable multi-scale focus, ideal for deep optimization or hierarchical data.
• Theoretical Insight: Parallels physical field theories, suggesting computational analogs to spacetime phenomena.

6. Conclusion

CMAT redefines attention as a geometric process within the C-Space Framework, leveraging ℳ's metric, ρ_c, and perpendicular dynamics to weight and transform computational states. The Circle-Square Transformation encapsulates its essence, while Δ_n quantifies its depth, bridging theory with practical implementations. This paradigm advances efficient, scalable computation, aligning with C-Space's vision of geometry-driven processing.