Theorem 3.3 (Energy Conservation)

The semantic work required for cognitive operations follows the quadratic law:

m = ||n||² = n · n

where n is the feature vector in the cognitive manifold.

Proof

Let n = (n₁, n₂, ..., nₖ) be the feature vector representing cognitive features and/or dimensions (depth(s), weight(s), densities).

Step 1: In potential state P, the concept exists with k independent features.

Step 2: Transition to realized state requires activating all dimensional interactions. The interaction matrix has elements nᵢnⱼ, but only self-interactions contribute to the minimal work:

m = Σᵢ Σⱼ nᵢnⱼ δᵢⱼ = Σᵢ nᵢ² = ||n||²

Step 3: This is the minimal work because:

Each feature requires activation energy proportional to its magnitude Cross-feature interactions can be minimized through optimal pathways

The self-inner-product provides the lower bound

Thus cognitive work scales quadratically with feature dimensionality, establishing the neuromorphic law.