The Moran--Hutchinson formula in semimetric spaces

What the math gives to ML

The paper establishes a quantitative relation between finite systems of contractive similitudes, their attractor dimension, and branch separation. If contraction ratios r_i satisfy the similarity-dimension equation sum_i r_i^s = 1, then under the open set condition the attractor has positive and finite s-dimensional Hausdorff measure; conversely, positivity implies the open set condition in the paper's semimetric setting. This mechanism can transfer to branching recurrent or generative neural networks by controlling branch Jacobian contractions and penalizing overlaps. The resulting architecture has an explicit complexity target and predicts a measurable transition in attractor dimension when branch separation is lost.