The φ-Compression Law: Golden Ratio Optimization in Neural Architecture

We report a conservation law governing neural network compression: when layer widths follow golden ratio (φ ≈ 1.618) scaling via Lucas numbers, the product of efficiency gain and parameter retention equals unity: efficiency × retention = 1.0004 ≈ 1 Systematic experiments across 216 training runs and four neural network architectures reveal that φ-based (Lucas) scaling achieves 2.81 ± 0.07× parameter efficiency gains with 64.4 ± 0.9% parameter reduction in convolutional networks (SimpleCNN, ResNet-18, ConvNeXt-Tiny), while showing exactly no effect in Vision Transformers. The efficiency gain approaches Euler's number (e ≈ 2.718, 3.4% deviation) and the retention fraction approaches 1/e (≈ 0.368, 3.3% deviation), suggesting a deep connection between golden ratio architecture and information-theoretic optimality. The transformer null result is not a failure but a prediction: φ-scaling benefits hierarchical/cascaded processing, which CNNs possess inherently through width progression but standard transformers lack due to uniform layer structure. We propose and preliminarily validate the φ-Transformer architecture—with Lucas-scaled layer capacities—demonstrating that introducing hierarchy to transformers recovers the conservation law (efficiency = 2.76× ≈ e, product ≈ 1.15).