The authors introduce Formula-Driven Supervised Metric Learning (FDSML), a synthetic pre-training framework that replaces discrete classification labels with continuous relational supervision derived from the generative parameters of an Iterated Function System (IFS). Building on Formula-Driven Supervised Learning (FDSL), which pre-trains vision models on fractal images without real data, FDSML addresses the loss of continuity inherent in discretizing IFS parameters into class labels. The method employs FineFractalDB, a refined fractal dataset exhibiting higher intra-class variation and more stable structures, alongside a novel Parameter-Aware Triplet (PA-Triplet) Loss. This loss function constructs triplets by assigning positive and negative pairs based on Euclidean distances in the underlying IFS parameter space, thereby training embeddings to preserve parameter-derived relative similarity orderings. Evaluated on ResNet-50, FDSML demonstrates significant gains in transferability to image retrieval tasks: Recall@1 increases from 65.6% to 74.0% on Stanford Online Products and from 56.2% to 78.4% on Stanford Cars compared to standard FDSL pre-training. The approach remains competitive on downstream classification benchmarks while exhibiting improved robustness across multiple corruption settings, establishing parameter-driven metric learning as a stronger synthetic pre-training signal for tasks beyond recognition.