Do the GPU memory math for full fine-tuning a 7B model in bf16 with Adam, then redo it with LoRA.

LoRA freezes the base weight and learns a low-rank update BA, reducing trainable parameters and optimizer state. Rank controls adaptation capacity; choose it by task complexity, target modules, validation quality, and memory budget.

Use equations or tensor shapes where they clarify the claim, then name an experiment or ablation that would distinguish competing explanations.