Finite Interaction Lens: Structured Precision Sharing for Vision-Language Models

Identifying which visual interventions interact in a vision-language model requires comparing their joint and separate effects. The resulting four-output difference can be small enough for rounding error to obscure it. We introduce the Finite Interaction Lens (FIL), a mixed-precision method for evaluating finite intervention families. At each selected linear projection, FIL decomposes each candidate's input into a native state, two single-intervention components, and a full-dimensional mixed residual. It projects the single-intervention components at higher precision and reuses the same numerical contributions in the joint candidates, while retaining every mixed residual and nonlinear trajectory. We derive a local cancellation identity and a conditional bound on error propagation through a smooth nonlinearity. In one MLP of Qwen3-VL-4B-Instruct, FIL reduces final interaction MSE by 94.4% on 16 synthetic families and by 99.85% and 99.35% in two separate four-image COCO studies, relative to a control that has the same nominal BF16-product budget and higher-precision single interventions but projects each joint candidate independently. One image accounts for most of the absolute gain in each COCO study. Higher-accuracy multi-product BF16 arithmetic on all candidates gives lower aggregate error at similar total query times. Single-product BF16 already recovers the reference choices in both candidate-selection tasks we tested. On these development inputs, FIL improves interaction fidelity at a fixed nominal precision budget. Its practical value depends on a task that needs the extra fidelity.