Activation patching and weight-space ablation both claim a component is causally responsible for a behavior, yet they act on different objects: one forward pass versus the parameters behind every forward pass. We ask when they agree. We study an idealized model where a conditional computation is carried additively through a residual stream, $F(x)=F_0(x)+\sum_iα_i(x)v_i$, read out by a linear functional, and prove three exact results. First, deleting a subset of carriers collapses a matched input pair onto the same unconditional output \emph{if and only if} the removal is symmetric on the pair and leaves no outside contrast; the error is deterministic, and we give its exact form even when the two conditions hold only approximately. Second, patching a carrier moves the readout by its donor-receiver \emph{contrast}, while ablating it moves the readout by its \emph{absolute level}; neither bounds the other, and we construct pairs where every single-carrier patch flips the decision while no single-carrier ablation does. Third, for an attention head composed with its own layer's normalization and MLP, we derive an exact first-order interaction formula with a provably second-order remainder, vanishing identically when only the MLP is ablated but not, in general, when a head is. Small transformers trained on a synthetic conditional task illustrate all three predictions: across thirty-nine ablation configurations the measured interaction is strongly rank-correlated with the idealized model's predictive accuracy (Spearman $-0.83$), and a second task and architecture reproduces the same pattern, including a further polarity reversal. The single-block interaction result extends past one residual block, and the synthetic validation is tested against a real pretrained model, in a companion paper that takes this theory further along both axes.
Spatial reasoning requires both location-bound computation and location-invariant structure: agents must make local moves while preserving route, object, or constraint-level plans. We propose interaction locality, a task-geometry-aware framework for measuring whether information flow stays within nearby cells or semantic segments, or crosses them. We instantiate the framework with sparse-autoencoder feature ablations and finite-noise activation patching, with structural Jacobian and attention checks reported in the appendix, and apply it to HRM and TRM, two compact hierarchical and recursive reasoning models, on Maze-Hard, Sudoku Extreme, and ARC-AGI. Across these models, activation patching gives the clearest architectural fingerprint: high-level recurrent states tend to write information within nearby cells or same-segment units, while repeated recursive updates accumulate these local writes into broader solution structure. This pattern holds across maze paths, Sudoku constraints, and ARC-AGI object neighborhoods, with the strongest concentration in TRM. To test whether interaction locality extends beyond toy-yet-challenging grid benchmarks, we also apply it to MTU3D, a large-scale embodied 3D scene-grounding model. In this MTU3D setting, causal spatial locality appears primarily at the transition where visual scene features are handed to the downstream grounding module, rather than uniformly throughout the visual encoder. This contrast suggests that the local-to-global handoff observed in HRM and TRM is tied to explicit recursive reasoning dynamics, while embodied 3D models may concentrate causal spatial structure at module boundaries. Interaction locality turns the intuitive local-execution/global-planning story into a reproducible measurement framework for recursive and embodied spatial reasoning.