A Theory of Conditional Collapse under Low-Rank Weight-Space Ablations: I. The Single-Block Theory and Synthetic Validation

Title:A Theory of Conditional Collapse under Low-Rank Weight-Space Ablations: I. The Single-Block Theory and Synthetic Validation

Abstract: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, $F(x)=F_0(x)+\sum_i\alpha_i(x)v_i$, 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.

25 pages, 2 figures. Part I of a two-part series; see the companion paper "Cross-Layer Interaction under Weight-Space Ablation" (Part II)

Machine Learning (cs.LG); Artificial Intelligence (cs.AI)

arXiv:2608.03620 [cs.LG]

(or arXiv:2608.03620v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2608.03620 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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