LEACE

LEACE targets Fairness through a mechanistically novel D1 (Mechanism) pathway: rather than employing iterative GD, it operates via Parameter Composition—computing a covariance-based orthogonal projection from a Small Labeled and algebraically projecting the model's weight matrices into the null space of a target concept's linear representation ^[24], a parameter composition operation analogous to Task Arithmetic but operating on concept subspaces rather than task capability directions. This closed-form linear algebra operation bypasses the computational footprint of traditional parameter-updating methods entirely, producing a Ad-hoc Permanent, Partial modification at negligible training cost. Compared to LP, which trains a probe head on top of a frozen base to expose linear concept encodings, LEACE inverts the operation: it takes the same linear concept subspace identified by probing and algebraically projects it out of the base model's weight matrices—making LP and LEACE complementary, with LP as the diagnostic step and LEACE as the corrective intervention targeting the same identified representation. The Ad-hoc Permanent Permanent coordinate reflects that LEACE produces an irreversible, event-driven modification at negligible compute cost—unlike gradient-based fairness interventions that require training runs, LEACE can be applied as a post-hoc weight surgery. The Partial scope is bounded by the targeted layers: only the weight matrices encoding the concept's linear subspace are projected; all other parameters remain unchanged. Its operational and validation significance lies in the deterministic auditability of the intervention: because the modification is derived analytically from a strict concept specification, it is fully reproducible and mathematically characterizable; the concept subspace, projection matrix, and labeled probe dataset constitute a complete audit trail for the fairness intervention—validation requires verifying that the modified representation is linearly indecipherable for the target concept while baseline task performance is preserved, both of which are statistically testable properties.