J-Spaces in Transformers: Pointwise Causality, Corpus Averaging, and Sparse Geometry

Jacobian-based interpretability often assigns one name to several nonidentical objects. We separate four: the pointwise terminal causal Jacobian of a fixed report, a corpus-averaged residual-transition Jacobian lens, a layer-local probe Jacobian, and the sparse nonnegative J-set generated by a token dictionary. For fixed prompts, reports, sites, state conventions, and linearization points, terminal pullbacks satisfy an exact chain identity. Corpus averaging breaks product factorization by an explicit operator-covariance term, reconciling pointwise conservation with depth-varying averaged-lens geometry. The corpus dictionary D₌(WU E G→ L)^, defined before final normalization, is generally distinct from the average of pointwise terminal logit gradients. The two estimands can nevertheless be nearly parallel while yielding different sparse supports, because coherent overcomplete dictionaries make greedy nonnegative pursuit support-unstable. We therefore distinguish the operational union of k-sparse nonnegative cones from its unrestricted linear envelope and from metric-dependent reportability subspaces. The resulting framework gives coordinate- and output-covariance laws, Fisher--KL pullbacks, finite-intervention identities, pre-layer-normalization component derivatives, transport-aware comparisons, constrained steering, corpus active subspaces, and finite-sample recovery guarantees. The accompanying prespecified protocol binds each claim to a named estimand, disjoint data roles, retained artifacts, and executable decision rules. It includes Gram- and covariance-matched dictionary nulls, a complete subject-number by attractor-number factorial, frame-clustered inference, a real attractor-number placebo, a deterministic probe-to-terminal certificate, and a sparse-support identifiability gate. Crucially, a nonsignificant causal effect is never interpreted as evidence for selectivity: the selectivity-consistent verdict requires confidence intervals for the J, sign-reversed J, and placebo effects to lie inside a prespecified equivalence region while the full residual swap succeeds. All 48 theory tests and 92 experiment tests pass, together with controlled differentiable-network validation. The pinned Pythia and Qwen studies remain unexecuted; this release establishes a mathematical and methodological result, not pretrained-model evidence.