lin-0147

11.3 Influence and closure are separate observers

A population influence score for intervention \(i\) and coordinate \(j\) is

\[ I_{ij} = \left( \mathbb E_{P^{(0)}}[(v_i)_j^2] \right)^{1/2}. \]

Large \(I_{ij}\) proposes that regime \(i\) changes coordinate \(j\). It does not distinguish direct from mediated propagation without further conditioning or scoring.

Let \(\Pi _V\) project onto the visible span. The pairwise closure residual is

\[ r_{ij}(z) = (I-\Pi _V(z))[v_i,v_j](z), \qquad R_{ij}^2 = \mathbb E_{P^{(0)}}\lVert r_{ij}(Z)\rVert ^2. \]

This is the BRIDGE INC observer: it tests whether the fitted response family closes locally. The influence and closure blocks are kept separate because one proposes arrows while the other qualifies the adequacy of the visible field model.

Proposition 11.1 Geometry-screen retention

Suppose every arrow in a prespecified target set \(E^\star \) lies among the top-\(k\) population influences into its endpoint with margin \(\gamma {\gt}0\). If the estimated fields converge uniformly in the required \(L^2\) and \(C^1\) senses, then the empirical screen retains every member of \(E^\star \) with probability tending to one. Residual classes separated by a margin around a fixed threshold are also consistently separated.

Proof ▼

Uniform convergence of the fields gives uniform convergence of the finite family of influence scores. An error smaller than \(\gamma /2\) cannot reverse the top-\(k\) ordering at a target arrow. Continuity of \((u,v)\mapsto [u,v]\) in the chosen \(C^1\) norm, together with consistent projection, gives uniform convergence of the residual scores. The threshold claim follows from the separation margin.

Corollary 11.2 Conditional downstream consistency

If the screen retains the representation required by an estimand \(\mathcal G^\star \), and the downstream learner is consistent for \(\mathcal G^\star \) whenever that representation is available, then the two-stage procedure is consistent for \(\mathcal G^\star \).

The corollary deliberately inherits the estimand and assumptions of the downstream method. The screen supplies retention, not a new identification theorem.