lin-0150

11.6 SKFM: amortizing the geometry

Kernel BRIDGE can be expensive because local fields are repeatedly estimated. SKFM replaces them with a conditional flow-matching network

\[ f_\Theta (x,t,i) \]

indexed by intervention \(i\). Automatic differentiation computes pairwise brackets. Their projected residuals form a Gram matrix \(G\); a leading eigenspace \(E\) summarizes a low-rank visible footprint.

Spectral Kernel Flow Matching

  1. Fit amortized response fields to observational–regime transports.

  2. Verify held-out continuity, overlap, and influence calibration.

  3. Compute selected Lie brackets by automatic differentiation.

  4. Project brackets normal to the visible field span.

  5. Form the residual Gram matrix and estimate its stable spectral rank.

  6. Produce an influence mask and residual-footprint certificate.

  7. Either pass the mask to a causal learner or apply a scoped ordered extractor.

The spectral coordinates summarize structured nonclosure. They are not identified latent variables without additional factor-model assumptions. Likewise, the direct extractor uses a supplied or learned order and a soft triangular penalty; this is a modeling choice rather than a general equivalence between DAGs and solvable Lie algebras.