Logarithmic Interventions Suffice for Nonlinear Causal Representation Learning

Causal representation learning aims to recover latent variables and their causal graph from nonlinear observations. However, existing identifiability results leave open full recovery under hidden multi-node intervention targets and replacement laws that vary across target-environment pairs. In this work, we show that same-unit factual-interventional pairs separate target identification from causal ordering. A minimum-code criterion over candidate residual charts and binary target codes recovers residual coordinates and intervention signatures (binary environment-target patterns) by selecting a minimum-Hamming-weight code consistent with paired invariances. After this alignment, coordinatewise normalization of environment density ratios isolates causal-order information from replacement strength. We then introduce Conditional-foliation peeling (CFP), which compares the oriented-foliation structure of the resulting conditional CDFs to identify and recursively remove sinks, variables with no remaining children. For smooth scalar invertible-noise SCMs with an invertible decoder and fixed conditional supports, the paired laws identify endogenous variables up to permutation and componentwise strictly monotone transformations and recover the entire DAG, assuming positive paired-support densities and Terminal-response faithfulness (TRF). Any distinct nonzero signatures suffice, including codes of arbitrary rank and nesting. Matching upper and lower bounds show that the minimax number of labeled intervention environments, excluding the factual environment, is exactly $\lceil {\log }_2(d+1)\rceil$ for $d\ge 2$.