Erased, Rerouted, or Rescaled? Post-Training and the Causal Quotient of a Language Model's Belief State

Weihan Li, Tianshi Zheng, Junhao Wu, Xinlei Chen

What happens to information a pretrained model already encodes when post-training no longer rewards using it? The common language of representation compression conflates three fates: information may be erased, rerouted away from the decision while still represented, or rescaled to occupy less variance while still represented and used. We make these fates identifiable in models whose pretraining recovers Bayesian belief states. A reward that reads only a coarse function of the hidden state defines an exact reward-null kernel. The kernel lets us separately measure whether the information remains recoverable, whether decisions causally depend on it, and how much activation variance it occupies. Theory says what is protected: KL-anchored reinforcement learning preserves the reference policy's log-odds among equally rewarded outputs, supervised and unanchored objectives carry no such constraint, and spectral compression implies neither erasure nor loss of use. In controlled worlds, post-training mostly reroutes or rescales reward-null information and leaves it decodable. Without an anchor decisions can stop using it although the representation survives, and with one they keep using…