Recoverable but Unreadable: Exact Accessibility Laws in Noisy Latent Parity Codes and Beyond

A representation can contain nearly all information about a latent factor while exposing none of it to a simple probe. We formalize this gap by distinguishing recoverability, what the full representation reveals to the Bayes-optimal decoder, from accessibility, what is exposed through a restricted readout interface. We study this distinction in noisy latent parity codes, a fully analyzable benchmark family. Under the uniform prior, we prove an exact reveal law: the first order at which a target becomes readable is determined by a single combinatorial quantity of the mixing matrix, the minimum number of observed parities needed to synthe- size that target parity. This quantity also governs both local readout and low-order interaction readout. For balanced binary targets, every sub-threshold local view is exactly independent of the target, so Bayes-optimal prediction remains at chance. For random dense mixing below channel capacity, the full latent vector is decod- able with exponentially small error, yet each coordinate remains unreadable to every sub-threshold simple interface. In the BSC case, even the first nonzero wit- ness can be exponentially weak, so reveal need not imply learnability. We also use this family as an exact probing benchmark and show that accessibility-aware objectives can reduce low-order exposure while preserving strong global utility. A continuous learned bridge beyond the parity family exhibits the same qualitative pattern: strong global signal can coexist with substantially weaker local readabil- ity, and accessibility-aware shaping reduces low-order exposure there as well.