The Wall is Disjunction, Not Abstraction: Sparse Autoencoders on a Game-Playing Network

Sparse autoencoders (SAEs) trained on game-playing networks reliably recover per-cell board facts and reliably fail on relational concepts such as "this line is one move from completion.'' Whether the network lacks the concept or the dictionary cannot name it is undecidable from one layer with nothing to compare against. We measure a linear probe and an SAE dictionary on the same activations at two layers of a Quarto agent, plus a third measurement of where a concept goes when no single atom carries it. Across three agents, three SAE architectures, four dictionary widths and 546 ground-truth concepts: (i) the preferred layer partitions by concept family, so a pooled comparison averages effects of opposite sign and reports nothing; (ii) the probe overturns the natural reading of that split, since for state threats the convolutional layer is equally or more linearly available in 11 of 12 cells while its dictionary isolates $1.4$--$4.8\times$ less; and (iii) the residual failure has a specific logical shape. Holding basis, agent-relativity and base rate fixed and varying only logical form, single-atom capture drops from 81.2\% to 0.0\% the moment a concept is a disjunction; agent-relativity, the property we had previously blamed, costs 2 points. The dictionary instead rebuilds each 4-way disjunction from exactly its four disjuncts (modal cost 4 latents, and none of 266 concepts is cheaper), and an 8$\times$ larger dictionary does not move it. The wall is not missing information and not capacity; it is the absence of an atom for OR.