Independently trained transformers compute the same function in residual-stream bases that differ by a uniform random rotation on SO(dmodel). We call this phenomenon polymorphism: same function, mutually unintelligible interior coordinates. One matrix multiplication per model pair removes it: an orthogonal Procrustes fit on a single batch of activations transfers sparse-autoencoder feature dictionaries and steering vectors between independently trained models, with no retraining. The phenomenon is invisible to the standard SAE universality metric. Decoder-column cosine similarity matches across seeds at 98%, the SAE-universality headline number, while an SAE trained on one seed reconstructs another seed's activations at negative explained variance, worse than predicting the constant mean. The decoder columns align; the encoder reads from a rotated frame. A single Procrustes rotation R restores reconstruction to within 0.025 EV of the within-seed ceiling at every internal site. R is Haar-distributed: ∥R−I∥F matches the random-orthogonal prediction 2dmodel to 0.1% at dmodel=512, and a Kolmogorov-Smirnov test of R's eigenvalue spectrum against Haar SO(dmodel) returns p≈1.000 pooled and per-pair. Diff-of-means steering vectors transfer in three regimes by alignment with R's invariant subspace: clean when pinned by shared output weights, partial when overlapping the rotated subspace, inverted otherwise. With no shared I/O (Pythia), all three collapse to universally inverted. The same rotation account holds across training checkpoints within a single run. Validated on a 104k-parameter Dyck-3 transformer and nine independently-trained Pythia-70m seeds on The Pile, via a pre-registered four-bar operational framework. Frontier-scale (10B+) replication remains open.
We introduce \emph{universal transformers}: fixed transformers that can simulate any transformer in a given class via a suitable input embedding. Analogous to a universal Turing machine, the input embedding encodes a description of the target model while all internal parameters remain fixed. We provide explicit sparse constructions achieving universality when the embedding dimension is sufficiently large, and further show that universality is generic: randomly initialized transformers are universal almost surely, which aligns with recent empirical results of Zhong and Andreas (2024). We empirically validate our theory on the algorithmic tasks of parenthesis balancing and multi-hop reasoning. Our results suggest that much of a transformer's expressive power may reside in its input representation rather than its learned weights.
A transformer's answer lives on one axis: the direction its unembedding reads. Its intermediate states largely do not, and that off-axis position is usually treated as an obstacle to interpretation. We show it is functional. A 12-layer model computes in two phases. Through the first, every sublayer writes into a subspace held near-orthogonal to the read-out, attention 75 to 96 degrees off it at every depth. Moving attention's values onto the read-out is 64 to 84 times more damaging than a matched random rotation, and the damage is entirely in cross-token mixing: the subspace insulates composition from the vocabulary. Beneath it the frame itself turns rigidly with depth. In the second phase the answer arrives on-axis, late, and by addition rather than by turning accumulated content onto the read-out. Pressing every layer onto the read-out instead, as training for early exit does, matches the baseline on perplexity, LAMBADA and BLiMP while cutting the concept-phase workspace from about twenty-five effective dimensions to fourteen, a change none of those benchmarks register. The geometry can also be imposed, though not by asking for it. Prescribing it through the loss is a lottery: six of eight seeds collapse, because a model told to null its read-out projection obeys most cheaply by discarding dimensions. Inserting one fixed rotation at the phase boundary lands it instead, at baseline quality. A sparse rotation the surrounding weights can absorb converges on all nine seeds, against five of nine for ordinary training. Which rotation is immaterial: twenty-five runs across thirteen distinct ones reach the same quality, and two baselines from different seeds hold their concepts in near-orthogonal frames while agreeing on their read-outs. That freedom is usable: a basis drawn at random and prescribed before training is adopted across the concept phase, with quality unchanged.
Language models are thought to exhibit the phenomenon of superposition, representing many more features than dimensions in their residual streams. Sparse autoencoders (SAEs) are designed to recover such features post-hoc, but training models that are interpretable by construction has remained impractical, as a per-layer over-complete bottleneck is prohibitively expensive in both memory and compute. To overcome this issue, we introduce the ParityTransformer, a GPT-2-scale architecture whose intermediate representations are efficient and wide / sparse by design. At each layer, a Deep Parity Bottleneck (DPB) replaces a learned over-complete basis with a parameter-free algebraic dictionary, providing a deterministic incoherence guarantee and eliminating the memory requirements that have prevented per-layer interpretable bottlenecks at scale. A DPB is a hierarchically structured sparse bottleneck which efficiently enforces sparsity using a multi-level mixture-of-experts approach: a hardware-aware implementation that closes the cost gap between activation sparse and dense training to a manageable interpretability tax. Empirically, ParityTransformers perform at least as well as post-hoc SAEs on sparse probing tasks, while out-performing on measures of feature absorption, steering effectiveness, and fine-grained causal interventions. Because subsequent computation acts only on features that survive the sparse bottleneck, the ParityTransformer's features are native to the model's forwards pass by construction, addressing the question of whether SAEs probe features the model actually uses during computation. We see this as a step toward training models whose internal representations are interpretable by design rather than recovered post hoc.