Hierarchical Grading in Large Language Models

We introduce Graded Large Language Models (GLLMs), an algebraic framework that equips the representation space of a transformer with a grading and propagates the induced weighted scalar action through embeddings, self-attention, and the training objective. The construction extends the theory of graded neural networks and graded transformers to autoregressive language models while preserving expressive power, asymptotic computational complexity, and inference cost. The governing geometric picture is that of geometric invariant theory. The benefit of a grading is expressed by a Kempf--Ness functional on the grading torus; the grades that improve upon the uniform architecture form an open convex cone whose membership is decided by a Hilbert--Mumford-type criterion pairing a grade direction against two measurable profiles of the target and the data; the optimal grades are the coincidence point of two moment maps, given in closed form; and the ordinary transformer appears as a semistable isotropic point on the boundary of the cone: one member of a larger graded family rather than a distinguished optimum. Separately, for level-stratified targets we prove a minimax separation between the graded prior and its absence: over all estimators the risks of the graded and uniform target classes separate throughout an explicit window of sample sizes, by a factor that decays exponentially in the number of levels under geometric stratification. Both profiles are estimable offline, so the optimal grades solve a convex program certified before training begins. Because the grading is absorbed into the learned parameters after training, every GLLM compiles to a standard transformer of identical architecture and inference complexity.

How Token Influence Decays with Distance: A Green-Function View of Trained Language Models

We study how the next-token prediction of an autoregressive Transformer language model changes under small perturbations of earlier input token embeddings. Motivated by operator learning and iterative solvers for differential equations, we investigate how the influence of one token on another decays with distance in a trained model. In multilevel methods for differential equations, such as domain decomposition, multigrid, and multilevel preconditioning, one often exploits a separation between strong local interactions and weaker but essential global interactions. The latter correspond to the long tail of the Green's function and are typically handled by a coarse-level operator. Inspired by this perspective, we compute an empirical, distance-resolved gradient profile of token dependencies using autograd. Experiments on trained Pythia models and Qwen2.5-0.5B show that, over the measured distance range, the median Jacobian sensitivity is much better described by a power-law-type decay than by an exponential alternative: the diagonal-normalized profile is well described by $$\overline G(r) \approx γ+β(r+1)^{-p}$$ with exponents $p \approx 0.7$--$0.9$ (typically $0.8$--$0.9$). This behavior appears on coherent text from Gutenberg and WikiText-103. Token-shuffling experiments show that the power-law profile persists even when syntax and prediction quality collapse, whereas randomly initialized models do not exhibit it. The slowly decaying long-range sensitivity thus appears to be a learned property of trained autoregressive Transformer operators. These findings suggest that hierarchical or coarse-level mechanisms in language models may be able to exploit the long-tailed sensitivity profiles.

From Tokens to States: LLMs as a Special Case of World Models and the Continuous Path Beyond

The AI community has framed the relationship between large language models (LLMs) and world models as a dichotomy: LLMs predict tokens; world models simulate reality. Yann LeCun argues in 2022 that reaching general intelligence requires abandoning autoregressive token prediction in favour of latent-space architectures. This framing is unnecessarily binary. Two claims will be defended. First, LLMs are a degenerate special case of world models: the state space is the set of all token sequences, the only action is appending one token, and world models are therefore a strict generalisation of LLMs, not a replacement. Second, there is a natural continuous spectrum from NTP to JEPA, with multi-token prediction, future-summary prediction, and next-latent prediction as intermediate stations already populated by current research. Moving along this spectrum relaxes the LLM constraints one by one. It also progressively surrenders the two practical advantages that make LLMs trainable at scale: internet-scale self-supervised data, and a transformer architecture co-designed for discrete token prediction. Both are examined as open research questions: the data question (the cliff from self-supervised text to instrumented action-labelled environments) and the architecture question (whether the transformer generalises to continuous-state prediction, or whether a new primitive is needed).

Data-Efficient Autoregressive-to-Diffusion Language Models via On-Policy Distillation

We study the transformation of autoregressive models (ARLMs) into diffusion language models (DLMs). Rather than pretraining from scratch, prior work replaces the causal attention in ARLMs with bidirectional attention and then trains the resulting model using a DLM objective. However, these approaches incur two distribution shifts. First, transitioning from a next-token prediction objective to a DLM objective can discard knowledge acquired by the ARLM during training. Second, standard DLMs suffer from a train-inference mismatch, as the training loss is defined on randomly masked sequences rather than the trajectories encountered at inference produced by confidence-based decoding. To address both challenges, we introduce an On-Policy Diffusion Language Model (OPDLM) in which On-Policy Distillation (OPD) is employed for ARLM-to-DLM transformation. Specifically, OPDLM is trained via self-OPD, where the student, an ARLM with bidirectional attention, generates its own trajectories, and the teacher, the original frozen ARLM, distills its knowledge by providing target logits on these trajectories. By training directly in an on-policy manner, OPDLM eliminates the train-inference mismatch in DLMs, while distillation from the original model enhances knowledge retention from the ARLM. Empirical results demonstrate that OPDLM requires 15x to 7,000x fewer training tokens with strong performance across a wide variety of tasks. OPDLM avoids the prohibitive cost of DLM pretraining and positions DLM transformation as a form of ARLM post-training.