A continuous-time attention operator makes fixed LLMs behave continuously

- The Continuous Causal Transformer generalizes causal attention to an integral over a continuous-time embedding function and provably reduces exactly to standard discrete attention when tokens have uniform unit duration (Eq. 8), so any pretrained Transformer runs under it with no retraining. [marro-etal-2025-implicitly-continuous] - Continuously shrinking a repeated token's duration makes each of six pretrained LLMs emit multiple distinct token counts across the sweep rather than one fixed answer: normalized peak frequency averages 0.74 observed vs 0.26 under a discrete-model counterfactual (2.90x; up to 3.60x; e.g. Llama-2-13B-Chat satisfies the continuity property in 92.3% of cases). [marro-etal-2025-implicitly-continuous] - Linearly interpolating between two token embeddings yields smoothly varying next-token distributions (empirical Lipschitz / max-derivative estimate averaging 6.214, range 4.96-9.28), and off-lattice points still compose coherently (an apple-banana midpoint predicts "green"). [marro-etal-2025-implicitly-continuous] - Output is empirically shift-invariant (consistent with RoPE) but not scale-invariant, and the effect is weaker in GPT-2 (learned positional embeddings). Models: Llama-2-13B-Chat, Llama-3-8B, Phi-3-Medium-4k-Instruct, Gemma-1-7B, Gemma-2-9B, Mistral-7B, plus GPT-2 in a positional-embedding ablation. [marro-etal-2025-implicitly-continuous]