SL Algebra: A Control IR for Structured, Grounded, and Auditable LLM Inference

Large language model inference increasingly combines heterogeneous controls, including structured-output constraints, retrieval, tool execution, and runtime policies. Existing systems typically implement these mechanisms as separate prompts, decoding processors, or callbacks, making their composition difficult to reason about and optimize. We introduce State-Logit (SL) Algebra, a control intermediate representation that separates state operators, which modify auxiliary state or conditioning context, from logit operators, which transform next-token scores. This representation enables formal reasoning about control plans, semantic-preserving rewrites, alternative physical lowerings, and execution traces. We formalize non-anticipatory plans and multiple levels of equivalence, derive normalization laws for exact hard masks, and prove a tokenization-lift theorem for byte-level interface constraints. We further represent grounding as a logical join independent of its physical realization. Experiments across Qwen2.5 and Qwen3 models show that exact constraints achieve 100% structured-output validity, explicit grounding substantially improves QA exact match, and mask fusion provides a mean 1.55× runtime speedup. These results demonstrate that separating control semantics from physical realization enables both formal guarantees and practical optimization for controlled LLM inference.