Resonance Attention: A Quantum-Inspired Reformulation of Attention, Semantic Representation, Contextual Update, and Calibration in Large Language Models

Standard Transformer attention represents each token span, at each layer, by a single hidden vector, and compares spans through a learned bilinear pairing of projected vectors. The claim is not that a high-dimensional vector is incapable of encoding several semantic features or potential readings simultaneously; distributed contextual representations can do so. The narrower claim is that such multiplicity is not represented in a canonical, explicitly structured, mathematically typed, posteriorupdatable, and intrinsically calibratable form. We call the resulting architecture Resonance Attention (hereafter, Resonance). This paper develops a strictly methodological program that replaces three of those primitives with operator-valued objects taken from the mathematics of quantum information: (i) a lexical or contextual span is represented by a low-rank density operator ρ, a positive semidefinite unit-trace matrix whose spectral modes encode coexisting interpretive potentials; (ii) a query is represented by a positive effect E with 0 ≼ E ≼ I, so that attention relevance becomes the bounded, typed, directional pairing Tr(Eρ) ∈ [0, 1]; and (iii) contextual meaning change is represented by completely positive trace-preserving maps and instruments, so that the informal notion of "collapse of meaning in the sentence" becomes a well-defined conditional state update that can concentrate, reweight, or reorganize semantic modes without assuming entropy always decreases. No ontological claim is made that language is quantum, that the brain performs quantum computation, or that quantum hardware is required; the import is purely mathematical. We give an exact low-rank identity, Tr(CC^† BB^†) = ∥ C^† B ∥^2_F , showing that the effect-state score costs O (d*r^2) per pair for rank r ≪ d and decomposes as a doubly weighted sum of squared cosines between query modes and state modes, which connects the score to principal angles between semantic subspaces and to quadratic-kernel attention. A kernel-reduction proposition delimits what the score alone can prove: Tr(Eρ) is a linear functional in a lifted feature space constrained to the positive semidefinite cone, so any gain attributable to the score in isolation is evidence for a richer constrained kernel, and the discriminating content of the architecture is located instead in the components no fixed kernel can express: the parameterfree Lüders posterior, which the same factors deliver at no additional asymptotic cost and which can serve as an operator-native value channel; noncommuting instrument updates; and spectral functionals consumed downstream. A pre-registered primary experiment, the semantic state trajectory benchmark, tests four predicted entropy-trajectory signatures (resolution, garden-path revision, sustained multiplicity, conventional stability) against equal-capacity probes on matched classical hidden states. We show how the spectral entropy of the contextual state supplies an intrinsic, layer-local uncertainty signal usable for calibration, abstention, and temperature control, complementing output-level calibration methods. Tensor products and tensor-network coarse-graining supply composition and multiscale reduction; noncommuting update maps supply order effects for negation, scope, and discourse revision; positive constraint operators supply graded, separable penalties for morphosyntactic, semantic, and pragmatic adequacy. Prior art in probabilistic embeddings, density-matrix semantics, DisCoCat, quantum-inspired attention, Lindblad language models, and tensor-network language models is surveyed and constrains the novelty claim, which is deliberately narrow: the typed effect-state attention rule, its efficient factorized realization, the attention-coupled instrument update, the entropy-based calibration program, and a falsification protocol in which every operator-valued ingredient must beat a matched classical baseline or be abandoned.