Coverage-Gated Coalitional Coding for Multi-Modal Discrete Tokens

Discrete multimodal codes, such as semantic IDs, neural codecs, unified tokenizers, are distributed: each token observes only a subset of modalities, and synergistic targets vanish unless every constituent is present. Existing analyses typically treat the tokenizer as a centralized quantizer of a fused vector and report a single rate-distortion curve. That formulation conceals an operational constraint. A missing factor is information-theoretically invisible, independent of decoder skill; even after every factor is covered, recovery remains a finite-rate problem. We develop a coverage-gated theory of this constraint. For independent Gaussian factors and a product interaction, local channels with information budgets $r_i$ attain an exact one-letter MMSE given by a product of scalar Gaussian gains. Average-rate codes replace that product by its concave envelope, which for order $k\ge 2$ has a genuine setup phase ($0.906$ bits per modality at $k=2$). Allocating a global budget across an access hypergraph of such edges is coalitional water-filling, which strictly dominates order-scalar $V/\mathrm{rank}$ rules; finite tokens inherit a $2Lϵ$ plug-in guarantee. Seeded operational checks recover these predictions. On Flickr8k captions with frozen OpenCLIP residual tokens, exact-budget allocation is compared with MMQ-style shared/specific schedules and a peak-rate analogue of bursting.