Semantic Rate-Distortion for Bounded Multi-Agent Communication: Capacity-Derived Semantic Spaces and the Communication Cost of Alignment

When two agents of different computational capacities interact with the same environment, they need not compress a common semantic alphabet differently; they can induce different semantic alphabets altogether. We show that the quotient POMDP $Q_{m,T}(M)$ - the unique coarsest abstraction consistent with an agent's capacity - serves as a capacity-derived semantic space for any bounded agent, and that communication between heterogeneous agents exhibits a sharp structural phase transition. Below a critical rate $R_{\text{crit}}$ determined by the quotient mismatch, intent-preserving communication is structurally impossible. In the supported one-way memoryless regime, classical side-information coding then yields exponential decay above the induced benchmark. Classical coding theorems tell you the rate once the source alphabet is fixed; our contribution is to derive that alphabet from bounded interaction itself. Concretely, we prove: (1) a fixed-$\varepsilon$ structural phase-transition theorem whose lower bound is fully general on the common-history quotient comparison; (2) a one-way Wyner-Ziv benchmark identification on quotient alphabets, with exact converse, exact operational equality for memoryless quotient sources, and an ergodic long-run bridge via explicit mixing bounds; (3) an asymptotic one-way converse in the shrinking-distortion regime $\varepsilon = O(1/T)$, proved from the message stream and decoder side information; and (4) alignment traversal bounds enabling compositional communication through intermediate capacity levels. Experiments on eight POMDP environments (including RockSample(4,4)) illustrate the phase transition, a structured-policy benchmark shows the one-way rate can drop by up to $19\times$ relative to the counting bound, and a shrinking-distortion sweep matches the regime of the asymptotic converse.