Verified translation has two well-studied extremes: prove the translator once (certified compilation), or validate each run of one translator (translation validation). Both treat a single translation in isolation. We study translations as a graph – many source languages, several reasoning targets, multiple independently built routes – where the honest answer to "is this translation correct?" differs from edge to edge. We present a calculus of fidelity-graded translations: pairs of languages close commuting squares that are checkable per program and compose by pasting; declared fidelity grades compose by weakest link, are re-established per run by inline checking, and are exceeded by agreement between independently derived routes; and an end-to-end theorem isolates a fundamental asymmetry – witness-carrying answers are self-certifying at the source, while universal answers are where grades, branches, and certificates earn their cost. The compositional core is mechanized in Lean 4. The calculus is implemented in hurdy-gurdy, a platform of 13 languages and 13 pairs around two reasoning hubs, built as a two-directional experiment in LLM-generated correctness: independent LLM agents wrote every pair, largely unsupervised, with the architecture's cross-checks as the only semantic gate, and the platform's intended player is itself an LLM. All code, and most of this paper, is LLM-generated; the human contribution is the architecture. The same gate is the intended growth model: hurdy-gurdy scales in language support through pairs contributed by anyone – with LLMs, with agents, or by hand – admitted by architecture, not authorship. We report conjoined coverage, branch agreement, a compliance-derived benchmark with machine-derived ground truth, witness replay, certified unreachability, and escape-rate experiments for the gate.