Abstract
To answer a question about a program, move the program to where the question is decidable. Every such move is a translation, and every translation is a place to be wrong. We study translation as a graph -- many languages, a few reasoning targets, independently built routes of honestly different trustworthiness -- and give it a calculus: pairs of languages close commuting squares that are directional (exactness is the identity-embedding special case of over-approximation), checkable per program, and composable, a route's contract being the componentwise meet of its hops' contracts -- assurance class, direction, kept observables, measured cost. One asymmetry organizes trust: witness-carrying answers are self-certifying by replay at the source; universal answers are where grades, independent branches, and re-checked certificates earn their cost. The compositional core, lax telescope included, is mechanized in Lean 4. hurdy-gurdy implements the calculus as two planes meeting in one registry. The use plane reads declarations and produces evidence-carrying answers; its builders and its intended player are both LLMs, untrusted by construction. The evolution plane grows the graph: unmet questions are recorded as demand, pairs are recommended by evidence and registered by humans, and a ratchet keeps every prior verdict standing. Answers never write; growth never answers. Run indefinitely, the loop converges on every reducibly decidable question, at fidelity that only rises. We measure the July 2026 snapshot -- per-construct conjoined coverage, dual-route branch agreement for two ISAs, source-level witness replay, certified unreachability re-validated by a formally verified checker, escape rates for the gate itself -- and report the defects the architecture caught in its own authors' work.
Blogger's Review: This paper explores the relationship between program translation and trust mechanisms, proposing a novel calculus to assess the reliability and fidelity of translations. This approach is significant in the context of formal verification, especially when dealing with information from untrusted sources. By integrating graph theory with program analysis, the paper demonstrates how to achieve composability and dynamic trust management in complex systems, offering broad application prospects.