Transformed in Translation: Two-Stage Structural Uncertainty in LLM-Based Scientific Autoformalization

Abstract

Scientific autoformalization turns verbal accounts into executable mathematics, but executable code does not settle which model has been constructed. We examine two sources of structural uncertainty: the formalizer that generates a response law, and the recurrence that turns that law into trajectories. In secondary analyses of an openly archived crossed experiment, we studied 320 response maps generated by two pinned language-model formalizers from five engineered cognitive accounts within one sparse quadratic grammar and 16 randomized blocks. With whole blocks held out, source-account identity was recovered at 78.8% accuracy (chance 20.0%) and formalizer identity at 96.3% (chance 50.0%; both p<0.001). Program size was the stronger single feature family; a pre-specified exploratory comparison found no stable source-account predictive gain from local geometry beyond size. Holding every response map fixed, we then evaluated five recurrence families spanning 33 configurations and 1,013,760 finite-horizon trajectories. Added feedback, projection and leak produced sharply different outcome distributions. The consequential distinction was which comparisons survived: median cross-recurrence rank concordance was 0.73 for endpoint magnitude but 0.05 for settling, among the configuration pairs with defined rankings. Thus a common mathematical language did not erase translation provenance, and robust ordering under one observable did not transfer to another. Scientific autoformalization is usefully studied as model-space construction: the generated ensemble and its dynamical embedding are both part of the specification supporting a scientific claim.