Invariant Discovery with Honest Refusal: An Operational Protocol and an Evolving Language

Abstract

Symbolic regression systems are optimized to answer: given data, return a formula. We argue that for scientific discovery, the more valuable capability is to refuse - to return no law when no invariant is supported, and to state why. We present CV-to-0, an operational protocol that adjudicates between three outcomes on the same search machinery: invariant (survives held-out validation, null calibration, and compression tests), predictive approximation (fits data but fails invariant criteria), and no signal (indistinguishable from noise). Refusals are diagnosed mechanically into four categories - insufficient data (DATA), inexpressible operation (GRAMMAR), fundamental unpredictability (NOISE), insufficient search depth (DEPTH) - each validated by a prescribed falsification procedure. On a 12-task benchmark spanning Feynman physics and real-world datasets, our implementation achieves 6 wins, 3 operational parities, and 0 losses against PySR under comparable per-seed wall-clock budgets, while reporting PREDICTIVE APPROXIMATION for an AUTO-MPG structure that PySR returns as a predictive expression (PySR has no invariant/refusal gate; R-squared parity, no false acceptances observed on 100 shuffled-target controls). Beyond adjudication, we demonstrate language ontogeny: on a cold start with an empty vocabulary, the system grows a reusable word from partial hypotheses discovered in data, transfers it across semantically distinct domains with a 3.8x speedup, and does so without a human-seeded transferred word. As a real-domain case study, we apply the protocol to lithium-ion capacity fade across three chemistries (273 cells, 56,377 points): the result is not a universal law but a hierarchy of invariance - power-law form holds across chemistries while the exponent is chemistry- and protocol-specific - and the adjudication layer catches an artefact that passes fit-based review while also refuting two published universal-degradation claims. We release a compliance matrix mapping every protocol criterion to implementation, tests, and verification scripts.

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