From Design Points to Machine-Checked Parameter Regions: Reusable Certificate Templates for Engineering Models

We present a reusable algebra for machine-checked certificates in declared engineering models, where supplied equations, inequalities, monotonicity directions, and perturbation bounds imply an encoded safety, lifetime, stability, or performance predicate. Four recurring rules describe meet, dominance, monotonicity, and margin perturbation across thirteen model domains and eight certificate geometries; a fifth deterministic uncertainty-penalty rule is implemented once as a generic conditional suite and is not instantiated in the thirteen domain suites. Stored manifests record 323 named conditional claims—189 in fourteen abstract suites and 134 in thirteen illustrative named-scenario suites—with zero sorry placeholders; the count includes direct condition restatements as well as derived arithmetic and algebraic consequences. Reported robustness quantities are slack budgets in margin units, not parameter-space radii without an additional sensitivity bound. The named values and comparisons are illustrative inputs, not empirical findings about the named products or systems. The proof tooling supports Lean 4 export, but none of the retained suites has a current-source Lean seal and this paper does not claim Lean-validated theorem equivalence. Formal checking establishes consequences of the declared assumptions; it does not validate the assumptions or the physical models. Keywords: formal verification, safety certificates, battery thermal safety, electromigration, battery degradation, catalyst design, phase stability, binding stability, screening bounds, uncertainty-penalized certificates, proof assistants, industrial design Maturity: Draft. Part of The Latent research program.