Deterministic Proof Synthesis: Methodology and Case Studies

This work presents a deterministic framework for proof synthesis in which language models are used as instruments rather than authorities. The methodology emphasizes strict constraint enforcement, invariant validation, counterexample supremacy, and explicit failure classification. Through multiple case studies—including Collatz-type dynamics and energy-function analysis—the paper demonstrates how deterministic reasoning kernels can surface partial truths, invalidate false conjectures, and prevent empirical results from being misclassified as proofs.