PRH | Aux | 4.3.4 • Collatz Proof Verification: Isabelle/HOL

This note gives a fully explicit, lemma-by-lemma specification of the Lyapunov-certificate method for the accelerated Collatz map. It is arranged to be directly transcribed into a prover Isabelle/HOL and avoids hidden assumptions. The only external finite input we assume is a verified residue certificate (for $k=13$, a set of $2^{12}=4096$ linear inequalities). We enforce a strict, conservative policy on the unique exceptional residue class $r^*$ : at $r^*$ we always take the minimal admissible valuation $v_2=k$ and the canonical successor $R_{\star}=\operatorname{odd}\left(3 r^*+1\right) \bmod 2^k$. Standard facts from elementary number theory are stated when used and referenced to standard sources. No results about global functional graphs or probabilistic heuristics are invoked. All related Isabelle/HOL theories, which cover the proof and beyond, are fully executable, and all sessions build cleanly in Isabelle (all proofs checked; everything green).