THE GOLDEN-RATIO AND COMPOSITE-FINITE-MACHINE FOR COLLATZ

We study the digital structure of Collatz iteration in golden-ratio representation, and construct composite finite machines, whose states are exact integer pairs, pruned bidirectionally through the real and conjugate embeddings, translating arithmetic assertions into emptiness certificates. Main contributions include: a proof of the gully-blocking master lemma with optimal threshold; machine certificates of cross-step prohibitions valid for all odd numbers; a complete classification of recharge cascades with a spectral radius bound below the golden threshold; and, at the resource layer, an 8-margin machine giving a strict upper envelope for the deficient-recharge count. Together with a constant-level equivalence, a record-domination lemma, and a payment pairing lemma, we close the global accounting budget, thereby discharging the conditional form of the master theorem under two explicitly stated external inputs. All machine certificates can be independently reproduced by the accompanying programs. The composite-finite-machine paradigm applies to analogous assertions in any Pisot base and is of independent interest.