HAM-CYCLE

Exports the honest shared graph encoding, ordered Hamiltonian-cycle semantics, the total raw certificate checker, exact checker semantics, quadratic certificate bound, and the total typed VERTEX-COVER reduction with a proved semantic equivalence. The typed construction is also lifted to a total raw map with exact all-input language semantics and a cubic output-length bound. The fixed verifier layer composes the reusable graph checks, target-field transformations, certificate distinctness pass, and cycle-edge lookup into a single polynomial-time TM2. Consequently the honest serialized HAM-CYCLE language is now proved to belong to NP. The fixed reduction layer now generates the nondegenerate header, the full internal widget-edge family, and the complete selector clique. It also computes the canonical per-vertex incidence rows with a fixed polynomial-time scanner and formats their successive references into the complete incidence-chain edge stream. It also extracts the first and last gadget ports of every nonempty incidence row with a fixed polynomial-time controller, repeats those endpoints for every selector, and formats the complete selector-endpoint edge multiset. The four families are now assembled by one fixed polynomial-time machine into an exact ordinary target, with a permutation bridge proving that edge-record ordering does not change Hamiltonian-cycle semantics. A fixed classifier and stream selector also totalize this target over the two degenerate typed branches. The reused raw syntax/well-formedness guard closes the all-input machine, so the honest serialized HAM-CYCLE language is now proved NP-complete.