Canonical Cylinder-Induced Transfer Operators for Quadratic Rigid Successor Systems and the Class-Level Weakened Spectral Conjecture

This paper formulates a class-level canonical transfer-operator package for the quadratic rigid successor test class introduced in the earlier manuscripts of the successor‑complexity programme. The immediate arithmetic motivation is the following. In the period‑two benchmark, the fixed‑order grouped transport data determine a canonical cylinder‑induced realization whose branch weights are forced on one grouped cylinder scale. That realization closes the benchmark verification problems appearing in the operator‑theoretic reduction of the main conjecture. The natural next question is whether the same construction can be organized uniformly for the entire quadratic rigid successor class. We isolate an abstract branch package for a system in the quadratic rigid successor class: a finite clopen branch partition, one‑step grouped cylinder compatibility, branch separation on a fixed grouped cylinder scale, and locality of the induced branch weights. From these ingredients we define a canonical cylinder‑induced operator directly from the finite‑order grouped transport operator, and we prove that the definition is well posed, unique among all admissible branchwise models, and completely determined by a finite combinatorial datum. We then show that once the branch package is available uniformly at rank one, the same Lasota–Yorke/quasi‑compactness/Perron‑isolation mechanism used in the benchmark case propagates to every such system. The main point is conceptual as well as technical. The passage from the weakened form of the uniform conjecture to the fully unconditional form still requires a separate arithmetic identification theorem asserting that the canonical operator coincides with a natural grouped return operator. However, the present class‑level canonical package removes the need to solve that arithmetic identification before carrying out the full functional‑analytic architecture. In this sense the paper turns the benchmark construction into a systematic class‑level replacement theory: for every system in the class satisfying the explicit branch axioms, there is an analytically complete canonical operator, and the weakened form of the main conjecture follows from that operator alone.