MSP² Generator Table: The 3x−1 Mirror System, Coverage Mechanisms, and Exceptional Cycles

This research note by Mario Sousa Pereira and Enzo Mazzoni extends the study of the MSP² Generator Table to the mirror system defined by 3x−1, in comparison with the classical Syracuse (3x+1) system. The document examines how changing the sign affects the recursive construction of the Generator Table, including parity substitutions, coefficient transformations, and additive constants. Particular attention is given to the structural coverage mechanism and to the phenomenon of "return before coverage" in the 3x−1 system, illustrated by the exceptional cycles involving 5 and 17. The analysis also identifies a central question for further investigation: establishing rigorously why return before coverage cannot occur in the positive 3x+1 system, and determining whether the identified exceptional cycles exhaust the corresponding phenomena in the mirror system. The deposit contains research notes in English and French, intended to support independent mathematical examination and further formalization. This work complements the authors' earlier research on MSP² and the Generator Table, including the structural recurrence methods and supporting tables. Authors: Mario Sousa Pereira and Enzo Mazzoni.