Polynomial recognition and inflation of Hessenberg skew Schur classes

We recover the connected skew Schur class of every ribbon composite over a zero-free inner whose near-concatenation powers are rigid up to rotation. For three zero-free blocks with a rigid total merger, the connected class has one, two, or four members; commuting blocks admit legal factor-reversal presentations without individual block rigidity. Recognition of constant-gap Hessenberg powers is complete through five rows. At a balanced Hessenberg inflation threshold, the lowest component determines the outer cuts up to reflection, and every inner with at most four rows admits a complete orientation argument. Consequently positive constant-gap Hessenberg inners through four rows have complete inflation classes and legal generation at every nonnegative time. Further criteria apply at unbounded height and beyond Hessenberg matrices. Higher-row orientation and general three-block legal generation remain open.