  We show how quantized transport can be realized in Floquet chains through
encapsulation of a chiral or helical shift. The resulting transport is
immutable rather than topological in the sense that it neither requires a band
gap nor is affected by arbitrarily strong perturbations. Transport is still
characterized by topological quantities but encapsulation of the shift prevents
topological phase transitions. To explore immutable transport we introduce the
concept of a shiftbox, explain the relevant topological quantities both for
momentum-space dispersions and real-space transport, and study model systems of
Floquet chains with strictly quantized chiral and helical transport. Natural
platforms for the experimental investigation of these scenarios are photonic
Floquet chains constructed in waveguide arrays, as well as topolectrical or
mechanical systems.
