VIII. Return map, rigid regime, and invariance gap in the 2-adic Collatz dynamics

This work proposes a structural reduction of the Collatz dynamics on ℕ to an autonomous dynamical system in the ring of 2-adic integers ℤ₂. By formalising the rigid regime — the phase in which the binary support of the initial parameter is exhausted and the cylindrical constraints trivialise — it is proved that the persistence of an infinite orbit is contingent on a return map F: ℕ × ℤ₂× → ℕ × ℤ₂×. A fundamental dichotomy is established for the survival of any non-trivial orbit: the escape regime (ℓⱼ → ∞) and the stationary regime (ℓⱼ = ℓ*). Both scenarios impose 2-adic convergence to the absorbing point −1, generating a norm tension that forces sustained exponential Archimedean growth. The main gap of the conjecture is localised in the non-invariance of the resonant cylinders Rₛ(ℓ*), and the relationship between this gap and equidistribution within the ergodic components of ℤ₂× is discussed. This paper is part of a series of six works on the Collatz conjecture. In reading order: I. 2-adic structure of tails and survival sets in Collatz dynamics https://doi.org/10.5281/zenodo.18831439 II. Cylinder collision, bit non-reusage, and effective non-degeneration in 2-adic Collatz dynamics https://doi.org/10.5281/zenodo.18831527 III. Arithmetic obstruction to indefinite survival in 2-adic Collatz dynamics https://doi.org/10.5281/zenodo.18831690 IV. Arithmetic obstruction to mixed orbits in 2-adic Collatz dynamics https://doi.org/10.5281/zenodo.18831791 V. The ϕ function and the extension of the 2-adic budget argument to arbitrary k0 in Collatz dynamics https://doi.org/10.5281/zenodo.18831874 VI. Structural reduction of the Collatz conjecture: stretches, portals, and 2-adic survival sets https://doi.org/10.5281/zenodo.18831607 VII. Structure of entries to C1 and the rigid regime https://doi.org/10.5281/zenodo.18879276 VIII. Return map, rigid regime, and invariance gap in the 2-adic Collatz dynamics https://doi.org/10.5281/zenodo.18879361