VII. Structure of entries to C1 and the rigid regime

This technical note identifies precisely the point where the remaining obstacle in the 2-adic argument applied to the Collatz conjecture is concentrated. It is proved that any infinite orbit initiated in ℕ must eventually enter a rigid regime in which the initial parameter ceases to play a role and the dynamics are governed by an autonomous recursion on 2-adic residues. In that regime, the cylindrical constraints trivialise and the remaining problem becomes purely dynamical. The obstacle is reformulated in terms of 2-adic convergence towards a target point, and a natural sufficient condition is identified — the bi-Lipschitz 2-adic regularity of the entry parametrization — that would allow the argument to be closed. The document delineates precisely the boundary between what is proved and what remains open. 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