V. The function φ and the extension of the 2-adic budget argument to arbitrary k₀ in the Collatz dynamics

This paper completes a series on the 2-adic analysis of the compressed Collatz dynamics by extending the global budget argument to arbitrary initial classes. Prior works proved convergence for the class with valuation 2; here, the function φ is introduced, defined on odd integers, half-integers, and even integers. It is shown that φ descends the canonical representatives and acts as a 2-adic read head: each excursion in a class of valuation k consumes exactly k bits from the initial parameter, independently of prior excursions. This mechanism, established via induction without relying on class-specific results, allows the budget argument to rule out infinite orbits for any starting valuation. The paper also proves that cycles and convergence under the compressed dynamics and the original Collatz function are equivalent for positive odd integers, yielding the corollary that all such orbits converge to the 1-cycle. The analysis is purely arithmetic, building on prior results without new proofs. Version 1.1: Minor corrections and editorial revisions. 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