Extends the budget argument under Collatz dynamics for arbitrary initial values, indicating all odd integers converge to the 1-cycle.
Key Points
To extend the budget argument in 2-adic Collatz dynamics for arbitrary valuations and show that all positive odd integers converge to a specific cycle.
Introduced the function φ for odd integers, half-integers, and even integers.
Analyzed the descent of canonical representatives via induction.
Established a mechanism to prevent infinite orbits based on valuation without class-specific results.
Demonstrated equivalence of cycles in compressed and original Collatz functions for odd integers.
Proved that for any class of valuation k, k bits are consumed from the initial parameter per excursion.
Established that all positive odd integers converge to the 1-cycle under Collatz dynamics.
Confirmed that cycles and convergence are equivalent in the original and compressed dynamics.
Cite This Study
Miguel Cerdá Bennassar (2026) studied this question.