A p-Adic Lever for Two-Block Mechanical Families in the Collatz Carry Equation

This series develops a time-axis approach to structured sectors of the Collatz cycle problem. Instead of studying only the numerical orbit values, the work studies the valuation path formed by the powers of two removed at each odd step. This path records how a possible cycle moves relative to the critical slope (\log_2 3).

The main idea is that highly structured time-axis patterns, especially repeated two-block mechanical patterns, compress the Collatz carry equation into a much smaller arithmetic object. Once this compression occurs, exact (p)-adic divisibility and Diophantine approximation can be used to obstruct wall contact. In this setting, the obstruction is not statistical: the structure forces arithmetic constraints that become too rigid to satisfy except in known negative-cycle calibration cases.

The series shows that the known (-17) negative cycle sits naturally inside this two-block framework and can be completely classified: within its infinite two-block family, the only wall contact is the known (-17) contact itself. Broader finite-depth two-block tests show the same mechanism recurring, with known negative cycles appearing as calibration exceptions rather than new phenomena.

The contribution is not a proof of the Collatz conjecture. Rather, it is a structural obstruction program for finite-block time-axis families. It shows that when Collatz valuation patterns are sufficiently regular, the carry equation becomes rigid enough for exact arithmetic methods to act. This provides a foundation for studying increasingly complex finite-block sectors before confronting the irregular, positive-entropy part of the problem.