Global Fixed-Depth Finite Reduction in the Collatz Carry Equation

This draft develops and applies a finite-block obstruction method for the accelerated odd-only Collatz carry equation, focusing on alternating four-block valuation families. For a valuation word (D) of length (L) and valuation sum (S), wall contact is the divisibility condition
[
2^S-3^L \mid C_L(D),
]
where (C_L(D)) is the associated carry sum. The note studies the explicit alternating family
[
C(1/3)^a(4)^bC(1/6)^c(3)^d,
\qquad a,b,c,d\ge1,
]
whose block slopes have sign pattern ((- , + , - , +)) relative to the critical slope (\log_2 3).

The main result is a complete classification of this family: no wall contacts occur for any exponent tuple or phase choice. The proof combines a four-block telescoping identity, cyclic rotation invariance, a rotated (2)-adic lever, a corrected spread comparison, certified continued-fraction bounds for (\log_2 3), and an exact finite computation. The finite-reduction argument confines all possible contacts to an explicit box, and a padded exact check of (2{,}304{,}000) candidates finds zero contacts.

The draft also abstracts the mechanism into a broader fixed-depth finite-reduction framework. For fixed finite-block families with nonsingular blocks and slopes bounded away from the critical line, rotated (2)-adic levers and peeling arguments reduce any wall-contact search to a finite computation. The result does not prove the Collatz conjecture; rather, it identifies a rigorous obstruction mechanism for structured finite-block candidates and locates the remaining difficult regimes: near-critical block slopes, unbounded block depth, singular blocks associated with known cycle structures, and positive-entropy valuation words.