Near-Critical and Grouped Structures in the Collatz Carry Equation: Finite-Block Reduction, 2-Adic Windows, and Tower Products

This four-note sequence studies a narrow zero-entropy and grouped-tower sector of the accelerated odd-only Collatz carry equation through finite-block, Sturmian, and (2)-adic methods. The first note shows that fixed nonsingular near-critical mechanical block families remain effectively reducible, while the infinite continued-fraction tower forms a genuine renormalizing boundary. The second note identifies a native (2)-adic Sturmian carry constant governing the residue dynamics of standard Christoffel blocks. The third note develops an exact nonsingularity ledger for pure standard blocks by separating the obstruction into a shrinking (2)-adic window and an independent odd-prime divisibility layer, with a positive size-collision corollary tied to externally verified convergence bounds. The fourth note extends the program to the first grouped tower-product family (X_5^aX_6^b), proving a prefix-agreement saturation theorem: for (a\ge2), the (2)-adic window stabilizes at depth (83), so one explicit centered residue certificate nonsingularizes the family throughout its certified bounded-size region. The sequence does not claim a proof of the Collatz conjecture; it isolates and certifies specific structured sectors and formulates the remaining obstruction as a precise arithmetic problem involving (2)-adic windows, odd-prime divisibility, and renormalizing tower products.