Structured Sectors of the Collatz Carry Equation: Christoffel Towers, 2-Adic Windows, and Shell Residents

This six-note sequence studies structured zero-entropy and grouped-tower sectors of the accelerated odd-only Collatz carry equation through finite-block, Sturmian, continued-fraction, 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 fifth note formalizes the bit analyzer (\ell(D)=v_2(Q(D)+\Xi_\alpha)) as a reusable diagnostic, showing that it detects saturation depths, window recycling, and exact Christoffel/Stern--Brocot degeneracy identities across several structured families. The sixth note proves a general adjacent-pair saturation taxonomy for standard tower products (X_n^aX_{n+1}^b): upper-leading pairs saturate at depth (p_n-1), while lower-leading pairs saturate at depth (p_n+p_{n+1}-1) for (a\ge2), with the (a=1) branch governed by standard Christoffel factorization and semiconvergent landings. 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, residue genericity, and the behavior of non-mechanical or positive-entropy valuation words.