The Dual Automaton Structure of Collatz Realizability: From the Δk Automaton to the Integer Realization Automaton

This paper gives an accelerated-valuation formulation of a classical correction/residue correspondence in 3x+1 dynamics, organized as a dictionary between a forward dynamical automaton (the Δk Automaton) and a backward residue construction (the Integer Realization Automaton, IRA). Both automata are shown to share one arithmetic correction quantity Bk, giving a Common Correction Core linking forward orbit correction and backward finite-prefix compatibility residues. The finite-prefix congruence at the heart of this correspondence is the accelerated-coordinate counterpart of the classical parity-vector theorem of Terras (1976), surveyed by Lagarias (1985); the forward/backward reconstruction it formalizes is a finite-level relative of the 3x+1 conjugacy map of Bernstein (1994) and Bernstein-Lagarias (1996); and its generalization to an+1 dynamics for odd a sits inside the broader generalized-Syracuse literature of Möller (1978), Matthews-Watts (1984), and Bernstein-Lagarias (1996). No new arithmetic content is claimed in this correspondence; a precise comparison with this classical literature is given in the paper (Section 1.2). What this paper adds: (i) an explicit, exact-arithmetic accelerated-coordinate dictionary between the two automata, organized around the shared correction quantity Bk; (ii) a reproducible computational layer (bounded-drift correction-ratio envelope, exhaustively verified up to N ≤ 3×10^6, with accompanying code and data); and (iii) a quantitative, explicitly conjectural, discussion of a restricted bounded-drift candidate class for Collatz counterexamples, kept strictly separate from the proved correction algebra. This paper does not prove the Collatz conjecture. Part of a four-paper program on Collatz valuation realizability, together with "Tail Dependence and Finite-Address Obstructions in Collatz Valuation Dynamics" (DOI: 10.5281/zenodo.21468378), "Integer Stabilization and the 2-adic Digit Tower in Collatz Valuation Dynamics," and "Ghost Languages and Unbounded Prefix Memory in Collatz Realizability." Accompanying code and data (exact-arithmetic verification of all identities, N ≤ 3×10^6 correction-ratio envelope, generalized-a drift statistics) are included in this record.