Pair Predicate Decomposition of Collatz-type Maps (xn+1)/2^d and the Structural Closure of 3n+1

This paper expresses simple arithmetic in dense formal notation. A plain-language summary with worked examples and trajectory visualizations is strongly recommended as the primary entry point: English: onigirito.github.io/collatz-m4m6/docs/summary_en.html 日本語: onigirito.github.io/collatz-m4m6/docs/summary_ja.html We decompose the binary representation of a natural number n into 2-bit pairs and introduce a system of 16 Boolean predicates defined on each pair, with the left-bit projection m4 and right-bit projection m6 serving as basis. Within this framework, we decompose the additive structure of Collatz-type maps T(n) = (xn+1)/2d and classify the carry behavior at each pair position as Generate/Propagate/Kill (GPK). The resulting computation is a 4-state deterministic finite-state transducer (DFST); all properties derived in this paper follow from exhaustive analysis of its 16-entry transition table. Our main result is the following. For x=3, the GPK of the upper stage (m4-stage) of the addition coincides, at every pair position, with the intra-pair predicates m2(AND) and m7(XOR) of the input n itself (Theorem B). That is, the carry structure of 3n+1 can be read directly from the encoding of n. We call this property structural closure -- in other words, the information needed to compute 3n+1 is already encoded in n itself. In contrast, we establish that this coincidence fails for all x ≥ 5 (Theorem C). From these results it follows that the m4-stage carry structure of the map closes within the predicate space only for x=3 (Classification Theorem). This per-pair closure, combined with exhaustive finite case analysis of the scan and re-pairing transitions, yields a deterministic finite-state system in which the activation condition for carry generation (G-dominance) and the activation result (carry-induced destruction of G-blocks) are inseparably coupled, rendering divergence of 3n+1 trajectories structurally impossible (Proposition 5.4). This paper has not been submitted for peer review, nor is there any plan to do so. Independent verification, citation, and reproduction are welcomed without restriction. Files in this record: m4m6_xn.pdf -- Main paper (LaTeX-compiled PDF) m4m6_xn.tex -- LaTeX source collatz-gui.exe -- Verification tool v0.4.0 (Windows x86_64) gui_verify_*.txt -- Raw GPK measurement data for 3n+1, 5n+1, 9n+1 Source code: github.com/onigirito/collatz-m4m6 | Project page: onigirito.github.io/collatz-m4m6