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

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). 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 contrast, we prove that this coincidence fails for all x ≥ 5 (Theorem C). For x ≥ 5, the GPK depends on correlations between adjacent pairs, but since all 16 predicates are per-pair independent, they cannot express inter-pair correlations. From these results it follows that the carry structure of the map closes within the predicate space only for x=3 (Classification Theorem). Whether this closure property contributes to the dynamical difference between 3n+1 and 5n+1 remains open, but the structural singularity of the parameter x=3 has been characterized within the predicate framework. Read online (HTML with interactive MathJax formulas): English version Japanese version / 日本語版 Project page Source code: github.com/onigirito/collatz-m4m6 Files in this record: collatz_paper_en.pdf — Main paper (English, viewable on this page) collatz_paper_en.html / collatz_paper_ja.html — Interactive HTML with MathJax TOOL_SPEC.html — Tool specification performance_report.html — Performance analysis