New Theorems on Accelerated Collatz Map Parity Vectors and Paradoxical Sequences — E8 Intelligence Research

FINDING: The Rozier-Terracol accelerated Collatz map study (arXiv:2605.13886v2) proves three theorems on parity vectors and paradoxical sequences, building on their 2025 preprint (arXiv:2502.00948), with a numerical addition — but the YouTube search results are mostly tangential (popular expositions, Lagarias survey, logistic map, operator proof), not the primary proof itself. | MATH: Accelerated Collatz map: T(n) = (3n+1)/2 for n odd, T(n) = n/2 for n even. Parity vector: v_k(n) = (n mod 2, T(n) mod 2, ..., T^{k-1}(n) mod 2). Terras' theorem: for almost all n, the parity vector determines the trajectory's length to reach 1. Lagarias' 1985 bound: N(x) ≤ (log x)/(log 2) + O(1) for stopping time. Tao's 2019 logarithmic density result: almost all orbits have bounded logarithmic density below 1. The new theorems likely refine conditions for nonexistence of cycles via parity vector constraints — exact equations not extractable from the abstract alone. | CONNECTION: No direct geometric harmo Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com