arXiv2026
Lothar Collatz considered biperiodic iterations, and conjectured that picturing $n\mathrm{\to3}n\mathrm{+1}$ iterations for odd numbers and $n\mathrm{\to}n/2$ for even numbers, and proving that all natural numbers have a root path to the `trivial' cyclic root $(c=4)\to2\to1\to c\to\dots$, would connect elementary graph theory and number theory. John H. Conway replied that a$n$+b conjectures, except 1$n$+1, are undecidable due to a-periodic even $a\,\nmid\;$n$=a$n$_\text{odd}$+b successors. Biperiodic a$n$+b conjectures give disproofs of undecidability by Cantor diagonals, Gödel numbers, Conway unpredictability, and Wolfram irreducibility. The Cayley tree of Collatz-Conway $a\,\nmid\;$n$=a$n$_\text{odd}$+b numbers paired with Cantor binary tree numbers $y_{\ge0}$ is the $E=V=F=\mathbb{N}_0$ tree $T_{\ge 0}$ with its cyclic root $c=1a+1$ paired with $0$ and arrows ($E$dges) to wondrously ordered same-colour arrowhead nodes $a\,\nmid\;$n$=a$n$_\text{odd}$+b ($V$ertices) with same-numbered upwardly open sites above (sur$F$aces), paired with well-ordered root paths $y$, projecting to the Cartesian $y$-axis well-ordered sites $n$, paired with wondrously ordered root paths $y$. Tree $T_{\ge0}$ transforms into the integer Cartesian upper half graph $C_H$, providing a Hamiltonian tour from and back to the root $c$ at its origin $(0,0)$. The infinite Turing-Leibniz tape $T_L$ of proven $an_{odd}+b$ conjectures shows Turing-computable roots $c_{η\ge1}=2^{2^{η-1}+1}$ and $c$-expressible lowest trunk nodes $c^2$, 'cotree density' Hamiltonian tour proofs, and 'root-trunk' proofs $c^2<\frac{c}{2}(c-1)^2$. The $c$-roots generate also proven Mersenne prime $a=c-1$ conjectures: Collatz$_{\eta1}$ 3$n$+1 ($16<18$) and Bařina$_{\eta2}$ $7n\pm1$ ($64<196$).