作为自守Cayley着色图的Collatz函数：$an+b$猜想的可判定性，$3n+1$猜想的证明
Collatz/Bařina a$n$+b conjecture proofs by 16/64: Cayley a$n$+b tree not Cantor-Gödel-Conway-Wolfram undecidable
AI总结：
将Collatz图转换为3-正则自守Cayley着色图及4-正则Middle Pages图，利用欧拉回路证明了$3n+1$猜想，并给出了$an+b$猜想的可判定性准则。
摘要：
Collatz猜想指出，在奇数上重复执行$n\mathrm{\to }\mathrm{3}n\mathrm{+1}$步骤，在偶数上执行$n\mathrm{\to }n\mathrm{/2}$步骤，相当于在一个连通Collatz图的“平凡”循环根$4\to 2\to 1\to 4\to \dots $中，沿着根路径走到分支数$c=4$。带有反向箭头$n \to 2n$和$n \to (n-1)/3$的Collatz图可以转换为3-正则自守Cayley着色图$T_{\ge 0}$，其节点为除以18余4或16的分支数，构成同余类$[4,16]_{18}$。对于$k=1,2,3,\dots$，在二进制数轴上用$2^k$个二进制数标记$2^k$个广度优先排列的根路径，并将它们与这些根路径的$2^k$个输出数字配对，得到$2^k$个配对数。这些配对分支数的3-正则Cayley图可以转换为4-正则Middle Pages图。这个4-正则图为来自同余类$[4,16]_{18}$的所有配对分支数提供了一条往返于平凡根数对{0,c=4}的唯一欧拉回路。这证明了Collatz的$3n+1$猜想。一个特定的$an+b$猜想是否为其所有配对分支数提供欧拉回路，可以通过它是否为小于$2a^3$的配对分支数提供这样的回路来判定。
英文摘要：
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$_{\eta2}$ $7n\pm1$ ($64<196$).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
