No Non-Trivial Collatz Cycle Has Twenty-Three or Fewer Odd Steps: Cycle Elimination, Descent Analysis, and Residue-Class Structure

We develop a systematic method for eliminating non-trivial cycles of the Collatz map \(T(n) = n/2\) if \(n\) is even, \(T(n) = 3n+1\) if \(n\) is odd, and complement it with a stopping time analysis toward the full conjecture. Previous cycle bounds are conditional: Simons and de Weger [4] showed that cycles with \(j \le 68\) odd steps must start above \(10^{10}\), but this does not exclude such cycles. Our main result is unconditional: no non-trivial cycle with \(j \le 23\) odd steps exists at any starting value, regardless of cycle length. This is, to our knowledge, the first universal odd-step-class elimination result.

Cycle elimination. For a hypothetical cycle of length \(k\) containing \(j\) odd steps in parity pattern \(P\), the cycle equation yields \(D \cdot n = Q(P)\) where \(D = 2^{k-j} - 3^j\) and \(Q(P)\) is a pattern-dependent constant. When \(D \nmid Q(P)\) for every valid pattern, no cycle exists at that \((j,k)\).

We prove three cycle elimination results:

1. Universal \(j\)-closure for \(j \le 23\): No non-trivial cycle has twenty-three or fewer odd steps, regardless of cycle length. For \(j \le 7\), this combines Diophantine pattern elimination with asymptotic convergence. For \(j = 8, \ldots, 20\), it relies on orbit verification (\(n = 2, \ldots, 9200\) all reach 1) together with asymptotic threshold bounds \(M_j \le 2781\). For \(j = 21, 22, 23\), Diophantine elimination covers the small-\(k\) ranges (35.67 billion patterns total, zero divisibility) and the asymptotic bounds \(M_{21} \le 3599\), \(M_{22} \le 9179\), \(M_{23} \le 9027\) are all covered by orbit verification to \(n = 9200\).

2. Cycle length lower bound \(\ge 63\): Exhaustive computational verification of 35.67 billion parity patterns across \(j \le 23\) and \(k\) up to 80. Zero divisibility events found. The bound follows from \(j\)-closure through \(j = 23\) combined with the power bound \(3^{24} > 2^{38}\), which rules out any \(j \ge 24\) cycle at \(k \le 62\).

3. The \(D \nmid Q\) elimination method: A general, scalable framework where each \((j,k)\) pair requires checking finitely many patterns, making universal closure decidable for each odd-step class.

Stopping time analysis. We analyze the stopping time \(\sigma(x) = \min\{r : S^r(x) < x\}\) of the Syracuse map \(S(x) = (3x+1)/2^{v_2(3x+1)}\) to address the complementary problem of divergent orbits. Three independent analyses establish:

4. Universal descent to \(10^8\): All odd \(x \le 10^8\) have finite stopping time, with maximum \(\sigma(63{,}728{,}127) = 237\).

5. Exponential escape cone decay: The density of odd integers with \(\sigma > d\) decays exponentially as \(P(\sigma > d) \approx 0.921^d\), with the theoretical (independent-\(v_2\)) model matching empirical data through depth 22 and a f