How many odds could a Collatz counter-example cycle have, relative to its length?

General
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Summarizing the answers to this thread’s original question, and taking into account these posts …

k = cycle length
x = number of odd terms

  1. For general integer cycles … k < \lceil x \log_2 3 \rceil + \log_2 x.

  2. For flat integer cycles (ratio of largest member to smallest member <7) … k = \lceil x \log_2 3 \rceil.

  3. For general rational cycles (with all members >1) … k < 2x.

  4. For rational circuits (with all members >1) … k = \lceil x \log_2 3 \rceil.

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