I read your links and looked at your graph, I think what you are seeing is the 27.32 day lunar declinational cycle....

If you resort your data by Lunar declination, you will see a combined result much better than Lunar Phase relationships give. ( the improvement from {27.32 day periods}5:1 to 14:1 {for the 109.28 day period}signal to noise results above) as compared to a 1.9:1 ratio for lunar phase relationships.
If you look at dates close to in sync with consideration of the 18.6 Mn signal you will see an even greater signal to noise enhancement.

What I have done is look at this set of data I am using (~22,000 daily reporting stations) for the whole USA, and pulling out the temperatures, or precipitations, for the date 13,550 days ago, and for the dates 6,558 days ahead and behind the center date. roughly 19 years ago, 37 years ago, and 55 years ago, and averaging the three date's Temperatures together for the same reporting stations, making a grid and contour map from the compiled data, to generate a forecast of the current cycle.

This is effectively a daily weather forecast with a 19 year lead time, because of the sorting method, the Lunar Declinational tides effect is modulated by the Synodic conjunctions of the Inner planets, and the dates I have chosen puts the lunar phase, declination, perigee and apogee cycles all back into sync to this cycle for the forecast day, they are all figured in synergisticly.

With the exception of the Synodic cycles of Earth with the gas giants. Where the interference of these planets throws the patterns off and the forecast "misses for the three week period" (as noted above, paragraph 6) centered on the Heliocentric conjunctions of Earth with any of the Gas giants.

The details you are asking for are in my last post you quoted from, with the addition of the size of the data set I am using. The data set is comprised of the TD 3200 daily 24 hour observations of ~22,000 extended stations, from 1885 to 1998.

The proof for me is the accuracy of the "Forecast" that this process generates.