so, just to be clear, you're acknowledging that these predictions are based on tables of coordinates of the sun and moon as described by these two ephemerides, yes? doesn't that kind of undercut your argument that they're based on saros cycles?
The Saros cycle just gives a time when to expect the next lunar eclipse to appear on the face of the moon. You will also need to know if you will be able to see the moon from your particular location. This is where the equations to find the coordinates of the moon come in.
The Lunar Eclipse is visible for anyone who can see the moon, and you will need to know whether the moon will be over your area at that time, not merely the time of the lunar eclipse.
The Solar Eclipse is visible for only a narrow path beneath the moon, and knowing the coordinates of the sun is necessary for knowing whether a solar eclipse will be visible in your area, not merely the time it will occur.
The coordinates of the sun and moon over the earth have nothing to do with computing the time of the lunar eclipse. It does not say that in the
Five Millennium Canon of Solar Eclipses PDF. The only points it brings up is a
direct statement that the eclipses are predicted based on the Saros cycle in section 1.2.2 "Saros Series Number"
"Each eclipse belongs to a Saros series (Sect. 4.2) using a numbering system first introduced by van den Bergh (1955)."
In 4.4 "Saros Series Statistics" we read:
"Eclipses belonging to 204 different Saros series fall within the five millennium span of the Canon."
The document also states the coordinates for the sun and moon are used in section 1.3 "Solar and Lunar Coordinates":
"The coordinates of the Sun used in these eclipse predictions have been calculated on the basis of the VSOP87 theory
constructed by Bretagnon and Francou (1988) at the Bureau des Longitudes, Paris."
"For the Moon, use has been made of the theory ELP-2000/82 of Chapront-Touzé and Chapront (1983), again of the
Bureau des Longitudes."
The section says nothing about the coordinates of the sun or moon being used to predict when the eclipse will occur. The coordinates of the sun is used in the eclipse predictions, but the knowledge of coordinates is only useful because it will help create a map showing where the eclipses will be visible from, which is what the map sections of the document is about.
You seem to be saying that there is some special math not mentioned in the document where the time of the eclipse can be predicted without the use of the Saros cycle, despite the document stating that each eclipse is based on a Saros cycle series.