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So while exploring some new ideas, I decided to figure out if I could use variables in the known set to determine the bounds of variables in the unknown set.

The variables in question are algebraic identities derived from the semiprimes, so you already know where this is going.

The existing known set is 1194 identities.
And there are, if I recall, roughly two dozen unknowns.

Many knowns have the unknowns as their factors. The d4 product set for example is composed of variables d4a, d4u, d4z, d4z9, d4z4, d4alpha, d4theta, d4omega, etc.
The component variables themselves are unknown, just their products are known. Anyway.

What I've found interesting is if you know the minimum of some of these subsets, for example d4z is smallest out of the d4's for some semiprimes, then you know the upperbound of both the component variables d4 and z.

Unless of course either of them is < 1.

So the order of these variables, based on value, changes depending on the properties of the semiprime, which I won't get into. Most of the time the order change is minor, but for some variables they can vary a lot between semiprimes, rapidly shifting their rank in the known set. This makes it hard to do anything with them.

And what I found myself asking, over and over again, was if there was a way to lock them down? Think of it like a giant switch board, where flipping one switch lights up N number of others, apparently at random. But flipping some other switch completely alters how that first switch works and what lights it seemingly interacts with. And you have a board of them thats 1194^2 in total. So what do you do?

I'd had a similar notion a while back, where I would measure relative value in the known set, among a bunch of variables, assign a letter if the conditions were present, and generate a string, called a "haplotype."
It was hap hazard and I wrote a lot of code to do filtering, sorting, and set manipulation to find sets of elements in common, unique elements, etc. But the 'type' strings, a jumble of random letters, were only useful say, forty percent of the time. For example if a semiprime had a particular type starting with a certain series of letters, 40% of the time a certain known variable was guaranteed to be above a certain variable from the unknown set...40%~ of the time.
It was a lost cause it seemed.

But I returned to the idea recently and revamped the entire notion.

Instead what I would approach it from a more complete angle.
I'd take two known variables J and K, one would be called the indicator, and the other would be the 'target'.
Two other variables would be the 'component' variables (an element taken from the unknown set), and the constraint variable (could be from either the known or unknown set).

The idea was that relationships between the KNOWN variables (an indicator and a target variable) could be used to indicate the rank relationship between the unknown component variable and the constraint variable.

You'd think this wouldn't work either, but my intuition was there were so many seemingly 'random' rank changes of variables in the known set for any two semiprimes, that 1. no two semiprimes ever shared the same order for every variable, and 2. the order of the known variables had to be leaking information about the relationships of the unknown variables.

It turns out my intuition was correct.

Imagine you are picking a lock, and by knowing the order and position of the first two pins, you are able to deduce the relative position of two pins further back that you can't reach because of the locks security features. It doesn't let you unlock the lock directly, but by knowing this, if you can get past the lock's security features, you have a chance of using information about the third pin to get a better, if incomplete, understanding about the boundary position of the last pin.

I would initiate a big scoring list, one for each known element or identity. And then I would check it in tandem like so:

if component > constraint and indicator > target:
indicator[j]+= 1

This is a simplication, but the idea was to score ALL such combination of relationship, whether the indicator was greater than the target at the same time a component was greater than a constraint, or the opposite.
This worked out to four if checks and four separate score lists.

And by subtracting one scorelist from another, I could check for variables that were a bad fit: they'd have equal probability of scoring for example, where they were greater than the target one time, and then lesser than it for another semiprime.

So for any given relationship, greater or lesser between any unknown variable and constraint variable, I could find any indicator variable and target variable whose relationship strongly correlated to the unknown's.