"I got 35 from here" ++ Yes, that 35 are the also mentioned 31 may be plausible,
but does not take transpositions into account.
It often happens that moves A and B are available, but at the next moves AB or BA are available, collapsing to the same position.
A position should be considered once only, regardless of how it was reached.
Transpositions are no exception, but the rule.
10^38 = 31^25
With a non transposing branching factor of 31 all chess games would be over in 13 moves.
There can be no more chess positions than there are chess positions.
That is the pigeonhole principle
It's funny how quickly you go from "Chess has branching factor of 35? Where did you get that from?" to "Yes, chess has a branching factor from 31 to 35, but [blah blah blah...]". You routinely get caught out in these types of mistakes, and then have to rapidly integrate them into your narrative.
Bringing up the pigeonhole principle as if it somehow cements your new rationalization is like adding "...and remember, ice is cold" to the end of an argument about glaciers melting...it only impresses the children in the audience that don't know any better and will happily accept anything written at a certain level of vocabulary.
You can either evaluate the pigeonhole boxes standalone, going backwards from mate, or going in a forward direction, you can evaluate them, but only with information in hand that you don't have until you can perfectly evaluate a single position (and that perfection going backwards from mate is why tablebases are created that way, such perfection of single position evaluations does not exist in the opening or middlegame). Whether AB or BA are actually identical in result is not something you can determine without calculation. If you are calculating to determine whether a transposition is actually the same in either order, then you are not actually working with 10^44, you are working with the larger game tree. Move order matters...ask any super GM that has forgotten their prep...
Here's where I think your fundamental mistakes lie...at each step of your reductions in magnitude, you double, triple, and possibly quadruple count the elimination of positions. 10^44 already is a reduction to *unique positions only*. Saying that you can take another square root out of that number (or worse, from your dubious 10^34 which is where the triple and quadruple counting occurs) because of transpositions is incorrect, because there are no transpositions in a set of unique positions...they were already eliminated to reach the set of unique positions. Your argument tries to apply it's largest reduction twice and then toss in some vague fuzzy reductions on top, all to get you to the 10^17 number you had in mind before you started. That's the opposite of the scientific method, cherry picking and fudging numbers/results to fit your hypothesis.
nah bro tygxc is downvoting all of your comments lol 💀
Yes the down votes don't even matter I only down vote when I see a bunch cause I like seeing numbers go up