Though I demonstrated a hard upperbound on the D(10) dedekind in the link here (https://devrant.com/rants/8414096/...), a value of 1.067*(10^83), which agrees with and puts a bound on this guy's estimate (https://johndcook.com/blog/2023/...) of 3.253*10^82, I've done a little more work.

It's kind of convoluted, and involves sequences related to the following page (https://oeis.org/search/...) though I won't go into detail simply because the explaination is exhausting.

Despite the large upperbound, the dedekinds have some weirdness to them, and their growth is non-intuitive. After working through my results, I actually think D(10) will turn out to be much lower than both cook's estimate and my former upperbound, that it'll specifically be found among the values of..

1.239*(10^43)
2.8507*(10^46)
2.1106*(10^50)

If this turns out to be correct (some time before the year 2100, lol), I'll explain how I came to the conclusion then.

This morning I was exploring dedekind numbers and decided to take it a little further.

Wrote a bunch of code and came up with an upperbound estimator for the dedekinds.

It's in python, so forgive me for that.

The bound starts low (x1.95) for D(4) and grows steadily from there, but from what I see it remains an upperbound throughout.

Leading me to an upperbound on D(10) of:
106703049056023475437882601027988757820103040109525947138938025501994616738352763576.33010981

Basics of the code are in the pastebin link below. I also imported the decimal module and set 'd = Decimal', and then did 'getcontext().prec=256' so python wouldn't covert any variable values into exponents due to overflow.

https://pastebin.com/2gjeebRu

The upperbound on D(9) is just a little shy of D(9)*10,000
which isn't bad all things considered.

Probably pure coincidence but if you look at the deconstruction of the dedekinds like so:

>>> decon(6)
offset: 1, exp: [[Decimal('2'), Decimal('1')], [Decimal('3'), Decimal('1')]]

>>> decon(20)
offset: 2, exp: [[Decimal('2'), Decimal('2')], [Decimal('5'), Decimal('1')]]
offset: 1, exp: []

>>> decon(168)
offset: 3, exp: [[Decimal('2'), Decimal('2')], [Decimal('5'), Decimal('2')]]
offset: 2, exp: [[Decimal('2'), Decimal('2')], [Decimal('3'), Decimal('1')], [Decimal('5'), Decimal('1')]]
offset: 1, exp: [[Decimal('2'), Decimal('3')]]

>>> decon(7581)
offset: 4, exp: [[Decimal('2'), Decimal('3')], [Decimal('5'), Decimal('3')], [Decimal('7'), Decimal('1')]]
offset: 3, exp: [[Decimal('2'), Decimal('2')], [Decimal('5'), Decimal('3')]]
offset: 2, exp: [[Decimal('2'), Decimal('4')], [Decimal('5'), Decimal('1')]]
offset: 1, exp: []

>>> decon(7828354)
offset: 7, exp: [[Decimal('2'), Decimal('6')], [Decimal('5'), Decimal('6')], [Decimal('7'), Decimal('1')]]
offset: 6, exp: [[Decimal('2'), Decimal('8')], [Decimal('5'), Decimal('5')]]
offset: 5, exp: [[Decimal('2'), Decimal('5')], [Decimal('5'), Decimal('4')]]
offset: 4, exp: [[Decimal('2'), Decimal('6')], [Decimal('5'), Decimal('3')]]
offset: 3, exp: [[Decimal('2'), Decimal('2')], [Decimal('3'), Decimal('1')], [Decimal('5'), Decimal('2')]]
offset: 2, exp: [[Decimal('2'), Decimal('1')], [Decimal('5'), Decimal('2')]]
offset: 1, exp: [[Decimal('2'), Decimal('2')]]

>>> decon(d('2414682040998'))
offset: 13, exp: [[Decimal('2'), Decimal('13')], [Decimal('5'), Decimal('12')]]
offset: 12, exp: [[Decimal('2'), Decimal('13')], [Decimal('5'), Decimal('11')]]
offset: 11, exp: [[Decimal('2'), Decimal('10')], [Decimal('5'), Decimal('10')]]
offset: 10, exp: [[Decimal('2'), Decimal('11')], [Decimal('5'), Decimal('9')]]
offset: 9, exp: [[Decimal('2'), Decimal('9')], [Decimal('3'), Decimal('1')], [Decimal('5'), Decimal('8')]]
offset: 8, exp: [[Decimal('2'), Decimal('10')], [Decimal('5'), Decimal('7')]]
offset: 7, exp: [[Decimal('2'), Decimal('7')], [Decimal('5'), Decimal('6')]]
offset: 6, exp: []
offset: 5, exp: [[Decimal('2'), Decimal('6')], [Decimal('5'), Decimal('4')]]
offset: 4, exp: []
offset: 3, exp: [[Decimal('2'), Decimal('2')], [Decimal('3'), Decimal('2')], [Decimal('5'), Decimal('2')]]
offset: 2, exp: [[Decimal('2'), Decimal('1')], [Decimal('3'), Decimal('2')], [Decimal('5'), Decimal('1')]]
offset: 1, exp: [[Decimal('2'), Decimal('3')]]

the powers in the 2's column go:
1, 2, 2, 2, 3, 3, 2, 4, 6

which are predicted by:
https://oeis.org/search/...

Again, probably only a coincidence, but kinda beautiful.

Exploring dedekinds and kronecker products (script at - https://pastebin.com/dDuT3dTp)

and the thing I immediately notice, if you output the matrix is that it is a lower triangular matrix. I don't know a lot about the kronecker or matrices in general, but if dedekinds can be generated in this manner, shouldn't some standard approaches like back substitution or forward substitution be applicable here or am I off in left field on this?

Established a new *much* tighter bound on the value of the D(10) dedekind.

Pasted code here:
https://pastebin.com/xYSND9NN

It's almost beyond doubt to be located somewhere between 10^75 and 10^77.