This post has content warnings for: venting, (about math).
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Sen pro Kyne. Emotionally overavailable.
Yeah I mean https://cohost.org/sen-guide but I'm going through some things right now
I keep going back to it to give it another chance at solving some basic problems, and every time it disappoints. It's particularly pernicious, because it confidently throws out nicely-formatted formulas with no regard for what those formulas actually mean. It's so confident that I had to run the formulas it spat out just to make sure it wasn't saying anything novel.
I'm very happy with my progress here! There is currently only a single remaining open problem that I'm working on relating to sieved matrices, which I discussed in this chost. Despite its roots in the INVERT transform and linear recurrences in general, it is wholly its own beast with its own eccentricities and related problems.
It's been really surprising to find how detailed the relationship between the signature function and base sequences, with so many identities directly relying on the exactness of the signature function. The only area which is lacking is the presence and application of signature convolution, but I've come to terms with the fact that I'm mortal and thus only capable of doing so much. But this definitely isn't the end!
This section of my paper deals with the result of a weird generalization of a construction adjacent to Cantor's diagonal argument. The lower generalization (62) has caught me in traps multiple times when I've gone to verify that I didn't miss something or mess up.
The funny thing is that the identities ultimately aren't as important as the "sen" matrix itself. That's where the Cantor relationship comes in. It's actually based off of a funky logarithmic process that I came up with to try and reframe CDA in terms of matrix multiplication, which eventually succeeded in transforming an arbitrary d into d + 1. That kinda made me go "Huh. Maybe the Continuum Hypothesis is False" and the rest is history.
I've reached a point where whenever I go to review these formulas, I have to remind myself that they've been correct every previous time I've evaluated them. What usually happens is an off-by-one error sneaks into my test code and makes it look like it's wrong. The last time I verified it it took over an hour, but thankfully resolved to True.
The major hurdle for most of SNR is that these formulas aren't proven, not even algebraically - they just exist and in every case I've tested they work. But as you may have wondered, these structures are very axiomatically heavy, and so proving these core relationships at such a low level is a huge (if not impossible) task. It's further complicated by the relationship with another near-ring (a right one this time) which connected sequence addition to the method of matrix multiplication which I use. There's a lurking, hidden beast residing at the interesection of the right and left near-rings, propelled by the signature function.
Of course, some code:
from cxr import Seq, Matrix
d = Seq(1, 1)
l = 10
g = [Seq(1), Seq(2, 1), Seq(1, 1)]
S = Matrix.g_matrix(Matrix.sen(d, l), g, l)
output2 = Seq([0 for _ in range(l)])
output = Seq(output2)
for n in range(l):
for k in range(n+1):
p = S ** (k+1)
r = S ** (n-k)
output[n] += p[k][0]
output2[n] += r[k][0]
print(output)
right_side = d + sum([e.sig() for e in g]).seq
print(right_side.f())
print(output2)
right_side = Seq(-1) + sum([e.sig() for e in g]).seq
right_side = right_side.sig() + (d * d.neg().f()).sig()
print(right_side.f().f())
I tend to get carried away when it comes to math, so having a separate page for that sort of focus is really nice 
So if you find yourself interested in my math-based musings, I suggest you follow @radix.