Visualizing the Collatz Conjecture as a Phase Transition(mathinspector.com) |
Visualizing the Collatz Conjecture as a Phase Transition(mathinspector.com) |
Galois factory, matter protocol... all just technical jargon slop that sounds fancy but carries basically no semantic meaning. The repo’s readme is even worse [0].
As the cherry on top it even closes out with an “it’s not just X, it’s Y.”
starting with 3=1+2 we have (1+x)P(x)=3P(x) when x=2. so we lift the problem from n to P(2)=n. this is a known technique of lifting the problem to a polynomial setting. after each itteration of the Collatz map i make sure all coeffients are either 0 or 1 by applying carry operations when a coefficient overflows. since the coefficients are unary strings, this makes it like a fluid dynamics problem (each character in a unary string is analogous to one unit of mass in a list of buckets where the buckets can overflow and spill unary characters over into their left neihgbor)
when x=2, multiplying x by P(x) is a left shift, whereas dividing by x, P(x)/x, is a right shift. (when P(2)=n is even the constant term in P(x) is zero)
the +1 term in 3n+1 effectively induces a non linear carry propoagation.
the new technique i used is based on a realization that the polynomial representation of the Collatz map behaves like an LFSR implementation of a finite field with a missing modulus. in LFSR a finite field is implemented where each element is an array of bits of fixed size corresponding to a polynomial and multiplication of elements is polynomial multiplication taken mod Q(x) where Q(x) is an irreducible polynomial. unlike the finite field LFSR the Collatz map in polynomial form as i have described allows the degree of the polynomial (size of the array of bits) to grow unbounded.
the surprise is when i subtract these two objects the sierpinski gasket appears and this fractal is not destroyed by itterations of the collatz map
this document[1] is a prior result showing a connection between fractals and collatz that i found after posting the OP
[1] https://upcommons.upc.edu/server/api/core/bitstreams/9bad675...
lesson learned! i will never post an ai slop blog post on here ever again. thanks for the feedback i needed to hear it.
I guess Grok didn't get the memo that fluid dynamics is math.
there is more to this post than just ai slop. there is a real experimental result here.
if you or anyone would like to see the non ai slop version i posted over on math stack exchange without any ai at all
https://math.stackexchange.com/questions/5121753/why-does-th...
Some simple wasm would work a treat here, no idea why this would be doing server side stuff.
Ended up not waiting several minutes despite being pretty interested :/
https://base-1-srrnmbhh3rkmnk8ygcxhvb.streamlit.app/
It seems that waking up the demo from the iframe doesn't always work, but directly visiting the embed url can wake it up.
By the way I just finished completely rewriting the entire blog post.