Visualizing the Collatz Conjecture as a Phase Transition(mathinspector.com)
mathinspector.com
Visualizing the Collatz Conjecture as a Phase Transition
https://mathinspector.com/papers/collatz-polynomial-entropy.html
7 comments
I also assume AI here. There is no visible scientific connection, no references and it all sounds nonsense. I have a PHD in fluid dynamics but cannot relate.
OP here. this has been a valuable learning experience for me. i was so excited to share what i was working on and i blew it. i will rewrite the blog post and readme later. let me at least briefly explain what i did as a reply to your comment
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.
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.
> The Collatz Conjecture isn't just a math problem. It's a fluid dynamics problem.
I guess Grok didn't get the memo that fluid dynamics is math.
I guess Grok didn't get the memo that fluid dynamics is math.
OP here, you completely caught me. I used ai to generate that blog post and lightly edited it. lesson learned! moving forward i will type up from scratch any blog post i post on here. sorry about that.
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...
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...
> Hmm... looks like it's taking longer than normal. Check back in a minute or two!
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 :/
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 :/
You can find the live demo here:
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.
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.
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.”
[0]: https://github.com/calhoun137/Base-1