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
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
My experience leads to the same conclusion that the models are very good at math reasoning, but you have to really know what you are doing and be aware of the blatant lies that result from poorly phrased queries.
I recently prompted Gemini Deep Research to “solve the Riemann Hypothesis” using a specific strategy and it just lied and fabricated the result of a theorem in its output, which otherwise looked very professional.
> Does replacing that lengthy text with "if you aren't sure of the answer say you don't know" have the same exact effect?
i believe it makes a substantial difference. the reason is that a short query contains a small number of tokens, whereas a large “wall of text” contains a very large number of tokens.
I strongly suspect that a large wall of text implicitly activates the models persona behavior along the lines of the single sentence “if you aren't sure of the answer say you don't know” but the lengthy argument version of that is a form of in-context learning that more effectively constrains the models output because you used more tokens.
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.