calhoun137

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

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

> Is there any way to deduce which invariance gives which conservation?

Yes. See Landau vol 1 chapter 2 [1].

> I'm looking for the fundamental reason, as well as how to tell what will be paired with some invariance when looking at some other new invariance

I'm not sure there is such a "fundamental reason", since energy, momentum, and angular momentum are by definition the names we give to the conserved quantities associated with time, translation, and rotation.

You are asking "how to tell what will be paired with some invariance" but this is not at all obvious in the case of conservation of charge, which is related to the fact that the results of measurements do not change when all the wavefunctions are shifted by a global phase factor (which in general can depend on position).

I am not aware of any way to guess or understand which invariance is tied to which conserved quantity other than just calculating it out, at least not in a way that is intuitive to me.