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.
OP here, I just am getting back into my open source work and decided to write up all the scripts for my new video's in LaTex. You can find a video version of this paper on my channel here: https://www.youtube.com/watch?v=SWXDr6IlsbA&ab_channel=TheOn...
my understandinf is that conservation of momentum does not mean momentum is conserved as time passes. it means if you have a (closed) system in a certain configuration (not in an external field) and compute the total momentum, the result is independent of the configuration of the system.
The key point is that energy, momentum, and angular momentum are additive constants of the motion, and this additivity is a very important property that ultimately derives from the geometry of the space-time in which the motion takes place.
> 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.
Very nice article! I recently had a long chat with chatgpt on this topic, although from a slightly different perspective.
A neural network is a type of machine that solves non linear optimization problems, and the principle of least action is also a non linear optimization problem that nature solves by some kind of natural law.
This is the one thing that chatgpt mentioned which surpised me the most and which I had not previously considered.
> Eigenvalues of the Hamiltonian in quantum mechanics correspond to energy states. In neural networks, the eigenvalues (principal components) of certain matrices, like the weight matrices in certain layers, can provide information about the dominant features or patterns. The notion of states or dominant features might be loosely analogous between the two domains.
I am skeptical that any conserved quantity besides energy would have a corresponding conserved quantity in ML, and the Reynolds operator will likely be relevant for understanding any correspondence like this.
iirc the Reynolds operator plays an important role in Noethers theorem, and it involves an averaging operation similar to what is described in the linked article.
I don't see any evidence here for "self awareness", among other things, ChatGPT is simultaneously answering a very very large number of queries, and the underlying hardware is just a bunch of servers in the cloud. Furthermore, what would it even mean for "ChatGPT to become self aware" and how could we measure if this had taken place or not? Without a solid definition and method of measurement, it's meaningless to talk about abstract concepts like "self awareness".
Nevertheless, a sensible definition for self awareness is some kind of neural network that becomes aware of its own activity and is in some way able to influence its own function.
After considering these issues for a long time, I came to the conclusions that
1. It's impossible for a program running on a normal computer to have self awareness (or consciousness), because those things are essentially on the hardware level and not the software level
2. In order to create a machine that is capable of self awareness (and consciousness) it is necessary to invent a new type of computer chip which is capable of modifying its own electrical structure during operation.
In other words, I believe that a computer program which models a neural network can never be self aware, but that a physical neural network (even if artificially made) can in principle achieve self awareness.
Since there is no way to 100% fingerprint a device, therefore there is no way to uniquely identify anyone with 100% confidence using pure fingerprinting techniques.
My view is that fingerprinting is a set of tools which can be used for "good or evil" if that makes sense. If you are gathering meta-data to determine the capabilities of the device, then this is part of the wider framework of data points which can, in principle, be used for fingerprinting a user. This data can be imported into a completely different system by a sophisticated adversary, so it needs to be treated as a security vector, imho
Thats true! But thats sort of like saying there was an accident on the highway because 2 cars crashed. There are lots of ways a configuration change could disrupt service, all I am getting at in my previous comment was having even more transparency when stuff like this happens instead of boilerplate statements about config changes, is something I would appreciate and this would give me more confidence in the dev team as well.
This must be a configuration update that went wrong. I am sure of this because thats always the PR angle. Does anyone know what is really going on here?
Since these vaccines got emergency use authorization, that means they did not follow the standard procedure for clinical trials.
Therefore, precautionary measures which respond dynamically to trends detected in newly available data, is the logical, ethical, and scientifically correct thing to do, imo.
> Doesn't even a week's delay in vaccinating predictably increase the number of deaths in a country
One of the many unknowns about these vaccines is the length of time they give immunity, this can only be determined with confidence by looking at the data after a sufficient amount of time has passed. If the immunity only lasts for say 10-20 weeks, then getting it one week early, would mean the immunity goes away one week early as well. So in this case, I'm not sure there would be a major measurable impact. If the immunity lasts for say 50 weeks, that would be a different story.
I think the reason mathematicians use single letter variables is because mathematics is about the way that people think about patterns, understand patterns, and the relationship between them. Therefore the letter is used to help our brain understand the connection between abstract concepts (Categories).
As opposed to programming languages, where the goal is to do something practical, in pure mathematics, our goal is to create a language capable of helping our brains understand the infinite complexities of nature.
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.