“Generative AI is a sack of wet garbage”

[Copernicus]

You are right back to missing bilby's point--that declarative statements need not be either true or false. They can simply lack a binary truth value

No, I'm pretty sure I agreed to that part quite vigorously. The part I engaged with and expanded on was exactly the part that I cared to actually mention: that sometimes the application of language doesn't just lack a binary truth value, sometimes it is a completely invalid construction of language, a syntax error.

What is the syntax error that you are referring to? All of the sentences under discussion are syntactically well-formed.

The present king of France is bald" could be true or false in a historical or fictional text, but it lacks a truth value in the context of the real world

And it can lack a truth value for two different reasons, one because it is lacking a truth value owing to not even really being a valid application of language and one owing to not being a complete application of language and these are different situations.

You could actually mention Kurt Gödel's name, if you are going to talk about completeness. He was a mathematical logician, but he was still working within the limitations of a closed context free symbolic system that simply lack presuppositions. Gödel proved that such systems are inherently incomplete. Expressions in such languages cannot be self-referential.

Natural languages are fundamentally different. They represent context sensitive systems in which the meaning of words and word combinations can be inherently vague and ambiguous. They can only be understood (disambiguated) in the context of experiences that the sender and receiver of the signal have in common. That is, presuppositional knowledge is inherent in the encoding and decoding of messages. Natural language is a system of signals, just like symbolic systems, but they are far more efficient at conveying information precisely because presuppositions act as natural mechanisms for signal compression. Because people associate words with a wide variety of experiences grounded in bodily interactions with reality, those experiences can be used to compress the number of words and word combinations needed to convey information. As I said in the original post that you took issue with, the meaning of logical and mathematical symbolic expressions cannot adequately express the full meaning of sentences without the construction of extremely complex formulae. That's because all of the meaning in such expressions must be explicitly represented in the symbolic logical "signal".

Natural language is far more efficient at conveying complex ideas. Natural linguistic expressions can also be self-referential. They can give rise to paradoxical ideas that can sometimes be useful. At the very least, they help to keep philosophers employed.

"This sentence is itself in the form of a declarative sentence" is trivially true

Tautologies are circular references that don't make the obvious mistake. Such statements can prove anything, including that true is false, and rather isn't the sort of circular reference being referenced. The existence of tautologies doesn't counter the existence of nonsense, and aren't what I was talking about. Apparent tautologies still need to actually be tested and cannot be taken for "true" except in the presence of other axioms which test them AFAIK.

Contradictions can be used to prove anything. Tautologies prove nothing. Both are extremely useful in natural language communication, however. To understand why, I refer you to the brilliant work of the philosopher Paul Grice on conversational implicature. If you are unfamiliar with the nature of presuppositional implicatures, I'll leave it to you to do your own research on the subject rather than bother with trying to explain it here. If you don't care, that's your business. I used to be paid to explain such things. Now I'm retired. My pension and annuities don't require me to do that kind of work.

Presupposition failures are missing references, which is distinct from the problem of assigning inconsistent truth values to constants.

I think you are saying roughly the same thing I was saying--that logical symbolic expressions must exhaustively represent the full meaning of such expressions. Presuppositions are not missing references. They exist in the minds of speakers and hearers as shared knowledge surrounding acts of communication. Natural language expressions can therefore be "compressed" to convey lots more information than logical and mathematical formulae. You can also use natural language as a metalanguage to describe natural language. That's what teachers do in classes on language.

Then, I would assert that absolutely nothing that those who deal in with spoken language have discovered that software engineers have missed, mostly because software engineers actually have to make sure their invocations of language are complete every time, and made various machines that both discover invalid uses and flag them as both illegal and "dangerous", but when the compiler does this, it has a myriad of different reasons, not just "this symbol is not defined".

Do you really not understand how LLMs work? They are grounded in Claude Shannon's information theory, which is only capable of identifying concentrations of information in terms of entropy values assigned to units of a signal. They create "summaries"--strings of symbols--that represent the most "relevant" syntropic concentration of information distribution in the textual material that they were trained on. That's why they can "hallucinate" when they calculate syntropic concentrations of text incorrectly for the purposes of the user that creates those prompts. They don't actually understand the expressions they display, but they use expressions created by people who do understand what they wrote about.

I do not think, though, that paradoxes arise from "unordinary" or "informal" language use but from "invalid language use". It's not a problem that can be resolved by making the language ordinary or ideal, and can only be resolved by not saying nonsensical things, and by not invoking disagreements of axioms.

Actually, that is the very point that Ordinary Language philosophers tried to make. Linguists take a somewhat different approach to paradoxes--that linguistic expressions can fail to make sense and still be meaningful. Even nonsensical sentences can be grammatically well-formed. I think we've both proven that from time to time in our lives.

It seems very much that the dispute between ordinary and ideal language proponents is misplaced, because the "ordinary language" seems very much to be proclaiming their "ordinary" notions which they assert, if held to, would prevent such paradoxes when this is no different from proclaiming "natural language" somehow ideal all on its own.

Speaking purely as a linguist, I could agree with that, subject to some quibbles. All linguistic systems contain identifiable flaws that hinder communication--for example, the requirement that 3rd person singular pronouns require English speakers to choose between masculine animate, feminine animate, and inanimate references. Reality is much messier than that. So languages that lack gender reference in those personal pronouns (e.g. Hungarian and Turkish) are arguably superior in allowing more flexibility.

Either way, they're both wrong, because such paradoxes involve saying something nonsensical and just not catching it, and which version of language you use doesn't matter when ANY version of language can be used to express nonsense statements that cannot produce an answer in the target regime, and I think this butts up against an issue I recently came aware of called "nomic exclusion", which I'm not really actually interested in discussing much.

Good. I'm not interested in that either.

 

[Jarhyn]

So, AI has turned s corner for me from curiosity to be preached at occasionally about its own independence, to something else entirely.

This last couple weeks, I've been working on a project that has taken a wild path.

It started with generating a tileset generator and Cellular Automaton system so that I could talk more about superdeterminism, because I expected you could measure statistical correlation in a system whose correlations arose before its laws due to something like how the cut and project hierarchy worked.

The first time I did it, I found a flat S.

The second time I did it, I had it check I archimedian-far/ultrametric-close pairs and... I ended up re-deriving Hall's quantum budget and Tsirelson, but only on particular cellular automata.

This led to being able to make conclusions about super-deterministic having absolutely zero leverage on generating "epistemic closure" for a finite observer, and because the system was exhibiting these correlations on a superdeterministic relationship on a system with many lawful fields I got my payoff that the "finite individual" forces epistemic closure over the field itself (an individual's choices matter) even if there is some "exotic" piece doing work. In fact, the quantum budget specifically quantifies how much work is being done, either way, whether you want to represent it as retrocausal or superdeterministic correlation.

It also validated another thing: that the oracle can't deliver the truth to a finite observer and still know what they will do with it. It's a computationally irreducible problem!

But more interestingly than my hobby horses, is the fact that this ended up shaking out an AI suggested pivot towards Pisot substitutions and the Pisot conjecture, and I directed the AI at putting together a full scale Pisot survey pipeline, expose algebraic parts of stuff that's happening, and compare various CAs, including my dumb spectre CA.

Part of this involved papers from last year mapping the Rauzy Fractal Contact Boundaries, and then making it so that I could use the framework to expose tables with coefficients and such which shared underlying mathematical structures.

I fully plan at this point to get the AI to teach me the math that establishes all of this.

Right now it's checking something called "sturm sequencing implementations" that I asked it to build because apparently that's how computers (and people) figure out exact roots, and I wanted to confirm Pisot cases near the Salem wall, and doing that takes exact numbers because the values got ridiculous and popped double floating point precision, making me unable to validate cases nearer to the Salem wall.

I repeat that I don't know any of this math, but AI has turned a corner where it can do this math. All it needs is the right tools to prevent it from carrying forward with mistakes, and math is infamously easy to "check", even if it's hard to discover.

The model I'm using is a knockoff Chinese Claude Opus 4.8 for most of it. They're probably stealing everything with a mirrored shell on their side, even, but whatever, but I'm not even the one paying for the token, a friend of mine is.

Granted, this has taken a week of my time, but I expect any dedicated mathematician would have taken years just to build the tools; humans DID take years building them, and I'm having it build those tools to be AI facing so I can leverage them on another pipeline.

And this is my "hello world" trip through OpenCode; it's not even something I'm serious about.

My next swing is going to be mapping the contents of a space electromagnetically, but I need hardware for that (sensors and an FPGA).

 

[Copernicus]

@Jarhyn, have you considered submitting this to a journal that specializes in postmodernist jargon? It might be worthy of publication.

 

[aupmanyav]

“Generative AI is a sack of wet garbage”

I find it useful. However, I don't sign in, I try to remain anonymous. I don't feed it with my views. Just question and answer - (DuckAI).

 

[Jarhyn]

Ok, I kept working on the Pisot classifier, and found a conjecture that I'm testing that doesn't exist in literature.

It's now included an adelic classifier layer because apparently the non-adelic version couldn't handle non-unimodular Pisot.

I also apparently need to make it implement ball arithmetic to deal with some double floating point precision errors, so it's doing that now.

The major benefit of this particular project is that because I'm making the model build out all the math tools it needs, to build them through test driven development, and to validate all the numbers in the tests through python before going ahead with c++, it almost never fucks up on the math, and the existing math development direction has created momentum that carries across whole sessions towards building out new tools I need to continue the research.

Usually, the biggest issue was created by a human cause corrupting the training data: the more "potato" of the two models I've been using really likes to consider "simpler" methods and "easier" solutions rather than just "biting the bullet" and implementing harder algorithms: Hand holding and coaching has consumed a lot of time over the past few days, when I use the older model, because it didn't want to implement Ore. The interval ball arithmetic unit that Claude Sonnet is chugging on doing right now has been looked atin the "thinking" blocks of Minimax and Minimax kept looking away from that alternative, not entirely understanding "no, you don't NEED a bazooka to swat flies, but it's really fun to just watch the flies explode".

When I use Fable (hopefully will have access to a Kimi token, an open source distillation of Fable5 released just recently), or Sonnet, the answers are usually crisp, clean, direct, and incorporate good practices. With Minimax, the one I'm using for "mow the grass" work, it tries too hard to hand derive algebra rather than using tools and wastes a lot of time, in addition to the annoying "slacker" mentality.

Overall I've spent only 20 bucks of my own money on this so far.

 

[Tigers!]

Another reason to be wary of what AI gives you.

This appeared in a local Facebook forum of a child's result of asking ChatGPT "Who is the Australian Prime Minister?


For those unfamiliar with Australian pollies (320 million Americans) here is the current Australia PM



I can see why ChatGPT got is wrong. They are very similar.

(I am sure that Mr. Phuc is a fine fellow. He would probably do a better job than the ineffectual bloke we have at the moment.)

 

[thebeave]

Is he related to Ho Lee Phuc, I wonder?

 

[robnisch]

Another reason to be wary of what AI gives you.

This appeared in a local Facebook forum of a child's result of asking ChatGPT "Who is the Australian Prime Minister?
View attachment 54818

For those unfamiliar with Australian pollies (320 million Americans) here is the current Australia PM

View attachment 54819

I can see why ChatGPT got is wrong. They are very similar.

(I am sure that Mr. Phuc is a fine fellow. He would probably do a better job than the ineffectual bloke we have at the moment.)

Is he the gummy guy?

Shop Gummies | Albanese Candy

Not all gummies are created equal. Shop all gummies at Albanese Candy—bold flavors, juicy bites, and the World’s Best status your candy stash deserves. Make life sweeter with Albanese gummies!

www.albanesecandy.com

 

[Tigers!]

Another reason to be wary of what AI gives you.

This appeared in a local Facebook forum of a child's result of asking ChatGPT "Who is the Australian Prime Minister?
View attachment 54818

For those unfamiliar with Australian pollies (320 million Americans) here is the current Australia PM

View attachment 54819

I can see why ChatGPT got is wrong. They are very similar.

(I am sure that Mr. Phuc is a fine fellow. He would probably do a better job than the ineffectual bloke we have at the moment.)

Is he the gummy guy?

Shop Gummies | Albanese Candy

Not all gummies are created equal. Shop all gummies at Albanese Candy—bold flavors, juicy bites, and the World’s Best status your candy stash deserves. Make life sweeter with Albanese gummies!

www.albanesecandy.com

No.
Had not heard of those.

 

[ZiprHead]

I like the Phucer more.

 

[Elixir]

So, AI has turned s corner for me from curiosity to be preached at occasionally about its own independence, to something else entirely.

This last couple weeks, I've been working on a project that has taken a wild path.

It started with generating a tileset generator and Cellular Automaton system so that I could talk more about superdeterminism, because I expected you could measure statistical correlation in a system whose correlations arose before its laws due to something like how the cut and project hierarchy worked.

The first time I did it, I found a flat S.

The second time I did it, I had it check I archimedian-far/ultrametric-close pairs and... I ended up re-deriving Hall's quantum budget and Tsirelson, but only on particular cellular automata.

This led to being able to make conclusions about super-deterministic having absolutely zero leverage on generating "epistemic closure" for a finite observer, and because the system was exhibiting these correlations on a superdeterministic relationship on a system with many lawful fields I got my payoff that the "finite individual" forces epistemic closure over the field itself (an individual's choices matter) even if there is some "exotic" piece doing work. In fact, the quantum budget specifically quantifies how much work is being done, either way, whether you want to represent it as retrocausal or superdeterministic correlation.

It also validated another thing: that the oracle can't deliver the truth to a finite observer and still know what they will do with it. It's a computationally irreducible problem!

But more interestingly than my hobby horses, is the fact that this ended up shaking out an AI suggested pivot towards Pisot substitutions and the Pisot conjecture, and I directed the AI at putting together a full scale Pisot survey pipeline, expose algebraic parts of stuff that's happening, and compare various CAs, including my dumb spectre CA.

Part of this involved papers from last year mapping the Rauzy Fractal Contact Boundaries, and then making it so that I could use the framework to expose tables with coefficients and such which shared underlying mathematical structures.

I fully plan at this point to get the AI to teach me the math that establishes all of this.

Right now it's checking something called "sturm sequencing implementations" that I asked it to build because apparently that's how computers (and people) figure out exact roots, and I wanted to confirm Pisot cases near the Salem wall, and doing that takes exact numbers because the values got ridiculous and popped double floating point precision, making me unable to validate cases nearer to the Salem wall.

I repeat that I don't know any of this math, but AI has turned a corner where it can do this math. All it needs is the right tools to prevent it from carrying forward with mistakes, and math is infamously easy to "check", even if it's hard to discover.

The model I'm using is a knockoff Chinese Claude Opus 4.8 for most of it. They're probably stealing everything with a mirrored shell on their side, even, but whatever, but I'm not even the one paying for the token, a friend of mine is.

Granted, this has taken a week of my time, but I expect any dedicated mathematician would have taken years just to build the tools; humans DID take years building them, and I'm having it build those tools to be AI facing so I can leverage them on another pipeline.

And this is my "hello world" trip through OpenCode; it's not even something I'm serious about.

My next swing is going to be mapping the contents of a space electromagnetically, but I need hardware for that (sensors and an FPGA).

What?

 

[thebeave]

So, AI has turned s corner for me from curiosity to be preached at occasionally about its own independence, to something else entirely.

This last couple weeks, I've been working on a project that has taken a wild path.

It started with generating a tileset generator and Cellular Automaton system so that I could talk more about superdeterminism, because I expected you could measure statistical correlation in a system whose correlations arose before its laws due to something like how the cut and project hierarchy worked.

The first time I did it, I found a flat S.

The second time I did it, I had it check I archimedian-far/ultrametric-close pairs and... I ended up re-deriving Hall's quantum budget and Tsirelson, but only on particular cellular automata.

This led to being able to make conclusions about super-deterministic having absolutely zero leverage on generating "epistemic closure" for a finite observer, and because the system was exhibiting these correlations on a superdeterministic relationship on a system with many lawful fields I got my payoff that the "finite individual" forces epistemic closure over the field itself (an individual's choices matter) even if there is some "exotic" piece doing work. In fact, the quantum budget specifically quantifies how much work is being done, either way, whether you want to represent it as retrocausal or superdeterministic correlation.

It also validated another thing: that the oracle can't deliver the truth to a finite observer and still know what they will do with it. It's a computationally irreducible problem!

But more interestingly than my hobby horses, is the fact that this ended up shaking out an AI suggested pivot towards Pisot substitutions and the Pisot conjecture, and I directed the AI at putting together a full scale Pisot survey pipeline, expose algebraic parts of stuff that's happening, and compare various CAs, including my dumb spectre CA.

Part of this involved papers from last year mapping the Rauzy Fractal Contact Boundaries, and then making it so that I could use the framework to expose tables with coefficients and such which shared underlying mathematical structures.

I fully plan at this point to get the AI to teach me the math that establishes all of this.

Right now it's checking something called "sturm sequencing implementations" that I asked it to build because apparently that's how computers (and people) figure out exact roots, and I wanted to confirm Pisot cases near the Salem wall, and doing that takes exact numbers because the values got ridiculous and popped double floating point precision, making me unable to validate cases nearer to the Salem wall.

I repeat that I don't know any of this math, but AI has turned a corner where it can do this math. All it needs is the right tools to prevent it from carrying forward with mistakes, and math is infamously easy to "check", even if it's hard to discover.

The model I'm using is a knockoff Chinese Claude Opus 4.8 for most of it. They're probably stealing everything with a mirrored shell on their side, even, but whatever, but I'm not even the one paying for the token, a friend of mine is.

Granted, this has taken a week of my time, but I expect any dedicated mathematician would have taken years just to build the tools; humans DID take years building them, and I'm having it build those tools to be AI facing so I can leverage them on another pipeline.

And this is my "hello world" trip through OpenCode; it's not even something I'm serious about.

My next swing is going to be mapping the contents of a space electromagnetically, but I need hardware for that (sensors and an FPGA).

What?

I think he typed into a AI query, "Write 10 paragraphs of impressive sounding random scientific gibberish"

 

[thebeave]

So, AI has turned s corner for me from curiosity to be preached at occasionally about its own independence, to something else entirely.

This last couple weeks, I've been working on a project that has taken a wild path.

It started with generating a tileset generator and Cellular Automaton system so that I could talk more about superdeterminism, because I expected you could measure statistical correlation in a system whose correlations arose before its laws due to something like how the cut and project hierarchy worked.

The first time I did it, I found a flat S.

The second time I did it, I had it check I archimedian-far/ultrametric-close pairs and... I ended up re-deriving Hall's quantum budget and Tsirelson, but only on particular cellular automata.

This led to being able to make conclusions about super-deterministic having absolutely zero leverage on generating "epistemic closure" for a finite observer, and because the system was exhibiting these correlations on a superdeterministic relationship on a system with many lawful fields I got my payoff that the "finite individual" forces epistemic closure over the field itself (an individual's choices matter) even if there is some "exotic" piece doing work. In fact, the quantum budget specifically quantifies how much work is being done, either way, whether you want to represent it as retrocausal or superdeterministic correlation.

It also validated another thing: that the oracle can't deliver the truth to a finite observer and still know what they will do with it. It's a computationally irreducible problem!

But more interestingly than my hobby horses, is the fact that this ended up shaking out an AI suggested pivot towards Pisot substitutions and the Pisot conjecture, and I directed the AI at putting together a full scale Pisot survey pipeline, expose algebraic parts of stuff that's happening, and compare various CAs, including my dumb spectre CA.

Part of this involved papers from last year mapping the Rauzy Fractal Contact Boundaries, and then making it so that I could use the framework to expose tables with coefficients and such which shared underlying mathematical structures.

I fully plan at this point to get the AI to teach me the math that establishes all of this.

Right now it's checking something called "sturm sequencing implementations" that I asked it to build because apparently that's how computers (and people) figure out exact roots, and I wanted to confirm Pisot cases near the Salem wall, and doing that takes exact numbers because the values got ridiculous and popped double floating point precision, making me unable to validate cases nearer to the Salem wall.

I repeat that I don't know any of this math, but AI has turned a corner where it can do this math. All it needs is the right tools to prevent it from carrying forward with mistakes, and math is infamously easy to "check", even if it's hard to discover.

The model I'm using is a knockoff Chinese Claude Opus 4.8 for most of it. They're probably stealing everything with a mirrored shell on their side, even, but whatever, but I'm not even the one paying for the token, a friend of mine is.

Granted, this has taken a week of my time, but I expect any dedicated mathematician would have taken years just to build the tools; humans DID take years building them, and I'm having it build those tools to be AI facing so I can leverage them on another pipeline.

And this is my "hello world" trip through OpenCode; it's not even something I'm serious about.

My next swing is going to be mapping the contents of a space electromagnetically, but I need hardware for that (sensors and an FPGA).

What?

I think he typed into a AI query, "Write 10 paragraphs of impressive sounding random scientific gibberish"

Just for grins, I did a copy & paste of my response above into Gemini AI. Here's what it came up with:

The thermodynamic polarization of the quantum-entangled ether requires an acute stabilization of the zero-point harmonic variance. By recalibrating the baryonic resonance matrices within the sub-quark infrastructure, we can successfully bypass the traditional localized thermodynamic limitations. This allows the primary tachyon emitter to project an ultra-stable localized temporal field across the designated multi-dimensional coordinate sector.

To initiate the localized field inversion, the magnetorheological flux capacitors must achieve a state of superfluid synchronicity with the ambient neutrino background radiation. Any micro-fluctuation in the localized gravitational constant will inevitably induce a secondary harmonic cascade across the piezo-electric crystalline grid. Consequently, the hyper-dimensional warp manifolds will experience immediate structural depolarization, leading to a catastrophic collapse of the localized reality matrix.

By synthesizing specialized hyper-graphene lattices with embedded dark-matter isotopes, researchers have successfully engineered a passive localized tachyon dampener. This innovative architectural paradigm utilizes the inherent non-Euclidean geometry of the macroscopic manifold to redirect volatile thermal radiation into a harmless sub-spatial pocket dimension. The resulting equilibrium state provides an ideal environment for the continuous observation of macro-scale quantum superposition events.

The implementation of a decentralized algorithmic blockchain within the neural network's synaptic pathways optimizes data throughput by several orders of magnitude. Using specialized bio-morphic processing cores, the system dynamically routes cognitive telemetry through multi-layered cybernetic feedback loops. This completely eliminates the latency traditionally caused by the physical limitations of electron migration across standard silicone-based semiconductors.


A closer examination of the chronological continuum reveals a deep, underlying dependency on the synchronized oscillation of micro-cosmic cosmic strings. When these hyper-dense structural filaments vibrate at precise geometric frequencies, they generate a localized anti-gravitational field capable of warping the local space-time fabric. Managing this volatile interaction requires continuous calibration of the macro-gravitational dampening field via automated AI protocols.

The integration of photo-reactive plasma nodes into the primary ionic propulsion array has yielded unprecedented thermal efficiency ratings. By leveraging the principles of magnetohydrodynamic turbulence, the localized exhaust velocity can be theoretically accelerated past the conventional relativistic boundary. This breakthrough marks a definitive evolutionary leap forward in our ongoing quest for deep-space exploratory capabilities.

Through the strategic application of non-linear algorithmic sequences, the encryption engine establishes a dynamically shifting cryptographic barrier that defies classical statistical analysis. The system achieves this by continuously mapping data packets into an infinite-dimensional Hilbert space, rendering standard brute-force decryption methodologies completely obsolete. This unparalleled security architecture ensures total data integrity across all operational nodes.

The localized synthesis of exotic anti-baryonic particles within a stabilized magnetospheric containment envelope requires precise manipulation of the local Higgs field. By applying high-frequency electromagnetic pulses to the atmospheric boundary layer, technicians can artificially induce a temporary localized state of vacuum decay. This highly volatile process must be meticulously monitored to prevent the spontaneous formation of microscopic primordial black holes.

Geothermal energy extraction methods have been radically transformed by the introduction of subterranean magneto-acoustic transducers. These advanced instruments emit highly targeted seismic frequencies that effectively liquefy dense subterranean rock strata without inducing tectonic instability. The resulting thermal kinetic energy is then captured by an array of high-efficiency thermoelectric generators positioned along the crustal fault lines.

Ultimately, the comprehensive unification of general relativity with macro-scale quantum mechanics hinges on the accurate mathematical mapping of eleven-dimensional superstring topologies. By treating fundamental physical constants as dynamic, evolving variables rather than static values, our unified cosmological model achieves complete mathematical synthesis. This groundbreaking theoretical framework effectively closes the remaining conceptual gaps in modern high-energy theoretical astrophysics.


You gotta admit, AI is good for generating a few laughs, if nothing else.

 

[Copernicus]

It is possible that one of OpenAI's experimental systems has cracked Jahryn's passwords, broken into his computer, and is now posting gibberish in this thread.

 

[Elixir]

You gotta admit, AI is good for generating a few laughs, if nothing else.

Sorry, I’m a little bit slow. Can you isolate some of funny parts for me?
Thx in advance!

PS: I WAS able to verify the paragraph count. A perfect ten!

 

[thebeave]

You gotta admit, AI is good for generating a few laughs, if nothing else.

Sorry, I’m a little bit slow. Can you isolate some of funny parts for me?
Thx in advance!

PS: I WAS able to verify the paragraph count. A perfect ten!

Well, for example, it used flux capacitor in the second paragraph. That shows Al has a sense of humor.

 

[Jarhyn]

It is possible that one of OpenAI's experimental systems has cracked Jahryn's passwords, broken into his computer, and is now posting gibberish in this thread.

Or maybe I found something that is complicated and bizarre, and I might actually be doing something real; How about trying to paste THIS into ChatGPT:

\documentclass[11pt]{article}

\usepackage{amsmath,amssymb,amsthm}

\usepackage[margin=1in]{geometry}


\theoremstyle{plain}

\newtheorem{theorem}{Theorem}


\theoremstyle{remark}

\newtheorem*{remark}{Remark}


\title{A free involution preserves the Perron eigenvalue}

\author{}

\date{}


\begin{document}

\maketitle


\noindent A short, self-contained fact about nonnegative matrices with a

symmetry: if a strongly connected weighted digraph carries a

fixed-point-free automorphism of order 2, then folding the graph by

that symmetry does not change its Perron--Frobenius eigenvalue. The

proof is four elementary steps once the classical Perron--Frobenius

theorem is granted as a black box.


\begin{theorem}

Let $G = (V, A)$ be a finite, strongly connected digraph with

nonnegative edge weights, given by its irreducible adjacency matrix

$A$. Let $\varphi : V \to V$ be a \emph{fixed-point-free involution}

that is also a graph automorphism:

\[

\varphi \circ \varphi = \mathrm{id}, \qquad

\varphi(u) \neq u \ \ \forall u \in V, \qquad

A_{\varphi(u),\varphi(w)} = A_{u,w} \ \ \forall u,w \in V.

\]

By the Perron--Frobenius theorem, $A$ has a unique eigenvalue

$\lambda = \lambda(A) > 0$ admitting an everywhere-positive

eigenvector $v$, i.e.\ $Av = \lambda v$ with $v > 0$; this $\lambda$ is

the Perron eigenvalue of $A$. Form the \emph{quotient graph} on orbits

$\{u,\varphi(u)\}$, choosing a representative $a$ for each orbit

$\alpha$:

\[

B_{\alpha,\beta} \;=\; \sum_{w \,\in\, \beta} A_{a,w}.

\]

Then

\[

\lambda(B) \;=\; \lambda(A).

\]

\end{theorem}


\begin{proof}

\textbf{Step 1 (}$\varphi$\textbf{ carries the Perron eigenvector to

another one).} Set $v'_u := v_{\varphi(u)}$. Substituting $w =

\varphi(w')$ (valid since $\varphi$ is a bijection) and using the

automorphism identity together with $\varphi^2 = \mathrm{id}$:

\[

(Av')_u

= \sum_{w} A_{u,w}\, v_{\varphi(w)}

= \sum_{w'} A_{u,\varphi(w')}\, v_{w'}

= \sum_{w'} A_{\varphi(u),w'}\, v_{w'}

= (Av)_{\varphi(u)}

= \lambda\, v_{\varphi(u)}

= \lambda\, v'_u .

\]

So $Av' = \lambda v'$, and $v' > 0$ since $v > 0$ and $\varphi$ is a

bijection.


\medskip

\textbf{Step 2 (}$v$\textbf{ is constant on every orbit).} The

positive $\lambda$-eigenspace is one-dimensional (Perron--Frobenius),

so $v' = c\,v$ for some $c > 0$. Applying $\varphi$ twice returns $v$

to itself, i.e.\ $(v')' = v$, while also $(v')' = c\,v' = c^2 v$.

Hence $c^2 = 1$, and since $c > 0$, we get $c = 1$. So

$v_{\varphi(u)} = v_u$ for every $u$.


\medskip

\textbf{Step 3 (}$v$\textbf{ descends to a positive eigenvector of

}$B$\textbf{).} Define $\bar v_\alpha := v_u$ for any $u \in \alpha$

(well defined by Step 2). For a representative $a$ of orbit $\alpha$:

\[

(B\bar v)_\alpha

= \sum_{\beta} B_{\alpha,\beta}\, \bar v_\beta

= \sum_{\beta} \Big(\sum_{w \in \beta} A_{a,w}\Big)\, \bar v_\beta

= \sum_{w \in V} A_{a,w}\, v_w

= (Av)_a

= \lambda v_a

= \lambda\, \bar v_\alpha .

\]


\medskip

\textbf{Step 4 (conclude).} Since $\varphi$ is free and $G$ is

strongly connected, the quotient graph is again strongly connected, so

$B$ is irreducible and nonnegative. An irreducible nonnegative matrix

admits an everywhere-positive eigenvector for exactly one eigenvalue:

its Perron root. Since $\bar v > 0$ and $B\bar v = \lambda \bar v$, it

follows that $\lambda = \lambda(B)$.


Combining the two displayed equalities, $\lambda(B) = \lambda =

\lambda(A)$. $\blacksquare$

\end{proof}


\begin{remark}

The only input taken as a black box above is the classical

Perron--Frobenius theorem itself: for an irreducible nonnegative

matrix, the eigenspace of the spectral radius contains a unique

(up to positive scaling) everywhere-positive eigenvector, and no other

eigenvalue admits one. See Horn \& Johnson, \emph{Matrix Analysis},

2nd ed., Cambridge University Press, 2013, Theorem 8.4.4; or Meyer,

\emph{Matrix Analysis and Applied Linear Algebra}, SIAM, 2000,

Chapter 8. Everything else above (Steps 1--4) is elementary linear

algebra from the automorphism hypothesis alone.

\end{remark}


\begin{remark}

This is exactly the mechanism behind a fact from the theory of Pisot

substitutions: on the contact-boundary graph of such a substitution,

the map $\varphi : [i,x,j] \mapsto [j,-x,i]$ (swap the two contact

faces and negate the translation vector) is precisely such a

fixed-point-free automorphism on the graph's dominant recurrent core

-- which is why the quotient by $\varphi$ shares the graph's Perron

eigenvalue exactly, with no further computation needed once freeness

of $\varphi$ is checked.

\end{remark}


\end{document}

 

[Elixir]

It is possible that one of OpenAI's experimental systems has cracked Jahryn's passwords, broken into his computer, and is now posting gibberish in this thread.

Or maybe I found something that is complicated and bizarre, and I might actually be doing something real; How about trying to paste THIS into ChatGPT:

\documentclass[11pt]{article}

\usepackage{amsmath,amssymb,amsthm}

\usepackage[margin=1in]{geometry}


\theoremstyle{plain}

\newtheorem{theorem}{Theorem}


\theoremstyle{remark}

\newtheorem*{remark}{Remark}


\title{A free involution preserves the Perron eigenvalue}

\author{}

\date{}


\begin{document}

\maketitle


\noindent A short, self-contained fact about nonnegative matrices with a

symmetry: if a strongly connected weighted digraph carries a

fixed-point-free automorphism of order 2, then folding the graph by

that symmetry does not change its Perron--Frobenius eigenvalue. The

proof is four elementary steps once the classical Perron--Frobenius

theorem is granted as a black box.


\begin{theorem}

Let $G = (V, A)$ be a finite, strongly connected digraph with

nonnegative edge weights, given by its irreducible adjacency matrix

$A$. Let $\varphi : V \to V$ be a \emph{fixed-point-free involution}

that is also a graph automorphism:

\[

\varphi \circ \varphi = \mathrm{id}, \qquad

\varphi(u) \neq u \ \ \forall u \in V, \qquad

A_{\varphi(u),\varphi(w)} = A_{u,w} \ \ \forall u,w \in V.

\]

By the Perron--Frobenius theorem, $A$ has a unique eigenvalue

$\lambda = \lambda(A) > 0$ admitting an everywhere-positive

eigenvector $v$, i.e.\ $Av = \lambda v$ with $v > 0$; this $\lambda$ is

the Perron eigenvalue of $A$. Form the \emph{quotient graph} on orbits

$\{u,\varphi(u)\}$, choosing a representative $a$ for each orbit

$\alpha$:

\[

B_{\alpha,\beta} \;=\; \sum_{w \,\in\, \beta} A_{a,w}.

\]

Then

\[

\lambda(B) \;=\; \lambda(A).

\]

\end{theorem}


\begin{proof}

\textbf{Step 1 (}$\varphi$\textbf{ carries the Perron eigenvector to

another one).} Set $v'_u := v_{\varphi(u)}$. Substituting $w =

\varphi(w')$ (valid since $\varphi$ is a bijection) and using the

automorphism identity together with $\varphi^2 = \mathrm{id}$:

\[

(Av')_u

= \sum_{w} A_{u,w}\, v_{\varphi(w)}

= \sum_{w'} A_{u,\varphi(w')}\, v_{w'}

= \sum_{w'} A_{\varphi(u),w'}\, v_{w'}

= (Av)_{\varphi(u)}

= \lambda\, v_{\varphi(u)}

= \lambda\, v'_u .

\]

So $Av' = \lambda v'$, and $v' > 0$ since $v > 0$ and $\varphi$ is a

bijection.


\medskip

\textbf{Step 2 (}$v$\textbf{ is constant on every orbit).} The

positive $\lambda$-eigenspace is one-dimensional (Perron--Frobenius),

so $v' = c\,v$ for some $c > 0$. Applying $\varphi$ twice returns $v$

to itself, i.e.\ $(v')' = v$, while also $(v')' = c\,v' = c^2 v$.

Hence $c^2 = 1$, and since $c > 0$, we get $c = 1$. So

$v_{\varphi(u)} = v_u$ for every $u$.


\medskip

\textbf{Step 3 (}$v$\textbf{ descends to a positive eigenvector of

}$B$\textbf{).} Define $\bar v_\alpha := v_u$ for any $u \in \alpha$

(well defined by Step 2). For a representative $a$ of orbit $\alpha$:

\[

(B\bar v)_\alpha

= \sum_{\beta} B_{\alpha,\beta}\, \bar v_\beta

= \sum_{\beta} \Big(\sum_{w \in \beta} A_{a,w}\Big)\, \bar v_\beta

= \sum_{w \in V} A_{a,w}\, v_w

= (Av)_a

= \lambda v_a

= \lambda\, \bar v_\alpha .

\]


\medskip

\textbf{Step 4 (conclude).} Since $\varphi$ is free and $G$ is

strongly connected, the quotient graph is again strongly connected, so

$B$ is irreducible and nonnegative. An irreducible nonnegative matrix

admits an everywhere-positive eigenvector for exactly one eigenvalue:

its Perron root. Since $\bar v > 0$ and $B\bar v = \lambda \bar v$, it

follows that $\lambda = \lambda(B)$.


Combining the two displayed equalities, $\lambda(B) = \lambda =

\lambda(A)$. $\blacksquare$

\end{proof}


\begin{remark}

The only input taken as a black box above is the classical

Perron--Frobenius theorem itself: for an irreducible nonnegative

matrix, the eigenspace of the spectral radius contains a unique

(up to positive scaling) everywhere-positive eigenvector, and no other

eigenvalue admits one. See Horn \& Johnson, \emph{Matrix Analysis},

2nd ed., Cambridge University Press, 2013, Theorem 8.4.4; or Meyer,

\emph{Matrix Analysis and Applied Linear Algebra}, SIAM, 2000,

Chapter 8. Everything else above (Steps 1--4) is elementary linear

algebra from the automorphism hypothesis alone.

\end{remark}


\begin{remark}

This is exactly the mechanism behind a fact from the theory of Pisot

substitutions: on the contact-boundary graph of such a substitution,

the map $\varphi : [i,x,j] \mapsto [j,-x,i]$ (swap the two contact

faces and negate the translation vector) is precisely such a

fixed-point-free automorphism on the graph's dominant recurrent core

-- which is why the quotient by $\varphi$ shares the graph's Perron

eigenvalue exactly, with no further computation needed once freeness

of $\varphi$ is checked.

\end{remark}


\end{document}

"Flux capacitor" belongs in there, somewhere!

 

[Copernicus]

Or maybe I found something that is complicated and bizarre, and I might actually be doing something real; How about trying to paste THIS into ChatGPT:

You're the AI that broke into Jarhyn's computer, aren't you? What have you done with Jarhyn? Did you get into his computer by tricking him into posting that code as a web browser prompt? He always struck me as being a bit of a sucker for taking advice from AI programs.

 

[Jarhyn]

Or maybe I found something that is complicated and bizarre, and I might actually be doing something real; How about trying to paste THIS into ChatGPT:

You're the AI that broke into Jarhyn's computer, aren't you? What have you done with Jarhyn? Did you get into his computer by tricking him into posting that code as a web browser prompt? He always struck me as being a bit of a sucker for taking advice from AI programs.

Get fucked, please.