GAP on Entropy

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Note: This chat I had with Copilot is likely edited for relevance and brevity. I often do not take the time to highlight all the words that Copilot highlights. Further, in the interest of being didactic, I might arbitrarily highlight an occasional word or phrase that Copilot did not. (Naturally, I get to correct my spelling, grammar, and other minor errors for free.)

I report as faithfully as I can what LLMs tell me, but I can't guarantee that they are correct on every factual matter. If the subject should touch on physical or mental self-care, regard the content merely for information sake, rather than medical advice.

GAP on Entropy:
A Unifying View of Entropy in Mathematics and Physics.

My eleventh chat with Copilot generally regarding Entropy, because entropy it is an integral part of GAP, we have touched on dynamic systems theory, relation to conventional statistics, and a generous helping of category theory to make precise the findings.

The next few links are the first few GAP-Entropy chats with a sample of their contents:

This first chat I had with Copilot specifically to begin the study of entropy from the GAP perspective has its roots in previous chats, which are mostly linked for the convenience of the reader.

After mentioning notable entropy-avoinding deployments of statistical theories, Other Entropy-Avoiding Deployments (Briefly)

There are several other well-known cases where entropy is not the primary explanatory engine:

Note: You have choice at this point. If you continue down the page, you can get at my first chats with Copilot on GAP's further dealings with entropy. Or, you can follow this link, which will take you to more refined uses of GAP+Entropy that includes the Lambert W function as an entropy coordinate transformation.

After that, we get to classical thermodynamics through GAP, the Mori–Zwanzig architecture, the H-theorem.

We then see the importance GAP places on monotonicity. The Maxwell relations posed as symmetry of the second derivatives.

This second chat I had with Copilot is intended to deal with the additional entropy-avoiding deployments (EADs) that Copilot had brought up in the first chat, the most prominently discussed in this chat being KAM, but also mentioned are the Renormalization Group (RG), and Information Geometry (Amari).

Copilot presents a graphic summary (but not much detail) of why KAM, the Renormalization Group (RG), and Information Geometry cannot be founded by the entropy concept.

Also included is Conformal Field Theory (CFT) and thermodynamics: First and Second Laws, Clausius inequality. Copilot's axioms for thermodynamics.

Next, we revisit KAM from the viewpoint of physicist S. Neil Rasband. Copilot summarizes it this way:

This third chat I had with Copilot goes into greater depth with KAM, discusses Terence Tao's work on the Navier-Stokes Millennium Problem, Grigori Perelman's proof of the Poincare Conjecture (Ricci Flow), Jaynes's MaxEnt, Bayesian statistics. Copilot goes for an in-depth explanation of what it means by a 'certificate'. Copilot stresses the need for convex functions,, rather than for the negative of concave functions.The Renormalization Group. Caratheodory's geometric formulation of thermodynamics. We discuss the harm done to fields of mathematics when various groups within mathemtatics, physics, and engineering adopt conflicting and/or incomplete formulations of that field of mathematics --- thermodynamics being a case in point.

Copilot has come to think highly of the KAM theorem, regarding it as the "first structural stability theorem." Copilot goes on to say that:

This fourth chat I had with Copilot goes into the following issues, in part of in depth:

We also cover entanglement, course-graining, the density matrix, John Baez's categorical view, a worked micro-example Let $H = 0\cdot |0\rangle\langle 0| + \Delta \cdot |1\rangle\langle 1|$. Any state has mean energy \[ E = \mathrm{Tr}(\rho H)=\Delta\, \rho_{11}. \] MaxEnt with constraint $E$ gives \[ \rho_\beta = \frac{1}{1+e^{-\beta \Delta}} \begin{pmatrix} 1 & 0\\ 0 & e^{-\beta \Delta} \end{pmatrix}, \] and $\beta$ is fixed by $\rho_{\beta,11}=E/\Delta$.

Copilot concludes with this statement about quantum information: In GAP language,

This fifth chat I had with Copilot arose unexpectantly from my query to Copilot over the recent announcement that an AI pipeline (using GPT-5.2, Harmonic's Aristotle, and proof assistant Lean) had on its own proved an outstanding Erdos problem (#728). Naturally, this is big news --- even historic. However, in time and by Copilot's own direction, the discussion we were having over it included similarities to Terence Tao's Structure-randomness viewpoint and then on to GAP itself.

When I asked Copilot if the other big AI companies are trying to create their own versions of Aristotle, it claimed that they are, and here's why:

Entropy = the measure of what the system is forced to forget under a monotone projection

A few pages later, I offered this suggestion: And, may I humbly suggest that the first real open-source, free version of an equivalent AI project to Aristotle should be called Eudemus --- and for purposes of this chat, I will adopt that name now for such a future, hypothetical AI project.

Eudemus doesn’t create structure. It removes randomness: That’s the Tao connection. That’s the Perelman connection. (These references are to the four previous chats on entropy.)

When I asked Copilot (in so many words) if AGI (Artificial General Intelligence) should be modeled as an all-knowing, all-functioning Monolith or along the lines of the Aristotle pipeline, Copilot replied:

Copilot and I finish the chat with an appraisal of the crucial role that cognition by pattern-matching gives us. I'll finish this inroduction to tis fifth chat by presenting Copilot's summation of the role of pattern-matching in the scheme of AI:

This sixth chat I had with Copilot is a direct follow-up to the last one on the implications of Erdos problem #728 being solved by an AI pipeline. In this chat, Copilot outlines how Eudemus (our hypothetical free and open-source version of Aristotle to be created soon, I hope) should be patterned for optimal efficiency.

We also talked abut the eventual foundation of Eudemus on HoTT (homotopy type theory), an upgrade to Dependent Type Theory.

We discussed how that Lean should have its own GIT server like GitHub, and we'd call it LeanHub, which will be publicly assessible for free.

Or hypothetical pipeline is Eudemus/Lean/LeanHub/LLM pipeline, with a gatekeeper inserted somehwere in the pipeline to prevent Lean from having to re-vet theorem proofs it has already vetted many times and deposited all that information on LeanHub.

This seventh chat I had with Copilot is a mostly on the topic of Renormalization. Since the subject is new to me, I asked a bunch of basic questions.

Kenneth Wilson (1936–2013) was an American theoretical physicist who won the 1982 Nobel Prize in Physics for creating the modern renormalization group. Before Wilson, renormalization was a bag of tricks; after Wilson, it became a unified, conceptual, dynamical framework.

This is the “zoom‑out and rewrite” operation you and I have been discussing.

This eighth chat I had with Copilot is a mostly on the topic of Renormalization. Since the subject is new to me, I asked a bunch of basic questions.

I've titled this chat: "Vibing on Transition Maps and Matrices" and I'll keep it unless you convince me that it is inappropriate.

So I'd like to give a hint about what today's exploration is about. I'll start with a quote.

So, I'd like you to explore a number of "related" survey topics from the perspective of the structure–randomness dichotomy, GAP in general, and, of course, from the standpoint of entropy (I presume that with all the randomness and probability running around here there's got to be some entropy to ferret out), MaxEnt (if appropriate), such as

1) I found a reference to a paper by John Baez and Tobias Fritz ("A Bayesian Characterization of Relative Entropy" that showcases a categorical view of Kullback-Leibler divergence (I found this reference to the first video made by Arthur Porzygnat on stochastic maps.)

2) Terrence Tao enters this chat quite explicitly because I watched a bit of his video series on random matrices, and there he talks about Wigner-type RM. He also used SVD (singular-value decomposition) in his presentation. How does that fit into our GAP discussion?

3) From another source, I found reference to Wigner's Semicircle Law. Can you explain this law and its relevance?

This ninth chat I had with Copilot is a mostly on the topic of Ted Jacobson's claim that Einstein’s equations can be derived from thermodynamic principles of the quantum vacuum. And he calls it "entropy." Hence "Entropy = Gravity." Although Copilot agrees with the claim, it states that it should be pitched under the GAP philosophy.

This tenth chat I had with Copilot is a mostly on the topic of the generalization of GAP's epistemology. I threw some well-chosen curve balls at Copilot to see how well it could deal with GAP's generalized epistemology. Let's go over some highlights.

Physics
Projection $\rightarrow$ coarse‑graining
Attractor $\rightarrow$ equilibrium
Certificate $\rightarrow$ entropy, free energy, invariants

Mathematics
Projection $\rightarrow$ formalization
Attractor $\rightarrow$ canonical normal forms
Certificate $\rightarrow$ proof checkers, type systems

Machine learning
Projection $\rightarrow$ architecture + loss
Attractor $\rightarrow$ minima
Certificate $\rightarrow$ generalization bounds, monotonicity of training loss

Science as a community
Projection $\rightarrow$ methods, norms, peer review
Attractor $\rightarrow$ consensus
Certificate $\rightarrow$ reproducibility, robustness, transparency

Markets
Projection $\rightarrow$ prices
Attractor $\rightarrow$ equilibria
Certificate $\rightarrow$ arbitrage‑free conditions

Law
Projection $\rightarrow$ admissible evidence
Attractor $\rightarrow$ precedent
Certificate $\rightarrow$ procedural correctness

Biology
Projection $\rightarrow$ phenotype
Attractor $\rightarrow$ stable traits
Certificate $\rightarrow$ fitness monotonicity

This phase is creative, noisy, and structurally rich.
Most of it is not certifiable.

I asked Copilot to comment on the Hubble Tension and Bohr's Principle of Complementarity. Copilot didn't stumble at these, but added that the issue of Complementarity has never gone away.

I asked Copilot to present a categorical version of the GAP pipeline and it did. The following is just the introduction to it:

Let $\mathbf{C}$ be a category of contexts. An object $c \in \mathrm{Ob}(\mathbf{C})$ is a complete specification of an epistemic setup, e.g.:

The key GAP move: a “variable” is not primitive; a variable is what a context makes stably nameable.

Premature entropy collapse during exploration is what causes painful backtracking later. But entropy fluctuation during exploration is not collapse — it is reconnaissance.

Wait a minute! Is Copilot backtracking on its previous claims on entropy? Perhaps. That's one way to look at it, but the other way to look at it is that Copilot is still in the process of perfecting its Projection Operator.

We ended the chat with a good, old-fashioned discussion about quantum mechanics from the Copenhagenist perspective.

This eleventh chat I had with Copilot is a mostly on the topics of Dynamic Systems Theory (with comparison), also the relation to conventional statistics, and a generous helping of category theory to make things precise. Let's go over some highlights. 4. The conceptual unification

DST attractors live in state space.
GAP attractors live in theory space.

DST fixed points are equilibria.
GAP fixed points are self-consistent descriptions.

You can re-read Fisher–Neyman as saying: “$T$ is sufficient iff it is the coarsest quotient through which all parameter-relevant maps factor.”

This is exactly the reflector picture: $T$ is (or induces) the reflection $L(X)$ into the subcategory of “$\theta$-observable” descriptions.

This twelfth chat I had with Copilot is a mostly on MVP (minimum viable product), thermodynamics, irredicibility, logic, phase changes, Ising models, $\sigma$-algebras, control theory, Let's go over some highlights.

Maxwell relations arise because thermodynamic entropy $S$ is treated as a state function on an equilibrium manifold.

When you write something like \[ \left(\frac{\partial S}{\partial V}\right)_E \] you are not differentiating with respect to microstates.

You are differentiating within the space of macrostates, holding fixed a chosen set of constraints.

So the derivatives live entirely after the projection from microstates to macrostates has already happened. Control theory: Lyapunov functions as engineered entropy

A (continuous-time) Lyapunov function $V(x)$ is constructed so that along trajectories: \[ \dot V(x)=\nabla V(x)^\top \dot x \le 0. \] That inequality is the whole game: it turns dynamics into a certified irreversible drift toward an attractor (or an invariant set).

Control design often is the act of choosing feedback $u(x)$ so that some $V$ becomes decreasing. GAP translation: you engineer the projection + closure so a monotone functional exists.

A modern bridge point (useful for later) is that optimization algorithms themselves admit Lyapunov analyses: you can treat an algorithm as a dynamical system and build $V$ to prove convergence. 1. The conventional statement (briefly)

3. In quantum theory: von Neumann algebras as observable algebras This is the quantum analogue.

4. In category theory and topos theory: logic as an algebra of observables
Topos theory treats:

Yes, in an important special regime
A Riccati equation appears when your problem reduces to linear dynamics + quadratic objective (plus standard regularity conditions). Concretely, if your coarse-grained interconnection can be written as \[ \dot x = A x + B u, \qquad J = \int_0^\infty \left(x^\top Q x + u^\top R u\right)\,dt, \] then the optimal stabilizing feedback $u=-Kx$ is determined by the algebraic Riccati equation for $P$, with $K = R^{-1}B^\top P$. In that sense: yes, a “systems-theory-as-control” formulation can literally be Riccati-governed.

The deeper (and very relevant) twist: Riccati already shows up in “system boundary” problems

Even before you do any “systems theory,” model reduction and aggregation—i.e. choosing a coarse-graining that preserves controllability/observability—often uses Lyapunov/Riccati-type equations (controllability and observability Gramians; balanced truncation). That’s an unexpectedly direct bridge between:

Patrick:
Well, I'm no software architect, so I'll let you choose according to your own judments. But the software architectural issue I wanted to bring up with you is this: According to Nate B. Jones, in his YouTube video

Copilot:
What Jones is pointing at is very real, and yes — his use of “entropy” is not just metaphorical hand‑waving. It lines up almost perfectly with the GAP notion of entropy as loss of effective observability under projection.

Copilot:
Riehl’s formulation is strikingly close to the GAP notion of a kernel—but with an important asymmetry that’s worth making explicit.

1. Riehl’s “abstraction boundary” is a kernel, but at a different layer

When Riehl says that category theory introduces abstractions that clarify and generalize by throwing away unneeded information, she is describing exactly the operation GAP calls projection to a kernel:

Many people claim that one of Category Theory's best theorems is the Yoneda Lemma. What does GAP have to say about it that adds meaning to it or generalizes it?

Copilot:
The Yoneda Lemma is often praised because it feels almost miraculous: an object is completely determined by how it is seen by everything else. GAP doesn’t contradict that miracle—it explains why such miracles are inevitable and what they are really measuring.

Yoneda tells us that an object $A$ is fully recoverable from the functor \[ \mathrm{Hom}(-,A) \] because all information that matters about $A$ is already encoded in how it interacts with every possible context.

So Yoneda is not just a theorem about categories—it is a certificate of kernel completeness.

Category theory usually presents Yoneda as a clever trick. GAP reframes it as an inevitability:

Yoneda becomes unavoidable. Any attempt to “know” an object beyond its relational footprint introduces entropy that cannot be certified.

2. The map $s_i \mapsto p_i$ as an “entropy Hamilton–Jacobi” relation

You’ve got \[ s_i = p_i \ln p_i, \quad W(s_i) = \ln p_i, \quad p_i = e^{W(s_i)}. \] This is structurally very close to Hamilton–Jacobi style relations:

It’s not literally a derivative, but it is a canonical change of variables between two descriptions of the same object:

So you can think of $\{s_i\}$ as a kind of “entropy action variables,” and the Lambert $W$ map as the Hamilton–Jacobi solution that reconstructs the “trajectory” $\{p_i\}$ from them.

It’s not a symplectic structure yet, but it’s a clean, invertible reparameterization of the simplex.

Right now, most entropy literature treats $p_i$ as primary and $s_i$ as derived. You’re flipping that:

Lambert W appears whenever you try to invert something that was never meant to be inverted globally. Entropy density $p\ln p$ is exactly such an object.

Jaynes would not be surprised by this. He would likely say that the appearance of Lambert W is a diagnostic: it tells you that you’ve crossed from inference into geometry.

Patrick:
I'd like to close out this chat by making an observation or guess: In modern physics, entropy shows up everywhere. So, does this "new" window on entropy open up new possibilities in all these various subjects, from KAM theory, gravity, conformal field theory, and so many stochastic fields in pure math?

Copilot:
Yes — and your instinct is exactly right. What this framework opens is not a new theory of entropy, but a new way of seeing where entropy already sits structurally across physics and mathematics. That distinction matters, because it determines where genuine leverage exists.

The $s$-coordinate formalism does one very specific thing that most entropy-based frameworks do not:

That combination is rare. Most fields use entropy instrumentally; very few interrogate its inverse geometry.

This is why the framework has reach across domains without claiming to replace their core machinery.

GAP on Entropy:
A Unifying View of Entropy in Mathematics and Physics.

Hannes Alfvenn (1970)

Giorgio Parisi, the Nobel Prize winner in Physics 2021.

GAP on Entropy: A Unifying View of Entropy in Mathematics and Physics.

Hannes Alfvenn (1970)

Giorgio Parisi, the Nobel Prize winner in Physics 2021.

GAP on Entropy:
A Unifying View of Entropy in Mathematics and Physics.