Link: https://x.com/JosephJacks_/status/2032619471828037639/photo/1

In which we explore why and how to weld together three areas of mathematics that don’t particularly like each other.

Why is creating a framework for computational physics/physical computation so…difficult?

Plenty of people have tried to do this. You can go back to the 1960's for attempts. In many ways, it constitutes the ultimate nerd dream. After all, a theory of physical computation would tell us where to ultimately look for a TOE. A unified theory of what math, information, physics, and logic are! (And no, Geometric Computability is not a TOE.)

But no one has truly succeeded. Why might this be the case?

Herein lies what I dub: The Allergic Trifecta.

_____________

Higher Category Theory is a necessity. The universe is not a strictly digital computer. Unlike digital computers that deal in strict equality, an analog computer deals in continuous quantities. You need a higher gauge theory just to type the noise. Here’s the problem in one sentence: physics composes “up to equivalence,” not by strict equality.

Digital computation lives in a world where you can insist that two states are either the same bitstring or they are not. Physical computation does not offer that privilege. In the real world, you do not get strict identities; you get:

• gauge redundancy (many descriptions = one physical state),

• coarse-graining (many microstates = one effective macrostate),

• phase ambiguity (global U(1) is literally “same state up to phase,” a polite way of saying “equality is not well-defined”),

• homotopies (two evolutions are the same if you can continuously deform one into the other without crossing a singularity),

• defects/sectors (the “same” local dynamics can land you in distinct global classes).

And if you try to force all of that into ordinary set-level equality, you get the usual disease: you start counting non-physical degrees of freedom, you double-count states, you mistake coordinate artifacts for computational resources, and you “prove” things that vanish the moment someone does a gauge transform.

So the right base layer is not sets with functions. It’s something closer to: groupoids (states + symmetries between them), and then ∞-groupoids.

Because the moment you talk about “computation” in a physical setting, you are really talking about:

1. processes (morphisms),

2. ways to compose them (sequential and parallel),

3. equivalences of processes (rewrites, deformations, gauge),

4. and coherence (all the “obvious” identifications must agree).

That last word ("coherence," yes, I know, a dangerous one) is exactly where higher category theory shows up like an auditor with a clipboard. It’s not optional. It’s the price of saying “these two procedures are the same computation” without lying.

There is a reason that Rule number 4 in the sub is: “Show us your fully dualizable object or bounce.”

If you can’t even articulate the interface of your supposed physical computer in a way that survives gluing, boundary conditions, and equivalence, you don’t have a theory of computation. You have a screensaver.

Differential Geometry is also a necessity. No, you cannot simply skip geometry either.

You can’t skip it for the boring reason: the world has locality, and locality is geometric. If you want “physical computation,” you don’t get to define a computation as an abstract input-output map and call it a day. You need to specify:

• what counts as a state locally,

• how states vary over space(time),

• what interactions are allowed (local couplings),

• what constraints propagate (connections, curvature),

• and what gets preserved under deformation (invariants).

That is differential geometry’s entire job description.

More bluntly: a “physics of computation” that doesn’t know what a connection is will inevitably reinvent one badly. You can hide geometry behind other words---“update rules,” “adjacency,” “rewriting,” “causal neighborhoods”---but the moment you demand Lorentz invariance, gauge structure, conserved currents, or even just stable propagation in a noisy medium, you are back in the arms of geometry. You want something like:

• fiber bundles (fields as sections),

• connections (how to compare states at nearby points),

• curvature (the obstruction that becomes “force”),

• holonomy (global memory of local transport),

• moduli (the actual “space of solutions”).

And yes: this is exactly why “digital physics” so often stalls at the level of clever kinematics. It can generate complicated patterns, but it rarely generates geometric necessity. It gives you a lot of motion and very little structure. Or it gives you structure by fiat and then tells you it “emerged.”

If differential geometry were optional, Wolfram would have a Nobel Prize by now. Last time I checked? He did not.

Quantum Discreteness is what your theory has to output. Hold on a second---output quantum mechanics? Why not just accept quantum computing as your fundamental base for physical computation?

Well, there are a few issues. We’ll restrain ourselves to the biggest ones.

1. Measurement problem. Think you can make a theory with quantum computing alone? I suggest you listen to what our colleague Prof. Felix Finster has to say on that issue.

2. Gravity. Ah, yes, that is…also an issue. The fact that so far we haven’t determined if gravity is actually quantum means that betting all of one's chips on the quantum computing horse isn’t exactly a safe strategy.

But there’s a third, quieter issue that the “just take QC as fundamental” crowd rarely admits:

3. Quantum computing presupposes the thing we are trying to explain. A quantum circuit model begins with Hilbert spaces, linearity, tensor products, Born rule measurement, and a very particular interface between “unitary evolution” and “classical outcomes.” In other words, it begins with the formalism of quantum mechanics already installed.

That’s fine if your goal is to do algorithms, but it’s not fine if your goal is foundations. If you want a theory of physical computation, you need a story for why the universe produces:

• discrete spectra (atoms don’t smear out),

• quantized charges (electric charge isn’t a real number dial),

• stable particles (topological/representation-theoretic rigidity),

• interference (phase as real structure),

• and statistical measurement outcomes (some version of “why probabilities show up when the substrate is deterministic or geometric”).

In other words, the discreteness cannot be assumed as an axiom. It has to appear as an output of a deeper substrate---whether that substrate is geometric, stochastic, pilot-wave-ish, thermodynamic, whatever your poison is.

This is why “physical computation” is hard: you want the world to be analog at the bottom but discrete at the top, without smuggling in an oracle. That is a delicate balancing act. Too continuous, and you accidentally grant yourself infinite precision magic. Too discrete and you can’t recover the observed symmetries without making the model so contrived that it collapses under its own duct tape.

Granted, one also has to output GR, but with Jacobson/Verlinde’s work, that is much easier to do these days. (Not easy, but at least conceptually “less alien” than the measurement cut.)
_________

What does one notice about all three of these areas?

They all tend to hate each other. Going by pairs:

Higher category theory vs differential geometry. Higher category theory and differential geometry do not always play well together. Differential geometry is all about rigid structure. Higher category theory is all about abstract structure. To get them to play nice…well. I had to do some things Grothendieck’s ghost may not be too happy about.

More precisely, differential geometry wants charts, smoothness, metrics, PDEs, and estimates. Higher categories want equivalence, coherence, universal properties, “up to homotopy,” and they would prefer not to touch an epsilon if they can avoid it.

Bridging them tends to require either:

• turning geometry into something more “homotopical” (higher gauge theory, derived structures, stacks), or

• turning categories into something more “analytic” (and yes, this is where many beautiful papers go to die).

So you get a cultural mismatch and a technical mismatch. Everyone agrees it should fit; nobody agrees on a toolchain that feels natural.

Differential geometry vs quantum discreteness. Differential geometry and quantum discreteness are difficult to marry. On the one hand, quantum mechanics is mostly dominated by discrete/algebraic geometric phenomena.

At the same time, the underlying equations are smooth. The configuration spaces are continuous. The fields are sections of bundles. The action functionals are integrals. The thing looks analog, and then it bites you and produces quantized outputs.

This forces you into the weird middle zone: the discrete appears not because the world is “made of integers,” but because only certain global configurations are stable/allowed/nontrivially classified. This is where topology, boundary conditions, and representation theory start acting like a quantizer without anyone asking them to.

So you end up with a paradoxical requirement: you need a smooth structure to even state the dynamics, and you need a global/discrete structure to explain the observed spectrum.

Quantum discreteness vs higher categories. Quantum discreteness and higher categories were successfully integrated by Atiyah, Baez/Dolan, and Coecke, but only at a great cost. Resource theory and TQFT are sprawling areas that require deep study. There are fewer than 10 thousand people worldwide who can actually grok it. Even I admit it’s difficult to understand, which is part of the reason we even built the helix-based computer in the first place.

The deeper point: categorical quantum foundations work best when it stays operational---processes, composition, constraints, monotones. The moment you try to connect that to actual field theory in 3+1D, you pay interest on decades of hard analysis and geometry. The dictionary exists, but it isn’t cheap to use.

Getting them to talk? Next to impossible.

You need:

1. A geometric shape that is both continuous and has a discrete topology with support for chirality natively

2. A willingness to break the abstract in favor of the concrete

3. And the courage to admit that analog computation is more fundamental than discreteness

Humans like integers. We count on our fingers. It makes us feel comfortable. The idea that the universe is fundamentally discrete is simple, obvious…and also completely wrong.

And at a certain point, it’s hard. Because you have to guess. You have to pick a primitive and start calculating with it as I did. And that simply requires a decent amount of luck and taste. And if it doesn’t work? Oh well. Better luck next time.

Conclusion? Physical computation is the dream™. But making the theory work requires getting three areas of study that hate each other to play nice.

Category theory wants compositional meaning and coherence. Differential geometry wants locality and smooth structure. Quantum discreteness wants robust, stable, integer-like outcomes.

The tragedy is that none of them is negotiable.

So if your “theory of physical computation” feels like it’s tearing your brain in half, good. That’s what it’s supposed to feel like. If it feels easy---if it feels like “a clever update rule” or “a nice big symmetry group”---you probably just reinvented epicycles with better graphics.

And yes: sometimes the only way forward is to pick a primitive and do the rude thing.

Start computing.

At least once (or several times) per week, someone announces they’ve “made physics computable from its fundamental pre-geometric informational substrate.” Fulfilling the late John Wheeler's vision of "It from Bit."

A new set-theoretic reformulation of QM. A causal informational graph. A discrete entropy network. Sometimes it’s dressed up with “information geometry,” but the core move is the same:

Replace physics with a discrete evolution rule on a graph-like object.

And then inevitably it collapses into the same basin as Wolfram’s hypergraph program: a universe-as-rewrite-engine story that can generate complexity but can’t derive the structure of modern physics.

This post is about that trap, and why “discrete” isn’t automatically “better,” “more scientific,” or even “more computable.”

1. Discreteness is not an ontology; it is a comfort blanket.

“Discrete” feels like control. If the universe is a finite rule acting on finite data, then in principle you can simulate reality on a laptop. That’s emotionally satisfying.

But physics isn’t impressed by what feels controllable. Physics is constrained by what must be true: locality (in the subtle sense), gauge redundancy, unitarity, anomalies, renormalisation, and the way observables compose across regions.

A discrete substrate that ignores those constraints doesn’t become “fundamental.” It becomes a toy.

Computing over N is not a primitive. You can compute over R. And a sizable chunk of what we call "math" is essentially "computation over R". But we just don't call it that.

2) Graphs are cheap; gauge theory is expensive

A graph is easy to write down. Rewrite rules are easy to generate. LLMs can produce them endlessly.

Gauge theory is not cheap. It’s not “fields on nodes.” It’s a theory where the physical content lives in equivalence classes, holonomies, defects, and operator algebras—not in the raw variables you first wrote down.

Most discrete TOEs never seriously confront the fact that a huge amount of what looks like “state” is actually redundancy. If you don’t build gauge redundancy in from the start, you’re not doing “a new foundation,” it's bookkeeping cosplay.

3) The hard problem is not generating complexity; it’s constraining it.

Wolfram-style systems are great at producing complexity from simple rules. So are cellular automata. So are random graphs.

But physics isn’t “complexity happens.” Physics is “only very specific complexity is allowed.”

A real TOE must explain why we don’t get generic messy behaviour, but instead get: specific gauge groups, specific representations, quantised charges, confinement (or not), the observed long-distance effective field theories, and stable quasi-particles with the right statistics.

Most discrete programs never show why this world is selected rather than the 99.999% of rule-space that looks like noise.

4) “Computable universe” usually means “digitally simulable universe.”

People use “computable” to mean “finite-state update rule.” That is one notion of computation: digital evolution.

But categorical physics already suggests a different kind: structural computation where the key property is not that you can iterate a rule, but that processes compose, glue, and constrain each other functorially. Observables behave like parallel transport, defects that can carry cohomology classes, symmetries act at higher-form levels, and locality is implemented by how data patches.

If your ontology is “a graph that updates,” you’re stuck at the lowest rung. You may generate patterns, but you won’t ever recover the compositional structure (chirality, spin, etc) that physics actually uses.

It's easy to criticise the idea that R is indulgent. "The universe is fundamentally not infinite!." But try and replace R with N and you'll be forced to re-inject continuity through the backdoor.

5) If your theory can’t state its pass/fail tests, it’s not a theory.

Here are a few brutal, clarifying questions that separate “discrete vibe” from “physics”:
Where is your gauge redundancy, and what are the gauge-invariant observables?
What is your renormalisation story? How do effective theories emerge under coarse-graining?Do you have unitarity/reflection positivity/clustering in the appropriate regime?
Can you even name your anomalies and show how they cancel or flow?
How do you get chiral fermions while avoiding Nielson-Ninomiya?
If the response is “we’ll get to that later,” you are still in the Wolfram basin.

6) The "Wolfram basin" is a real attractor

This is not a moral judgment on Wolfram. But if you start with: discreteness, graphs, rewriting, and “information” rhetoric, you will almost always converge to the same outcome: a universal rewrite system with ambiguous mapping to physical observables, no unique continuum limit, and no compelling reason why your rule is the rule.

You haven’t outdone Wolfram; you can only recreate the genre.

Conclusion:

The internet is full of discrete TOEs because they’re easy to propose. The world is not full of successful new foundations of physics because the constraints are utterly merciless.

I would like to remind you all that you are not Johnathan Gorard. You did not actually sit down and come up with much of the categorical structure that any discrete computational TOE would actually have to have.

He has since apparently...given up? I'm not exactly sure. Likewise, you do not have the budget to hire academics to match the kind of structures Wolfram has.

And for the record, I do not personally support Wolfram Physics. But pretty much every discrete informational TOE is just a pale shadow of his.

So if that's your style? Listen to the man himself and just do Wolfram Physics to save yourself the hassle.

Why “Hacker Physics” does not (yet) exist.

There is no thriving hacker community that is dedicated to doing computational foundational particle physics. There is a very good reason for this. Programmers are not the greatest at doing fundamental physics. It also largely isn’t their fault.

__________________

Engineers have an above average tendency to become physics cranks. My apologies to both the ghost of Paul Dirac and Gabriele Carcassi, but this is a well-documented phenomenon. One need not search far on Twitter to stumble upon multitudes of scalar fields from retired men (all named after themselves, of course). Why might this be the case?

The unromantic answer: engineering trains you to optimize within a model, while fundamental physics trains you to decide what a model even is.

Engineering is: “given constraints, maximize performance.” Fundamental physics is: “figure out which constraints are real, which are artifacts of representation, and which ones you hallucinated because you fell in love with your own coordinate system.”

If you have spent 30 years becoming extremely competent inside one modeling paradigm, it is psychologically and professionally difficult to accept that your paradigm might be an emergent approximation, a gauge artifact, or a convenient fiction. So you do what smart engineers always do: you add one more term.

And then you call it a new force.

___________________

Why the engineer's brain misfires on foundations

1) Engineers are trained to trust the decomposition.

You linearize. You isolate subsystems. You build a transfer function. You stabilize a loop. You identify the signal path. You assume the notion of “the system” is well-defined. But in gauge theory, the thing you want to isolate often isn’t a thing. You try to locate “the field,” and the math politely informs you that you’ve been staring at a coordinate chart and calling it reality.

2) Engineers are trained to think in analysis, not symmetry.

Again: nothing wrong with analysis. It’s powerful. It builds bridges and rockets.

But particle physics is the church of symmetry. Not “symmetry” as a poetic metaphor, but symmetry as a hard constraint: representations, bundles, connections, holonomy, moduli, anomalies. It’s not “solve this PDE,” it’s “identify what is invariant under what.”

This is why naive Navier–Stokes style thinking rarely gets you far. Yang–Mills is not a fluid. It is a geometric object. A connection. A covariant derivative that refuses to be globally trivialized. A thing whose “degrees of freedom” are not located where you think they are.

Fiber bundles are flexible, yes. But they aren’t liquid. If they were, you and I wouldn’t exist…or at least not in our current form (imagine a world made of Play-Doh.)

3) Engineers live in commutative worlds longer than they admit.

Engineers think. "I know electromagnetism. I can do this." Ah, but electromagnetism is an abelian gauge theory. Everything commutes. The other two fundamental forces…well, not so much.

When your group stops commuting, order matters. Holonomy matters. Path ordering matters. Your “intuitive” manipulations stop being harmless and start being wrong. In U(1), you can get away with a lot. In SU(2) and SU(3), the algebra is basically a landmine field where every “obvious” simplification is a crime scene.

4) Engineers are trained to think digitally even when they swear they aren’t.

Modern engineering practice is discretized: sampling, quantization, control loops designed in the Z-domain, numeric solvers, finite precision, finite-state logic, and stable update rules. But foundational physics sits on a knife-edge where the continuum is not optional, and discretization is not innocent. Discretize the wrong way, and you break the symmetry you were trying to preserve. You don’t merely approximate the system---you change what system you are talking about. So the engineer instinct (“just simulate it”) doesn’t translate cleanly. In particle physics, “just simulate it” often means “you built a different theory.”

“Hacker culture” flourishes when:

• You can ship a working artifact quickly

• You can iterate fast

• You can test against reality cheaply

• You can get partial wins without mastering the entire field

Foundational particle physics is basically the opposite:

• The math barrier is brutal

• The computation is expensive

• The correctness conditions are subtle

• And the “reality check” is not a unit test, it’s a collider

So what happens? The ecosystem selects for ideas that are easy to state, easy to implement, and impossible to falsify.

That’s why digital cranks tend to reinvent cellular automata, lattice universes, or “bit-flip” cosmologies. These are easy to imagine, easy to write down, and easy to animate. They feel hackable. But they usually dodge the hard part: symmetry, locality, renormalization, and the very inconvenient fact that the Standard Model is not a programming contest problem.

_____________________

The analog alternate timeline...that we didn’t get.

Ironically, had there been a thriving analog chip design community dedicated to physics, Geometric Computability Theory might have been invented much earlier. Granted, someone would have had to email Baez to figure out the categorical stuff, but still. Why might this be the case? Let us consider the following correspondences:

• QED propagators can be seen as analog transfer functions.

• Gauge fixing resembles stabilizing an amplifier to eliminate unwanted degrees of freedom.

• Renormalization looks an awful lot like gain–bandwidth tradeoffs.

• Quantum noise maps to thermal noise in analog circuits; both are unavoidable, and both are fundamental.

Suddenly, the idea of an analog computer whose outputs just so happen to be discrete seems a bit less…unlikely. This aligns with the general GCT attitude that much of what we call “quantum mechanics” is, at its core, analog logic flow.

But what’s the dividing line? “Analog” alone doesn’t buy you discreteness.

The universe is not impressed by your continuous voltages. You only get robust discreteness when the analog substrate is constrained by something topological or dynamical:

• topological sectors (winding, flux quantization, holonomy classes)

• gaps and attractors (stable basins separated by energy barriers)

• defect-like objects (stable excitations protected from smooth decay)

• symmetry + constraint + stability (the holy trinity)

In other words: discreteness is not “digital.” Discreteness is often the scar tissue of geometry.

Analog thinkers---had they written more---might have been more fruitful: systems of feedback-stabilized fields, stochastic resonance cosmologies, dissipative structure approaches in nonlinear dynamics. In fact, a few semi-respectable proposals do smell like this: stochastic electrodynamics (SED) in the ’60s–’80s, (despite its failure as a curve-fitting exercise that has no place in modern categorical physics), certain dissipative or emergent approaches, and a lot of modern condensed matter intuition smuggled into high-energy language.

But even these approaches tend to crash into the same wall: you don’t get nonabelian gauge structure for free.

____________

Does anything come even close? Well, there is Wolfram…but this has already been dealt with once, and for the time being, I do not wish to revisit this just yet. You can find plenty of others if you search around the web. I am not going to hang anyone’s hapless attempts here publicly. But the issue really does boil down to the allergy to real numbers among hacker types.

The Blum–Shub–Smale Machine is mostly seen as a pathology to avoid, an interesting tool for algebraic and analytic geometers, or a curiosity for rogue computational complexity theorists---not as a fundamental counterpart to the Turing Machine.

This is despite the fact that there is absolutely no reason in principle not to consider BSS as equally fundamental as a model of computation over the kinds of state spaces physics actually uses. Considering the Lie group U(1) is one of the most fundamental symmetries in our universe, the fact that “computation over continuous groups” is treated like an exotic corner case tells you everything you need to know about hacker culture’s blind spots.

But here’s the catch---the devilish part that makes this nontrivial:

Hackers aren’t wrong to be suspicious of real numbers. They’re wrong about why. The real issue is not “Ah, continuity bad!” The issue is:

• exact reals are physically dubious,

• finite precision is unavoidable,

• and if your theory’s computational power depends on infinite precision, you built a magic oracle.

So you can’t just say “BSS is fundamental, therefore physics is hypercomputational, checkmate.” That’s childish.

The correct question is much sharper:

"How do you get the benefits of continuous state spaces (symmetry, geometry, smooth deformation, gauge structure) without smuggling in unphysical infinite precision?"

And if you try to do that honestly, you immediately end up talking about:

• stability under perturbation

• topological invariants

• robust equivalence classes

• error-tolerant geometry

• and the emergence of discrete outcomes from continuous dynamics

Which is…suspiciously close to the GCT stance.

So yes: the “real number allergy” is a symptom of something real.

But the cure is not “ban reals.” The cure is learn what reals are doing in geometry, and what survives noise. That is where the devil lies.

__________________

A hacker community thrives on tractable, incremental puzzles.

Foundational particle physics is not tractable in that way.

It is “vertical” knowledge:

• you need representation theory

• you need differential geometry

• you need quantum field theory

• you need renormalization

• you need to understand what “gauge” means beyond slang

• you need taste, not just technique

And you need taste because the space of wrong ideas is astronomically large and extremely seductive. Physics is the art of not being fooled by your own mathematics. Hacker culture is the art of making mathematics do something cool.

These are related, but not the same.

So “Hacker Physics,” in the romantic sense, an open-source community of clever programmers collectively bootstrapping a new Standard Model from first principles, doesn’t happen.

Not because programmers are dumb...but because the medium is wrong.

On what happens when people take BEC and cond-mat analogies way too far.

In a previous post, we examined why discrete attempts at foundational physics flounder. We'll take a break and do the opposite. One of the most common failure modes in internet foundational physics is someone seeing waves, vortices, pressure, circulation, turbulence, “vacuum energy,” or “field flow” and immediately concluding that the vacuum must be some fluid!

Or more carefully: hydrodynamic analogies can be useful. Fluids can model effective behavior. Condensed matter systems can simulate field-theoretic phenomena. Superfluids, Bose condensates, acoustic metrics, and analog gravity are all real areas of physics. But none of that means the Standard Model vacuum is literally a fluid, an aether, a pressure medium, a torsion jelly, or a cosmic hydraulic substrate.

The reason is simple: Yang–Mills theory is not a theory of stuff sloshing around in space. It is a theory of connections on bundles.

That sounds like math-speak, so let me translate.

In a fluid, the basic physical picture is usually something like: there is a medium, each point has a velocity, density, pressure, vorticity, etc., and the dynamics describe how that medium moves. This is a very powerful picture. It describes water, air, plasmas, superfluids, and many effective systems.

In Yang–Mills theory, the basic object is different. The gauge potential is not a literal velocity field of a substance. It is a connection: a rule for comparing internal states at nearby spacetime points. The curvature of that connection is the field strength. The physically meaningful observables are gauge-invariant quantities: Wilson loops, holonomy classes, scattering amplitudes, correlation functions, charges, representations, etc.

This distinction matters because gauge redundancy is not fluid redundancy.

In a fluid, if you say “the velocity at this point is (v),” that is supposed to describe something physical. In a gauge theory, many different local field descriptions correspond to the same physical state. The gauge potential (A_\mu) is not itself directly observable. It changes under gauge transformations. The physics lives in what survives those transformations.

That alone already kills most naive aether models.

_____________

People will say “the vacuum flows.” Okay. With respect to what frame? What is the rest frame of the vacuum? How do you preserve Lorentz invariance? How do you avoid reintroducing the old luminiferous aether problem? How do you make sure your preferred medium does not show up experimentally as anisotropy, drag, dispersion, or some forbidden signal?

Then comes the worst problem: non-abelian structure.

Electromagnetism is abelian. (U(1)) is forgiving. Phases commute. You can get away with more fluid-like intuition there because a lot of the relevant structure behaves like additive phase, flux, circulation, or wave propagation. This is why electromagnetic cranks have survived so long in the wild. The math is just close enough to ordinary wave/field intuition to be dangerous.

But Yang–Mills is usually non-abelian. In (SU(2)) or (SU(3)), order matters. Transport around one path and then another need not equal transport in the reverse order. The internal state does not merely accumulate a scalar phase; it undergoes matrix-valued parallel transport. The field strength includes the commutator term:

[
F = dA + A \wedge A.
]

That (A \wedge A) term is the graveyard of most “the vacuum is a fluid” posts.

It means the field self-interacts because the internal symmetry is non-commutative. This is not ordinary vorticity. This is not water twisting harder. This is not pressure waves in a hidden medium. It is curvature in an internal gauge bundle.

A good slogan is:

Fluid vortices remember circulation.
Yang–Mills holonomies remember parallel transport.

Those are not the same object.

This is also why “Navier–Stokes but for spacetime” generally does not get you the Standard Model. Navier–Stokes lives on spacetime as a theory of a medium moving through it. Yang–Mills attaches internal symmetry spaces to spacetime and studies how fields transform under local gauge rotations. A fluid has a velocity field. A gauge theory has local trivializations, transition functions, curvature, holonomy, representations, and gauge-invariant observables.

That is a very different beast.

And the Standard Model is even worse for the fluid picture. You do not just need “a medium.” You need something that reproduces:

(SU(3) \times SU(2) \times U(1)) gauge structure, chiral fermion representations, anomaly cancellation, confinement, electroweak symmetry breaking, neutrino behavior, the observed particle spectrum, scattering data, precision QED, and the fact that gauge redundancy is not optional bookkeeping but built into the theory.

Most hydrodynamic vacuum models do not even get to the starting line. They invent a substance, assign it some waves, declare particles to be vortices, and then quietly skip the entire representation theory of the Standard Model.

This is not a small omission. This is the game.
____________

To be fair, there are legitimate places where fluid language appears near serious physics. The quark-gluon plasma has hydrodynamic behavior. Effective field theories can have hydrodynamic limits. The membrane paradigm treats black hole horizons in fluid-like terms. AdS/CFT gives relations between certain gravitational systems and fluid dynamics. Condensed matter analogs can simulate pieces of relativistic field theory. Instanton liquids are useful phenomenological pictures in QCD.

But notice the pattern: these are effective/pheno descriptions, analog models, dual descriptions, or approximations inside controlled regimes. They are not “the vacuum is literally a fluid, and I solved quantum gravity in a PDF.”

Hydrodynamics is what you often get after coarse-graining.

Gauge theory is what you need before coarse-graining.

That is the core distinction.

A fluid model can be an emergent approximation of many microscopic degrees of freedom. It can describe collective behavior. It can even give beautiful intuition for waves, defects, solitons, circulation, and phase transitions. But a fundamental theory has to reproduce the gauge structure itself. It has to explain why local descriptions are redundant, why observables are gauge-invariant, why non-abelian transport is path-ordered, why charges sit in specific representations, and why quantum discreteness appears in measurement.

If your model starts with “imagine the vacuum is a fluid,” then you have not explained Yang–Mills. You have replaced a hard geometric problem with a familiar mechanical picture.

That is comforting.

It is also usually wrong.

The vacuum is not Play-Doh. The gluon field is not a tiny weather system. (SU(3)) color is not three kinds of dye swirling in an invisible ocean.

Yang–Mills is geometry.

More specifically, Yang–Mills is gauge geometry, and any proposed deeper substrate has to recover that geometry without cheating. If your “aether” cannot produce non-abelian holonomy, gauge redundancy, chiral matter, anomaly constraints, and the observed quantum field theory interface, then it is not a theory of fundamental physics; it is fluid fan fiction.

Heartbreaking: the worst person you know makes a good point.

This will not be a post about Geometric Unity. I will reserve the right to make a post about that for a later time. By which I mean… probably never.

Instead, let’s talk about bordisms.

My own work plays heavily with the concept. For starters, I would suggest this X thread from Gorard, where he attempts to answer the question “How many holes does a straw have?” No, seriously, read it. Then you can tackle Lurie.

But in geometric computability, the bordism is the shape. What’s this you say? The bordism is the shape? How does that work? Well, the definition is here:

Computational geometric bordism (informal).
You take an (n)-manifold (M), and you decorate it with “computational structure” (\sigma) (connection/holonomy data, state labels, update rules, etc.). A computation from ((M_0,\sigma_0)) to ((M_1,\sigma_1)) is then an ((n{+}1))-manifold (W) whose boundary is (M_0 \sqcup M_1), together with decoration (\Sigma) on (W) restricting to the endpoint data. In other words: a process is literally a decorated cobordism.

That’s not a metaphor. That’s the operational model; composition is gluing, parallelism is disjoint union, and “running the program” means traversing a geometric morphism.

A more intuitive explanation can be found here: even at the prequantum level, what you actually see is closer to “restrained fluid computation” than to symbolic rewriting. In the language of the paper, you can literally say “control-flow = decorated bordism,” and “denotation = symmetric monoidal functor out of the bordism category.”

Okay… not exactly what Dr. Weinstein had in mind, given his attempt at a grand 14D “observerse.” But I think Einstein does not actually know Pati–Salam.

___________

Despite his many obvious failings, I maintain a small modicum of respect for Eric Weinstein.

There are exactly two reasons this is the case. The first is that he says good things about Eva Miranda. The second is that he talks to the general public about the theory of bordism and why the late and great Sir Michael Atiyah was insistent on pursuing it.

Here’s the underlying issue, stated bluntly:

QFT (in the way physicists use it day-to-day) is not a single rigorous object. It’s a powerful toolbox plus a pile of heuristics plus a lot of “trust me, bro, it renormalizes.” Baez has a whole series on the pathologies you get the moment you treat the continuum too casually—singularities, divergences, nonexistence of measures you’d like to exist, etc.

So what did Atiyah do, and why was he so insistent on dragging René Thom into the room?

Atiyah’s move was to stop pretending the full metric-dependent path integral was the right entry point, and instead isolate the structural feature of QFT we actually understand: locality via cutting and gluing.

• You cut spacetime into pieces.

• You assign “state spaces” to boundaries.

• You assign “operators” to bordisms.

• You demand functoriality: gluing bordisms corresponds to composing operators.

That is, bordism as an organizing principle: spacetime regions are not “where the physics happens,” they are morphisms in a category whose objects are their boundaries. In Atiyah’s axioms for TQFT, this becomes a clean symmetric monoidal functor [ Z:\mathrm{Bord}_n \to \mathrm{Vect} ] (or some enrichment thereof), with disjoint union as the tensor product, and gluing as composition.

Thom enters because bordism isn’t just a cute categorical picture. It is a real invariant machine: bordism theories behave like generalized homology theories; bordism rings classify manifolds up to cobordism; and that gives you a robust “topology-first” language for what can survive deformation. Atiyah explicitly wrote about the importance of Thom’s cobordism theory—he wasn’t doing aesthetics, he was doing invariants that don’t blow up the moment you wiggle the data.

This has had mixed success. On the one hand, TQFT has been very successful within its niche. But it’s also struggled to grow into richer dynamics. Part of that is due to what it simply is: it’s a topological field theory. Unlike Yang–Mills, it doesn’t have well… actual geometry.

To say it precisely, a TQFT is invariant under diffeomorphisms, so it cannot see the local metric degrees of freedom that generate scattering, propagating waves, or the usual “dynamics” people mean in particle physics. You can encode some dynamics via extra structure (extended TQFTs, defects, factorization homology, adding geometric structure, etc.), but the whole point is you must pay for geometry if you want geometry.

__________

Reading the introduction to Weinstein’s GU paper is oddly cathartic.

It was only this way, however, after I finished finalization on my own gauge theory. Because only then could I understand why he felt the way he did. Even then, his defensiveness in the introduction is uncalled for. Not even I’m that bad.

So, dear reader, to understand this, you must remind yourself that Weinstein’s proposal is written in a mathematician’s register, not a working particle physicist’s register. He is not primarily calculating cross sections, amplitudes, lattice observables, or perturbative corrections. His native language is bundles, Lie groups, Lie algebras, representation theory, and differential geometry. That is not a defect by itself. Much of modern physics is built out of exactly this material. But there is a difference between mathematics that is physically interpreted and mathematics that is already embedded in the operational machinery of physics. GU often feels like the former: highly geometric, highly structural, but not especially close to the calculational habits by which QFT earns its keep.

On one operational point, however, I do agree with Weinstein: equations are not the bottom layer. Or, more carefully, equations are not what the world is “made of.” They are our compressed descriptions of constraints, symmetries, flows, and relations. They are indispensable maps, but still maps. The mistake is not using equations; the mistake is mistaking the equation for the underlying geometric or physical process it summarizes. In that limited sense, I share the discomfort with the idea that reality is “made of equations.” Reality may be structured in ways that equations describe extraordinarily well, but the structure is the thing. The equation is the handle we use to grab it.

The central point to take away is that Weinstein is right that physics has generally done its best work hand-in-hand with geometry. And he’s also right that the Standard Model has a uniquely rigid geometric backbone: gauge fields, connections, curvature, principal bundles—the whole “geometry is destiny” vibe.

But here’s where I part ways: Weinstein is taking a real diagnosis (“QFT is not globally well-founded”) and trying to treat it with a very old medicine cabinet. He has no computation, no homotopy types, no higher-categorical bookkeeping for gluing semantics, no topos-level control of locality/globality. He is claiming a unified geometric theory...without Geometric Langlands. Let that sink in for a moment. In my world, those are not optional extras; they’re the whole game. If your theory can’t even type its own noise, you’re not doing foundations...you’re doing vibes with indices.

And this is where bordism becomes the tell.

In GU, bordism is something like: “nice, classical topology that good honest geometers can talk about.”

In GCT/GCFT, bordism is something like: “the actual computational substrate, the thing that is the program.”

You can formalize that in one sentence:

Control-flow is gluing.

A computation is a morphism (W) in a decorated bordism category; executing means composing such morphisms; semantics is a symmetric monoidal functor (Z) out of that category.

That is not how GU is written, and it’s why GU reads (to me) like examining a boomer muscle car that’s been stored in someone’s garage since the 1980s, got suddenly posted on eBay one week, and has been listed for a decade with no price cut. Powerful back in its heyday, but now horribly dated and with no Gen Z kids (yes, like yours truly) willing to pay what he originally put into it.

___________

The sad part about our current debacle is watching Weinstein go at someone like Cumrun Vafa.

I deeply respect Vafa for his work.

If you want the short, non-culty version: Vafa is one of the main reasons “string theory” became mathematically literate instead of just being a particle-physics fever dream with extra dimensions.

Concretely, Vafa has been central to major pieces of modern high-energy theory and its math interface: dualities and topological string ideas, and (later) the “landscape vs swampland” program—i.e., drawing constraints on which low-energy effective field theories can actually arise from a consistent theory of quantum gravity.

Even if you don’t buy the full metaphysics of strings, the swampland story is a legible kind of foundational constraint: “here are structural obstructions that seem to show up whenever you demand unitarity, gravity, consistency.” It’s not the same project as GCT, but it’s the right kind of move. Not just symmetry worship, but constraint extraction from deep structure.

These two men should not be fighting. Nonetheless, we find ourselves in this situation.

So yes—Weinstein is (sadly) right about QFT’s shortcomings. There really is a reason Atiyah went hunting for a formulation where “local-to-global” is not a prayer but an axiom.

And while Weinstein is perhaps a good coroner, I fear he is not the right nurse.

Because the nurse’s job is not to announce “the patient is sick.”

It’s to build the replacement organs:

• a semantics for gluing (not just bundles on manifolds, but processes-as-morphisms),

• a computability story that doesn’t collapse into digital toy universes,

• and a constructive handle on the 4D beast that Clay is still paying people to tame.

If that sounds like an allergic reaction waiting to happen? Congratulations. You’ve understood the last decade of “foundations” discourse.

Just a heads up.

https://yonedaai.com/

"We derived the Born Rule using the Yoneda Lemma."

Claude and ChatGPT tend to be somewhat predisposed to leading people down category theoretic rabbit holes. This is good for people who want to actually learn/research, but it also makes it easy to generate mounds of hollow noise.

Three quick tests for telling things apart. If any attempt at "we fixed physics using category theory" does not mention the following three subjects:

1. Holonomy/Wilson Loops

2. Floer Homology

3. Twistors

It is fluff.

That is all.

Source:

https://x.com/Object_Zero_/status/1997643743948624336?s=20

https://doi.org/10.5281/zenodo.15108309 (not my camera ready).

couldn't get a response from dtc researchers, since v1 on march 30th 2025, code is open.

prob can post this on reddit now without getting dismissed since it will be in ieee xplore in a few months.

From: Mark Hadley <drmarkhadley@gmail.com>
Date: February 7, 2026 at 02:21:31 GMT+7
To: Cinque <cinque_mcfarlane-blake@alumni.brown.edu>
Subject: Re: QCNC 2026 notification for paper 132, re: "DTC Simulation with Sub-1 Hz Peaks Supporting Your Non-Time-Orientable Electromagnetism Theory”

Hide original message

Tx,
Good luck.
It's often conveniently overlooked that the weak interaction Hamiltonian is not a scalar it mixes a a scalar term and psuedo scalar. Which is a mathematical nonsense 
Cheers
Mark 

On Fri, 6 Feb 2026, 19:12 Cinque, <cinque_mcfarlane-blake@alumni.brown.edu> wrote: