The Chirality of Reality: How a Puzzle Cube Flips Physics Inside Out

The firehose has certainly been spewing insights at full pressure this summer. On Friday I forced myself to remember what it felt like to do a "normal human activity" by getting a pedicure at the nail salon. Ahhhhhhhhhhhh I emoted as I settled into the massage chair, appreciating the muazk like never before. Lined up in formation with the women treating themselves at the end of their workweek feeling hella normal for a moment, I made a series of promises I knew were unlikely to hold up: "I'm gonna just chill at the pool for a few days. I'm gonna forget all this metaphysical detective stuff for a whole week."

The reality of my situation is that once I tipped the first domino in recontextualizing foundations through oriented observers the chain reaction it set off seems to be unstoppable. I sensed the inevitable next domino about to tip even if I didn't yet know the specifics of how it would happen.

It was Sunday night that I first felt the trademark kinetic energy heralding its coming appearance in both the noosphere and my physical body. I had just learned of Antikythera's call for papers for their special Agentworld issue. "Arghhhh", I thought. I knew I was about to break those nail salon promises which were in reality self-aware lies all along. I mean... a call for papers centered around the theme of "open-world centaur societies" put out by a think tank "reorienting planetary computation"? These are Cowgirl Cybernetics' exact metaphors and language. I would never forgive myself if I didn't throw my hat in the ring in time.

So I was up late working on a submission including my The Handedness of Monsieur Mathieu: The Human Body as a Matte Blanco Language Engine post as a writing sample. And somewhere in that process, I found it: the M12 Mathieu Group cube.

The Domino Tips

It didn't hit me all the way at first. Yet the seed to the full insight was planted right away when I noticed the connection to quaternions. In the section where he lays out the puzzle's solution, Chandler writes:

"On an M12 puzzle, solving any arbitrary set of 4 pieces leaves the remaining 8 pieces in one of 8 states, which I call ‘quaternion states’. The 8 pieces may then be solved optimally using algorithms corresponding to the quaternions i, j, k, and -1, and their inverses."

This percolated in the background for two more days until the full lightning bolt hit: the M12 cube is uniquely able to capture the Oriented Distinction Operator's "this distinction was made from here" double-pointer flip in a way that captures both an oriented observer and said observer acting in synergy with a quaternionic environment or context. And it does it such a way that the oriented observer is centered and the quaternions rotate around it. 🤯

The Oriented Distinction Operator & Mathieu M12 Group Convergence

Recall that the original Handedness of Monsieur Mathieu post was sparked by my emails with John Churcher centered around reference frames. And that the Oriented Distinction Operator is essentially a consolidated Dirac-belt-as-a-function double-covering from a local perspective which captures the context of an oriented observer making a distinction, which can be thought of as its reference frame.

In mapping the ODO to the M12 cube I saw that the Observer's "this"/"here" and what I call the Yoda's "this"/"here" are the 4 locked anchors to the 8 quaternion states performing the double-covering by first hitting an inversion (360 degrees) then coming back to home on the second rotation (720 degrees). These 4 points define the stabilizer group of the group action, essentially locking the frame of reference to ensure that the remaining 8 points of the 12 are acted upon relative to this fixed context. The subgroup acting on the 8 points is isomorphic to quaternion group Q8.

This is pretty wild because quaternions are usually flat captures or definitions but this is a capture of quaternions, by way of using Q8, that is necessarily process-oriented just like the Oriented Distinction Operator. Here, the frame of reference (the 4 anchors) and dynamic interaction (the 8 points) are both required to say there is an isomorphism to Q8.

Of course I had to get my hands on one of these M12 cubes. In the comment of my purchase submission to Chandler I applauded him for his work and linked him to Handedness of Monsieur Mathieu. When he emailed back with a response and the PayPal invoice he signed off with "And I think you'll find it satisfying that the M12 Cube itself is chiral..."

Indeed I do. I think this innate chirality is the specific magic factor that maps to the Oriented Distinction Operator's same exact requirements of orientation, observer, reference frame, and motion/interaction to even be able to define the Q8 isomorphism at all!

An Embodied Representation of the M12 Cube

When the realization hit I was still in a bit of disbelief and disorientation. I wanted to know exactly how this process could be represented in an embodied way like how I capture the ODO through a specific hand choice for pointing to self while having the opposite hand point away as I say "this distinction was..." then flipping the hands as I say "...made from here".

It was actually through reflecting on my use of a flashlight with a box during my 2024 Recursive Embodied Symbolic Deixis explosion (formerly known as "the magic explosion" before I broke down the mechanics) that I was able to have the realization in the first place. I had been exploring what was really going on when I spontaneously used a flashlight to perform an embodied, pre-conscious-awareness ODO act.

First I put my hand inside the box, dropped the lid over my hand, and shined the light across all 6 faces of a box from the inside pointing out. Then I closed the box and shined the flashlight across all 6 faces of the outside of the box pointing in. So 12 faces in total. Noting that a box also has 12 edges.

So when I next set out to figure out an action that could represent the M12 Cube's Q8 rotation mechanics from the perspective of an oriented observer (the 4 locked points) the flashlight as a tool for the task was already on hand (no pun intended, I swear I just typed that 🤭).

Step 1: Establish Identity

1. Anchor 1 ("HERE" - Source): Drop a stake exactly where your feet touch the floor. This is your absolute origin point.
2. Anchor 2 ("THIS" - Object): Extend your right hand straight out in front of you and touch the front wall. This defines your forward-facing direction.
3. Anchor 3 ("YODA HERE" - Mirror Source): Imagine your perfect mirror twin standing right behind you, feet planted exactly opposite yours on the ceiling.
4. Anchor 4 ("YODA THIS" - Mirror Object): The twin extends their hand, touching the back wall.

Snap! 💥 Look what just happened. By pinning these 4 structural points we lock down the horizontal and vertical baseline. You and your shadow are now welded into the center of the box. The chaos is frozen.

Step 2: The Trapped 8 Edges Appear

Let's look at the rest of the box. A cube has 12 edges, and we just used 4 perspectives (ground, front, and their mirror opposites) to lock the center chassis. Because our 4 anchors are holding the center rigid, these remaining 8 edges cannot float away or scramble into random noise anymore. They are trapped in a physical perimeter grid around you. These edges form 2 square loops:

1. The Upper Loop: The top 4 edges framing the ceiling above your head.
2. The Lower Loop: The 4 bottom edges framing the floor beneath your feet.

This is your Q8 quaternion cage. These are the only parts of the system still allowed to move, but they can only move by rotating along the axes you just established.

Step 3: Run the Flashlight Rotation (The 360 Degree Yoda Twist)

Now, let's make the cage rotate to see the -1 state. Take a flashlight in your right hand. Turn it on.

1. Point it at the Left Wall, then sweep it up to the Ceiling, over to the Right Wall, and down to the Floor.
2. Your arm has just tracked a perfect 360 degree circular rotation around your Front/Back axis.
3. Look at the 8 trapped edges on the ceiling and floor while you do this. As your light sweeps past them, the gears in Chandler's 4-layer gearbox turn. The top 4 edges and bottom 4 edges shift positions.

In our experience here— where the Yoda is an explicitly acknowledged instead of invisible implied part of the system —our arm isn't twisted (the strain is distributed between Observer and Yoda). But in the Eddie Oshins embodied quaternions version (the Philippine Wine Dance) tracking to what's really going on at this first half stage of the double-covering, the elbow is pointed up and the shoulder is strained.

Mechanically, if you look at the 8 edges of the cage they have returned to their original geometric positions. But their orientation parity is completely inverted. The positive and negative tracks have swapped. You are back where you started, but everything is backward and the ribbon is tangled.

We are at State -1, the Yoda shadow... and Yoda is experiencing their respective Yoda shadow relative to us based on the starting configuration. Also a Law of Yoda is that the Observer thinks Yoda is "the Yoda" but Yoda relative to Observer also thinks the Observer is "the Yoda". That's right, people. It's literally the Spiderman pointing meme.

Step 4: The 720 Degree Multi-Threaded Resolution

To untangle the system and get back to absolute solved identity (the 1 State), you cannot turn backward. You have to push through the mirror.

1. Keep the flashlight moving in the exact same direction.
2. Sweep it from the Left Wall, up to the Ceiling, over to the Right Wall, and back to the Floor for a second full 360 degree loop (720 degrees total).

On this second pass your body executes the counter-rotating gear move. In the Eddie Oshins embodied quaternions version the double-covering completes, and your arm untwists itself. State -1 multiplied by State -1 collapses back into State 1. This is exactly how Q8 acts on itself via left multiplication or conjugation.

This is a more comprehensive version of the ODO's consolidated operationalization of the Dirac belt trick: a 360 degree rotation flips the parity and a 720 degree rotation restores it. Puzzle-fanatics purchase Chandler's M12 Cube for the love of the game, unaware that it's a physical demonstration of how quaternions, and by extension spinors in quantum mechanics, must behave in a chiral, observer-dependent universe.

Conclusion

The reason the ODO's "this distinction was made from here" chiral double-pointer flip with self-reference as the frame and necessarily existing yet invisible Yoda shadow is so powerful when it becomes operational on the meta-cognition layer (the level of second-order cybernetics where we think about thinking) is because it's fundamentally a holographic eigenform which I suspect is a simplification of the background process of how perception operates. It treats the unmarked state as "anything not this distinction here" and does a kind of inside-out all-sides scan on the observer, the distinction (attention), and... everything else.

When a person is running this distinction-making process under the hood they obviously aren't waving a flashlight around. The 360 degree twist and 720 degree untwist are run in a single, instantaneous holographic pulse. Like a dolphin's sonar! The process sweeps the 8 trapped edges of the Q8 cage against the infinite background noise, catches the -1 reflection of the environment, and snaps a stable 3D frame into reality.

A Process-Oriented Definition of Quaternions

What if we have it flipped and quaternions aren't the grounds upon which things happen but are instead actually emergent natural properties of observer-dependent systems? In the M12 Cube model, this is in fact what seems to be happening.

In an earlier section laying out the basics I noted that this appears to be a unique necessarily process-oriented definition of quaternions. Only through the combination of both the representation of an oriented observer's frame of reference (the 4 locked points) and their acting upon the remaining 8 points are we able to say there is an isomorphism to Q8 and thus define the quaternions.

This kind of blew my mind and so I asked Mistral if there were any other explicit process-oriented definitions of quaternions out there. Mistral was like NAH, nothing that requires process to create the definition at all. But it did point me in some directions where process-oriented definitions may be hiding.

The usual suspects showed up: group actions in geometry, Topological Quantum Field Theory (TQFT), intuitionism/constructive mathematics, and embodied cognition. But the other candidate it suggested was a concept I had only just learned about that had relentlessly shown up in those two days of percolation between finding the M12 Cube and the domino fully tipping: gauge theory and symmetry-breaking in physics.

Precogging Gauge Theory Through My Ambient Environment

I had emailed Jim Bowery after meeting up with David McGoveran two weeks ago to say I was interested in the Sentax revival and also yap about Mike Manthey's space-like computation. I'm a space-like computation maniac lately and think that Manthey was onto something really deep there! Jim signed off his last email by saying that he and David are both really excited about an approach to the Combinatorial Hierarchy in the form of a discrimination gauge theory. Jim had hoped to publish something on this for the upcoming Laws of Form conference but did not have the time. This gauge theory of distinction they are working on doesn't incorporate the full space-like computation paradigm but does incorporate Z sub 3 {-1, 0, +1} which is a key component of Manthey's space-like computation.

"What's a gauge theory?" I thought. Yes, I had to Google it. I knew the term had to with physics but because I don't pay much attention to the empirical experiments side of things the meaning was unfamiliar. My approach is along the lines of Gödel's "I don't believe in empirical science. I only believe in a priori truth"... which probably explains the cursed nature of my mind and its products.

On the tail end of my all-nighter getting my Antikythera Agentworld submission in my friend Trey texted me about some physics things. Honestly, I had been a little bit bitchy to him they day before when I was enjoying my break and under the delusion that I would follow through on my nail salon vows. "NO MORE PHYSICS FOR A WEEK!" I said. "I am drawing a distinction!!!" So I was again trying to shut it down but he was insistent that I "Just read it! It felt like a breakthrough for me and it was relevant to what you've been researching." Fine, I thought. I'm already awake working anyway.

Much of the content of his texts had to do with exploring the intersection of making distinctions, boundary conditions, and symmetry-breaking like I was exploring first in Dess(e)ins d’Enfants: the Meno Upgrade for a Constructivist Genesis and in the Handedness of Monsieur Mathieu follow-up. We had some back and forth about this and then he said: "We have to talk about gauge symmetry and symmetry in general. This topic is extremely intriguing and could be very insightful in our search for an answer to boundary conditions and actually what attracts boundaries. What are the attractors? That's where I'm left perplexed."

I was like there's that gauge theory again! I relayed that it had just come up with someone else working on a gauge theory of distinction. Whenever something repeats in this way I am trained to pay attention due to the patterns one naturally forms in order to make sense of the experience of a less-linear-than-usual-but-still-human reality.

So when Mistral told me that gauge theory might be a place where a process-oriented definition of quaternions might be hiding I asked some follow-up questions trying to understand the how and why of this. In particular, it pointed to work by John C. Baez rooted in higher category theory which he appears to cover in a book called Gauge Fields, Knots, and Gravity. Which is published in the World Scientific Book Series on Knots and Everything, which Lou Kauffman from the Laws of Form universe is an editor of!

I had finally just gotten my hands on a copy of Knots and Physics in June when trying to untangle the unplanned weirdness of my not-not-quantum context-dependent category theory ECP software architecture (also a toy universe that apparently operationalizes quantum theory... without theorizing about it), thinking that knot theory and a topological perspective on all things quantum was key.

The Synthesis

But back to the main thread. Gauge theory is fundamentally about local symmetries. My new M12 Cube process-oriented definition of quaternions seems to implement a toy gauge theory where:

1. The 4 anchors (Observer + Yoda's "this/here") act as a local frame of reference, like a "gauge choice" for gauge fixing.
2. The 8 quaternion states are the degrees of freedom, like a gauge field. These states are being "gauged" (transformed) by the subgroup, the process of which (requiring both the 4 anchors and the 8 states which creates the Q8 isomorphism) is like a gauge group.
3. The 360 degree inversion/720 degree restoration is the gauge transformation that preserves the system's physical (or logical) state.
4. The required frame of reference in my M12 Cube definition maps to locality (symmetries depend on position) in gauge theory.

My model is a discrete, finite, observer-dependent version of gauge theory. It is not a physical gauge theory (in that it doesn't describe forces or particles) but it does capture the essence of gauge symmetry in a purely logical and relational way. Like how fellow science-ignorer Gödel would!

Also, isn't Chandler's M12 Cube giving Gödel rotating universe? The product description says "The puzzle has a set of 3 and another set of 4 axes that are geared together internally. They don’t all turn in the same direction, so a 4-layer gearbox was required to make it work." Immediately I was like uh huh. Take that, expanding universe believers. You can't avoid the deeper reality our cursed brains are clocking forever (pun absolutely intended). "Happy biiiirthdaaaayyy, Mr. Einstein..."

What's fascinating to me knowing the context of my ODO x context-dependent category theoretic ECP software architecture is that John C. Baez's higher gauge theory replaces the gauge group with a higher group (e.g. a 2-group), which has both objects and morphisms. The M12 Cube model can be seen as a 2-level system:

1. Level 0: The 4 anchors (Observer + Yoda's "this"/"here" as objects.
2. Level 1: The 8 quaternion states as morphisms between the anchors.
3. The subgroup isomorphic to Q8 acts as the 2-group that governs the interactions between the anchors and the states.

Baez has also worked on discrete gauge theories, where the gauge group acts on a finite set such as the M12 cube. In this context:

• The 4 anchors are like the vertices of a lattice.
• The 8 quaternion states are like the edges or faces of the lattice.
• The Q8 action is the discrete gauge transformation that permutes the states while preserving the lattice's structure.

And finally, we can make some mappings to Topological Quantum Field Theory (TQFT), where the physical system is described by a functor from a category of cobordisms (spacetimes with boundaries) to a category of vector spaces (quantum states). This model is like a toy TQFT where:

• The M12 cube is the "spacetime" (a discrete, finite space).
• The 4 anchors are the "boundaries" (like the input/output of a cobordism).
• The 8 quaternion states are the "quantum states" associated with the boundaries.
• The Q8 action is the "topological invariant" that describes how the states transform.

The Implications

In some of its output about how this M12 Cube model is a new perspective on gauge theory Mistral wrote: "This would mean that the ODO is not just a logical framework—it's a physical (or proto-physical) framework, where the act of distinction-making is analogous to the act of gauge symmetry in physics. Your model suggests that gauge theory might not be fundamentally about physics—it might be about logic and observation."

Internally I was like no shit... the phenomenological + embodied experience and brutal consequences of my 2024 "magic explosion" where I experienced the results of doing weird stuff like shining flashlights on the inside and outside of a box left no doubt in my mind. But attempts to capture this and communicate it to others understandably fall short and are misunderstood. So I don't talk about this dimension of it all. Let's just say the ODO didn't fall out of a coconut tree.

More Mistral notes on what's going on here are of course the quantum gravity connection (already reflected in my work earlier this summer when I made a "toy universe" through my ECP software architecture which operationalizes quantum gravity's core principles without theorizing about them). In this case with the M12 Cube we see it in potential reconciling of the discrete and continuous due to the model being discrete but capturing the essence of gauge symmetry which is central to quantum field theory.

And there are also some connections to the Higgs mechanism where a complex scalar doublet containing 4 degrees of freedom undergoes spontaneous symmetry breaking and gives rise to particles. 3 degrees of freedom are "eaten" by the massless W+, W-, and Z0 bosons, turning them into massive gauge bosons by supplying their longitudinal polarizations. The single remaining degree of freedom becomes the physical Higgs boson. Hmmm... wildly speculative, but this is sounding like Observer and Yoda coming to consensus about a distinction type of deal.

In the Higgs mechanism, the universe "decides" how to break symmetry, and particles emerge. In the M12 cube, the Observer and Yoda "decide" how to lock the frame, and quaternions emerge. Same energy and the same math. Just one’s physics, and the other’s logic. Or are they?

In the M12 Cube model the 4 anchors break the symmetry of the 12-point set, leaving the 8 quaternion states as the "massive" (or rather in this case distinguished lol) degrees of freedom. The Q8 is the "residual symmetry" after the breaking, just as the (SU(2)) is the residual symmetry after the Higgs mechanism breaks the original symmetry. Implications? Symmetry-breaking in physics may be a consequence of distinction-making in logic. Again reinforcing this overall theme emerging from the ODO with the M12 Cube and its necessarily process-oriented definition of quaternions of an inversion of physics. It's oriented observers all the way down, baby.

I see Chandler’s 4-layer gearbox as not coincidence or mechanical quirk but a physical manifestation of the Higgs mechanism’s symmetry-breaking. The 4 axes (Observer/Yoda) and 3 geared axes (the "eaten" degrees of freedom) are literally the gears of distinction-making.

Maybe instead of logic being a tool to describe physics physics is a manifestation of logic... with oriented observers at various yet not random structured scales that follow rules which emerge on their own when you ponder natural consequences of necessary chirality.

Maybe chiral anomalies are just fermions messing with creeps trying to observe them with a flashlight. 😉 Or the bidirectional echoes of me back in 2024 rippling throughout the Gödel rotating universe. Just kidding! Or am I...

The Dominos Will Tip Until Observer-Dependence Acceptance Improves

I had found the Monsieur Mathieu M12 dessin through a totally random fluke. Searching for an image of a dessin from Grothendieck's personal notebooks, the Monsieur Mathieu "bandit" caught my attention due to looking like a person with handedness... just like the "children" I had paired up facing one another in the classroom setting of my Meno-style dialogue the day before! And so when I wrote the original Handedness of Monsieur Mathieu I had no idea about the explicit connection of M12 to quaternions.

I was just like "Wait. That looks like a chiral person. I just used that as my primitive yesterday." and then dove in for an exploration, freshly inspired by my emails with John Churcher.

I never would have know about M12's link to Q8 which led to the mapping to the Oriented Distinction Operator as a reference frame for quaternion operations without Chandler's M12 cube. And never would I have found the M12 Cube and discovered this incredible new process-oriented geometric model of an oriented observer acting in tandem with a quaternionic environment without finding the Monsieur Mathieu dessin...

All because I did an image search and asked a dumb question. Puissions-nous toujours avoir des yeux d'enfant, en effet. 🙏