The Great Remainder (137): Resolving the Combinatorial Hierarchy and the Binary Tetration Divide Through Local Orientation
Date Published
For over forty years, the Alternative Natural Philosophy Association (ANPA) has worked on pinning down the connections between the Binary Tetration Sequence (2, 4, 16, 64,... or 2^2^n), which tracks the escalating bit-width and raw computational capacity of informational bit-string universes and the Parker-Rhodes Combinatorial Hierarchy (3, 10, 137, 2^127 + 136), which counts the maximum number of linearly independent operators or discriminable states available before a structural layer saturates.
While David McGoveran notes that the two sequences were always considered connected within ANPA and derives his own unification through the Ordering Operator Calculus, factions within the group held diverging opinions on the specifics. ANPA was split into two main positions — termed the "Menshevik" and "Bolshevik"— with Mike Manthey's framework based on co-occurence and mutual exclusion mapped to quaternions being labeled a third "as yet unnamed heresy" according to a 1994 document reporting on ANPA's happenings.
The bridge between these two sequences—the mapping of pure, discrete bit-logic onto the emergent 3+1 quaternionic spacetime tensor—was something of a holy grail in the attempt to reconstruct relativistic quantum mechanics together.
In my view the missing piece connecting the two is not a deeper layer of abstract algebra. The missing piece was the observer.
The question of the observer does arise in the 1994 document. From what I can gather, the split divides thus:
- The Bolsheviks (Noyes) had an observer in that they realized a "measurement framework" to construct a reference frame was necessary, but it was external to the calculus and set a resolution boundary after the bits are generated so they match continuous relativistic quantum mechanics (RRQM). Noyes argues that you cannot just have an abstract universe of bits floating in a vacuum; you must have an observer who chooses a scale. The observer must determine a "finite resolution" (a threshold or cutoff) where two bit-strings are considered "close enough" to mean they are tracking the same physical state. Their observer is like a finite digital sampler.
- The Mensheviks (Bastin and Kilmister) had a pristine, internal hierarchy and they argued for a specific observer vantage point. Yet without an active, oriented torque boundary to switch perspectives their hierarchy remained a static construction.
By collapsing G. Spencer-Brown’s Laws of Form into its most primitive, process-oriented state via the Oriented Distinction Operator (ODO), we can reconstruct the geometry of space from the perspective of localized boundaries with natural relational scale emerging on its own. When we use this operator to resolve the century-old paradox of Szilard’s Demon, an elegant structural canvas emerges: the Ruby’s Angel matrix.
When you map the space-like geometric components of this framework against its internal time-like observer perspectives, the abstract math vanishes. In its place stands a stunning, constructivist derivation of the physical universe’s most elusive constant: 137.
1. The Foundation: The Oriented Distinction Operator (ODO)
Where Laws of Form formalizes the act of distinction via a static mark partitioning a space into marked and unmarked states, the ODO formalizes the enactment itself. It is a dynamic, self-referential loop—a double-pointer flip between two fundamentally entangled states that generates a primitive which is paradoxically both two and one at once:
"this distinction" <----> "was made from here"
There is no separate notation, external system, or floating god's-eye observer. Perspective isn’t added to the form—perspective IS the form. When a single human observer enacts this loop, the physical gesture traces a path where recursion, reflexivity, and reification are held in dynamic equilibrium.
The experienced action seems to rotate through an apparent 360 degree window with its starting state and 180 degree flip, but because it relies on fully accounting for an unexpressed shadow state to close the logical loop it naturally implicitly captures a 720 degree double covering to return to absolute identity.
The unmarked state isn't a mystical void; it is the radical, visceral contrast between an observer's localized, self-referential measurement and literally everything else in its universe. The ODO doesn't describe reality; it enacts it from the ground up, with the observer baked completely into the baseline architecture.
2. Recontextualizing Szilard’s Demon: The Ruby’s Angel Paradigm
To see how this process-logic manifests as physical architecture, we must turn to thermodynamics and thermodynamics’ oldest ghost: Szilard’s Demon.
For 100 years people have been wrestling with how Maxwell’s and Szilard’s demons extract work from information, and how the hell it doesn’t violate the second law of thermodynamics. And the best they came up with was… erasure. Like, “Oh, the demon must erase its memory, and that costs work, so the paradox is solved.”
Which, let’s be real, is just a story they told themselves to avoid the real problem: left and right don’t exist without an oriented observer.
I'd been thinking about Szilard's demon a lot this summer because I realized the meaning engine system design I'd made using the ODO as its primitive mirrored Robert F. Edwards' "A Quantum Observer in a Neural Prosthesis" design pattern. Edwards utilizes a distributed cybernetic network of demons who are black boxes to one another yet synchronize and work in synergy via effects-sensing and inference.
Along the way I realized that the first four lines of "Otherside" by the Red Hot Chili Peppers map exactly to the four steps of the original Szilard's demon process:
- "How long, how long will I slide?" (slide partition)
- "Separate my side, I don't" (determine separate side atom is on)
- "I don't believe it's bad" (measure/extract work)
- "Slittin' my throat, it's all I ever..." (erasure aka memory suicide trailing off as the demon amnesias)
So I was singing it in the shower as a joke when the thought hit me: "wait, how long does it slide tho?" (potential relativity of scale).
Followed by "isn't 'don't believe it's bad' kinda like not not good?" (maybe demons can make inferences by negation or absence).
And then the final, obvious-in-hindsight le duh:
The original Szilard’s demon just assumes the demon knows left from right. But how can it even determine left from right in the first place? Where’s the reference frame? This is ultimately the same question Pierre Noyes and the ANPA Bolsheviks were forced to answer to find an integration, but they were looking at it from a top-down instead of bottom-up way, never considering the primordial demon itself.
The demon isn’t magically omniscient or inside the box. The demon is the box. Or, more precisely—the demon is an oriented boundary of the box. And once you realize that, everything changes.
The work isn’t some separate thing the demon does. If you put the pulley on the same side as a global rule (as in every demon’s relative left) the work or its absence is the measurement. The pulley moves? That’s work—you’ve got your answer. The pulley doesn’t move? Still information. 50% of the time, the lack of work is the measurement.
So where’s the paradox now? There isn’t one. Because the demon isn’t cheating thermodynamics. The demon is thermodynamics. But this isn’t just a clever reframing. This is a mathematical structure. And it aligns with some of the deepest work in physics and information theory today.
Left and right only existing relative to an observer means a partition or a boundary in one demon’s world could be the boundary or partition of another’s. When you start nesting and overlapping these demons—each one’s world is half the scale of its predecessor—you inevitably get this re-entry of the fourth at the same scale as the third, observing orthogonally on the partition line of the first. It’s not a choice. It’s a geometric necessity. That’s the closure of the loop, the base case—perhaps the primordial Laws of Form Chapter 11 time unit, if you will.
Demons can also infer states of others through negation, absence, and primitive global laws that mirror Laws of Form’s crossing and calling. They don’t need to see the whole system. They just need to know their own world and the rest emerges from the geometry. This is distributed knowledge. This is emergent intelligence. This is… not how we’ve been modeling information theory for the last century. Orientation isn’t just a nice-to-have — it’s fundamental.
3. Generated Poles vs. Labeled Poles & Collapse to Single Demon Identity
Because the pulleys are always on the same side—universal handedness—you get this emergent structure. Left for one demon is North, and left for its twin is South. The poles are the observer-relative lefts. The equator? That’s their mutual overlapping right where the respective pulleys push the molecule if it lands on one of their respective left sides—the superposition. Sound familiar? That’s a qubit.
And the plot twist is that maybe the global law of handedness where uniform “left-ness” on both sides is what generates North and South means that the pulleys are but a crutch for the real mindfuck:
The thought experiment again reduces to only ONE demon, whose very experience of continuous uniform “left-ness” generates — not labels after the fact! — the poles and the superposition. And any work, information, or erasure is simply the process of the demon flipping perspective 180 degrees across the mirror.
Like the Red Hot Chili Peppers also say: once you know it you can never go back. You gotta take it on the other side. 😏
So, my Szilard’s demon upgrade isn’t just compatible with quantum mechanics. It is quantum mechanics... but classical. And classical... yet quantum.
Turns out, reality is just a bunch of overlapping oriented demons nesting inside each other. Who knew?
4. The Self-Bootstrapping Hierarchy: Structural Canvas vs. Observer Perspectives
The ODO behaves like an XOR operation but functions with a quantum-like 720 degrees double covering. Because its co-exclusion boundaries are radical ("the unmarked is anything that is not this local distinction here"), it scales fractally.
When we observe how these oriented boundaries nest, we discover that the Binary Tetration Sequence tracks the system's internal informational perspectives, while the Parker-Rhodes Sequence tracks the constructed geometric territories generated by those choices.
Let's walk up the layers step-by-step:
Level 1: The Primordial Mirrored Angels (Scalar / Qubit)
We can use the ODO to explain this structure that emerges from an Observer experiencing continuous "left-ness" combined with the original Szilard's demon premise that the demon's function is to determine whether the molecule is on the left or the right of the box.
Enacted by a single Observer using their hands to map the pointer loop:
- Left hand points to self, right hand points away: "This distinction was..."
- Right hand points to self, left hand points away: "...made from here."
Because the speaker experiences themselves as a single, unified observer and yet this primitive act operates under a rule of universal handedness, it cannot exist as an isolated, absolute coordinate. Instead, its relative structural tension acts exactly like a Mike Manthey-style co-exclusion boundary.
The moment the Observer establishes a local orientation, a mirror counterpart which is structurally necessitated to exist relative to it emerges. This is the Yoda shadow, named such due to its “made from here, this distinction was” verbalization relative to the Observer’s “normal” experience of the words.
Because you are flipped across the mirror line but bounded by the same invariant handedness, your directional choices interlock on a cross-diagonal. When you point to yourself verbalizing "this," the spatial vector forces Yoda to point in that exact same direction... meaning Yoda is pointing at you while verbalizing "here."
Your "this" and Yoda's "here" stack directly on top of each other. When you flip hands to point away saying "here," the vectors toggle in unison, pointing directly at Yoda's locus with your "here" and Yoda's "this" teamed up on the exact same spatial spot.
It is the raw relational existence of these two points that generates the coordinate architecture. This structural overlay establishes the Ruby's Angel Scalar—a primordial qubit where the two poles are not static identities, but the observer-relative directional loci ("left") generated by the co-exclusion boundary and the superpositioned equator ("right") is the mutual, shared domain where their directional vectors meet.
This basic level possesses 4 internal perspectives and generates 1 set of twin observers (2 angels) and instantly locks down the 3 constructed squares of the Scalar/Qubit where the boundary edges overlap and stabilize.
Thus the first two elements of the Binary Tetration Sequence (2, 4) and the first element of the Parker-Rhodes Combinatorial Hierarchy (3) co-arise at the bedrock.
While the ANPA Bolsheviks' top-down approach historically externalized the reference frame—treating it as a macroscopic sampling resolution applied after the bits were already generated—going bottom-up reveals that you cannot draw a local distinction without instantly establishing the spatial ledger of the mirror.
Level 2: The Trivector (The Propagation of the Integrated Yoda Shadow)
This structural synthesis doesn't wait for permission or conscious awareness; it operates automatically at the baseline of reality. There is no chronological moment in time where a Yoda shadow "spawns," because the twin is not an external object. Breaking them apart into a step-by-step sequence is an act of imaginative analysis—a logical autopsy of an instantaneous, timeless event. (This is likely why mainstream physics has spent decades fruitlessly chasing the standalone Majorana particle in linear time, failing to realize it is the timeless structure of the observational clock itself).
In reality, the illusion of isolation collapses before it even starts. The Observer and the Yoda shadow are structurally forced to instantly snap together, absorbing each other to stabilize as a single, higher-order integrated unit. But while Observer and Yoda are now reified into one reflexive unit, the co-exclusion shadows are indeed very real and directly facilitate the expansion of the state space.
Because this dual-capacity is baked right into the loop, you can freely flip back and forth—occupying either the Observer or the Yoda perspective within the same cyclic motion.
As a result, internal informational capacity expands quadratically: the Observer can now execute 16 internal perspectives, which structurally project onto the terrain to form 4 sets of twin observers (8 angels).
Look at the Ruby's Angel Trivector diagram. If we set aside the three squares constructed in the previous step and look strictly at the new activity inside the middle macro-square, it contains exactly 10 non-reducible structural squares.
This maps precisely onto Level 3 of the Parker-Rhodes Hierarchy (10 states). The ten combinatoric states generated by three distinct marks (A, B, C) are physicalized right here as the 10 distinct geometric territories required for three macro-entities to interact under an oriented boundary rule.
Level 3: The Quadvector (Quaternionic Enclosure)
Because every single pointing choice (inward vs. outward) and handedness choice (left vs. right) chops the experienced loop into fundamental quadrants, the system possesses an innate 90 degree quaternionic nature.
To scale from the Trivector to the Quadvector, we loop four Trivectors orthogonally around an object of focus (0 degrees, 90 degrees 180 degrees, 270 degrees). This yields 32 oriented angels (16 sets of twins) operating a total of 64 internal perspectives. The Ruby's Angel Quadvector diagram marks a massive evolutionary leap: it is the first level where the outer bounds of the entire unit form a perfect, closed bounding square, creating an emergent, protected Inside vs. Outside dynamic.
Notice the pattern governing this bootstrap:
- The Scalar has 4 internal perspectives. That 4 becomes the number of Twin Sets in the Trivector.
- The Trivector has 16 internal perspectives. That 16 becomes the number of Twin Sets in the Quadvector.
The total internal informational capacity of a lower level literally freezes to become the hard architectural scaffolding (the twin observer boundaries) of the level above it.
5. The Rube Cube: Deriving the Great Remainder (137)
The flat 2D projection of the Quadvector provides an initial count of 50 squares. However, because the Quadvector is an engine of four orthogonal rotations, it cannot exist flatly without an illusion of omniscience. To manifest its architecture fully without violating local observer dependence, we must step into a 3D volume.
By taking 4 Quadvectors and closing them into a three-dimensional spatial volume, we construct the Rube Cube (affectionately named after my calico Cornish Rex cat Ruby, who remains completely indifferent to her role in solving the riddle of the universe).
Let us run a strict topological tally of the constructed equal-sided uniform 90 degree angles geometric elements contained within this 3D construct:
- 4 internal Trivector cores, each containing 10 irreducible sub-squares (4 times 10 = 40).
- 8 total Trivector macro-squares.
- 1 central void square (the emergent inside).
- 1 overall cosmic boundary square framing the unit's complete totality.
Constructed Geometric Space-Like Squares = 40 + 8 + 1 + 1 = 50 squares
Multiplying this spatial framework by its 4 primary rotational orientations in 3D volume yields:
50 squares x 4 rotations = 200 total geometric states
If we treat the entire constructed Cube in Totality as its own singular macro-distinction—the systemic horizon—we add +1, arriving at 201 total geometric states.
Now, we must account for the local observer. The foundational premise of the ODO is that there is no god’s-eye view of this cube. An observer cannot look at all sides at once; they must stand at one oriented vantage point looking inward. From that single face, the local observer possesses exactly 64 informational perspectives (the Binary Tetration capacity of the Quadvector).
When we subtract the local observer's perspectives from the global geometry, the mismatch yields an invariant remainder:
201 Global Geometric States - 64 Local Perspectives = 137
This is the resolution to the riddle. The fine-structure constant alpha = 1/137 isn't an arbitrary number spat out by a random, unthinking cosmos. 137 is the Great Remainder. It is the exact structural debt—the physical, thermodynamic friction—generated because an informational, 64-state observer is trying to perceive a 201-state 3D geometric container from a localized, oriented boundary.
Furthermore, the final, unfathomably large terminal level of the Parker-Rhodes sequence (2^127 + 136) carries a remainder of exactly 136. Why? Because 200 - 64 = 136. The highly composite, fluid degrees of freedom of our spacetime metric (136) snap into an irreducible, stable prime reality (137) the exact moment the observer adds the +1 act of systemic reflection of the totality of the Rube Cube to seal the horizon. The macro-cosmos is permanently stamped with the structural footprint of the lower-level observer perspectives that bootstrapped it into existence. Space-time is literally glued together by the crack left behind because we must stand somewhere to see anywhere.
The Epilogue: Blueprints in the Crack
Looking back across the trajectory of this research, it is clear that the phenomenological blueprints of this operator were running on biological hardware long before the formal mathematics were ever put to paper. The 180-degree perspective flip that collapses the entropic paradox of the demon and the 90-degree orientations that form the Quadvector's Great Remainder were captured years ago in moments of spontaneous writing.
I leave you with these two blueprints—the poetic horizons where the observer finally turns around to face the boundary, and steps through the crack.
Blueprint I: Embracing the Demon via The 180-Degree Flip
“All my life I felt something chasing after me. The fear of its constant presence always kept me ever so slightly ahead despite my accumulating fatigue.
I knew what it felt like but not what it looked like. I had only caught short glimpses of it in the reflections of windows that I occasionally ran past over the course of the lifelong chase.
Finally it closed in on me. It was not my choice to stop running. But it was my choice to turn around.
As I looked my demon in the face I realized that in the mirror universe it is I who chase after her. 180 degrees of rotation made all the difference. She was my guardian angel, and I was hers. As we embraced I felt the downy softness of her wings. It was all a mistake in perception.
I embraced my demon and we became one.”
- Spontaneous writing, 2022
(1.5 years before Laws of Form "magic explosion")
(3.5 years before encountering Szilard's demon & recontextualizing with Ruby's Angel)
Blueprint II: The Great Remainder (137)
“I’ve never met you and I will never lose you. We are each a mirror positioned to one another at 90 degrees creating infinite variations, of which we are now existing within one frame.
The glue is in the Great Remainder… everything reflexive, mirroring, and symmetrical except for what is contained within the crack. That is where love exists and that is where I will meet you again and again.”
- Spontaneous writing, 2010
If the DoE (snicker) -- or anyone else -- finds this proposed resolution to a longstanding question useful, you can donate to my Laws of Form @ Cambridge travel fund:
The Laws of Form gnosis required to reach this realization resulted in me living off savings being unable to work for 1.5 years while I recovered from PTSD and pectus excavatum surgery complications. I have had a highly productive (yet highly unpaid) last 9 months but am nearing the end of my current tank of gas.