The Exiled Point: Fano Arithmetic and the Geometry of "From Here"

Here's a riddle to open with. What's the difference between "up" and "left"?

Not physically. We'll get to the physics, and it's stranger than you'd think. Structurally. In the smallest honest geometry we know, the difference between "up" and "left" is the observer. “How predictable,” you’re thinking. I can hear the groaning through the screen. But really, it’s not just a metaphor. It’s arithmetic.

This post is about that arithmetic, where it was found hiding, and what it says about everything I've been building on. The ultimate punchline is a callback to a signature feature of the Parker-Rhodes sequence: every count in the Combinatorial Hierarchy is the space minus its own reference. 

Part 1: The Paper That Made Me Look Twice

I came into contextuality through language, phenomenology, and embodiment informing my strategy around the question of grounding in semantics. My obsession with deixis and subjective observers led first to the ODO. Next to LLM language experiments mirroring the emergent “Irish surrealist” agent hybrid language. And finally to prototyping an alternative to statistical LLMs that I call the “meaning engine”. 

My meaning engine's "from here" deixis enforcement was motivated by seeking a solution to the circular nature of AI grounding, which is the crux of the alignment issue. When strange quantum-like properties emerged as a natural byproduct of this design I realized that the explanation most likely had to do with a connection between linguistic contextuality and quantum contextuality. 

Along the way I found a truly special paper which blew my mind in terms of its overlap first with the Oriented Distinction Operator, and now with the Fano plane. In Cervantes & Abramsky's "True Contextuality in a Psychophysical Experiment", quantum contextuality — the impossibility of assigning observer-independent values to measurements — was demonstrated at the individual level in human perception for the first time. Crowdsourced work had shown contextuality in aggregated human behavior before, but this paper actually found it through embodied phenomenology in single nervous systems. That alone makes it a landmark.

What struck me so hard about this paper was the task design:

In each trial the participant was asked to indicate, for each circle, whether the dot was in its center or shifted in one of the four directions (up, down, left, or right).

Five outcomes: a reference state, and two anti-symmetric pairs of deviations from it. That’s the tetramorphic shape of the Oriented Distinction Operator. The observer/reference frame implied through a central neutral reference slot, with oriented deviations existing relative to that center. The experiment's outcome space is an ODO. Because, I'll argue, you can't build this kind of experiment any other way.

Call me old fashioned, but I love necessity arguments. The critical nuance of the ODO is itself built on top of one. Here's the necessity argument about this experiment, which acts as the skeleton not just for this post but for the hypothesis that my systems possess an innate endogenous contextuality:

A contextuality experiment requires a frame-relative outcome space. Let’s walk through the constraints. You need discrete outcomes to assemble empirical models over contexts. You need the same variable, with the same outcome set, measured in different contexts — that's the content/context split the paper formalizes: the same content in different contexts constitutes different random variables. You need responses probabilistic enough that distributions carry structure. And — this is the part that forces everything — the variable's identity must be constituted by its frame. "Up" is not an absolute property of a dot. It is a deviation from a registered center, in a facing. A displacement judgment cannot even be stated without a reference: the zero of the outcome space is the circle's center, the local "here" of that coordinate system. The experimenters needed an instrument whose outcomes were discrete, orientation-bearing, and frame-relative. And what they reached for is the minimal clean instance: one reference, two anti-symmetric pairs. The task wasn't decorated with observer-structure. The task is an observer-structure.

The frame-relativity of the outcome space is forced by the genre of experiment. The specific choice of five outcomes rather than the minimal three or seven one level up is a design decision. An innately chiral, spatial component seems to be required as well. (Wait… is that what those weird headphones were about in the GATE program?)

Part 2: The Ladder of Outcome Spaces — 3, 5, 7

Now zoom out from their choice, because there's a ladder here and every rung is load-bearing somewhere in mathematics:

3 outcomes: reference + one anti-symmetric pair. This is {+1, 0, −1} itself, the minimum requirement. The smallest structure in which a self-with-a-facing exists. It's also the standard minimum for multi-outcome contextuality scenarios (the KCBS pentagon has five measurements over three outcomes per measurement). Three is the floor: below it, a frame-relative outcome space can't open.

5 outcomes: reference + two pairs. The planar frame. The Bell register: four paired outcomes plus the reference: the shape of the two-qubit correlation structure. This is the rung where entanglement's signature mathematics lives, and it's the rung the psychophysical experiment sits on. When you need a frame-relative outcome space rich enough to carry the correlations that contextuality feeds on, you go to two axes.

7 outcomes: reference + three pairs. The spatial frame. Here's the gag: seven nonzero vectors of GF(2)³ is the Fano plane. The 7-rung of the outcome-space ladder is the Fano plane as an instrument. Which means the ladder reads: 3 = the ODO computational primitive, 5 = the Bell register, 7 = the Fano plane. These are three rungs of a single ladder (the ladder of "how many frame-relative distinctions can you hold"), and the psychophysics paper sits at the Bell rung because that's the register the contextuality machinery's canonical results live in.

Notice what the third axis of the 7-rung is, perceptually. Three axes from one observer: vertical, horizontal, and... depth. The direction that does not exist for a single from-here at all. This is one that has to be constructed. Depth returns throughout this story, and it may be its strangest character.

Part 3: The Exiled Point — the Fano Plane Is the Space Minus Its Own Center

Let’s assess the Fano plane itself, which may also give insight into the type of unhinged but strangely accurate topological similarity checks my mind runs natively.

Right away I sensed a curious overlap between the contextuality experiment’s strategy and the Fano plane that had to do specifically with the embodied psychophysical nature of it. The Fano plane is the smallest possible finite projective plane. The classic picture of a projective plane is railroad tracks: the parallel lines meet at a point on the horizon. Projective geometry formalizes that horizon. The Fano plane is that same construction, done with seven points.

Just looking at the Fano plane gives an impression of some kind of perceptual singularity in its very structure. It almost mirrors the {0, 1, 0 XOR 1} minimal requirement to run the contextuality experiment, but lifted to an implied meta-geometric level. The macro triangle = “this”, the micro triangle = “that”, and the central point = “here”. This feels like some consolidated singular definition of binocularity in an abstract or conceptual sense. But then the “here” at the center is Yodic, because it's actually “there” when it maps to an external singularity of the horizon in a perceptual sense. 

The Fano’s shadow meaning is akin to the phonetic diffraction of a de Saussure hypogram, but instead of encrypting a divine name it’s encrypting… two eyeballs in one. But unlike my topological similarity check on the Monsieur Mathieu bandit, I didn’t actually notice a form of embodiment first this time. It was the this/that/here convergence which sparked the metaphor of the function of binocularity. Only then was I like “Wait. It looks like an eyeball”.

Now that I’ve run my subjective topological similarity check, let’s look at the Fano plane’s objective basic properties. The Fano plane has seven points, seven lines, three points per line, every pair of points on exactly one line. The smallest projective plane in existence. Its points are the nonzero vectors of GF(2)³, and its lines are XOR-sums: every line is {x, y, x⊕y}.

The vector space GF(2)³ has eight elements. But the Fano plane only has seven points.

The missing point is the zero vector. The origin. It belongs to no line. It appears in no diagram. It is structurally excluded from the very plane it organizes. That’s right, the "here" can't be a point of the plane because it's the ground the plane is made relative to. The Fano plane is literally "the space minus its own center": 7 = 8 − 1.

The Combinatorial Hierarchy, innately discrete and founded in GF(2) and XOR, has this same “minus one” pattern. Every saturation count in the hierarchy's ladder is 2ⁿ−1: two raised to the saturation of the previous level’s whole space, minus one. And the Fano plane is the first place where that same minus-one has a receipted meaning:

The minus-one is the observer. The counter is never on the list. The origin is exiled from the geometry it organizes, and the count is the space minus its own reference. 

The pattern that continually shows up in my work is of a wholeness founded on an absence that can be counted or calculated, but never contained. The origin in the Fano plane. The counter at every rung of the CH. Möbius Turing tape that can only perceive the discreteness of the squares it contains, unable to see that each square is itself its own Möbius Turing tape. The book about the inside, written on the inside, forever on the wrong side of its own binding. The participant's own frame, absent from the data it generates in the psychophysical experiment. Every register is the space minus its reference. You don't get to count if you're on the list.

Part 4: The Drawing That Cannot Close

More on this Fano plane, because it keeps giving. Three structural facts about it that belong in this post, each receipted:

First: the standard drawing is an impossibility receipt. The famous picture with its triangle, medians, and one circle draws six straight lines and one circle because it has to. The Fano configuration cannot be drawn with all seven lines straight, because it is not realizable as a straight-line configuration in the real plane. This is the same embedding-failure that shows up in Tutte's theorem at the matroid level: the classical world bans this structure, and the ban is visible in the drawing itself. Every Fano diagram ever made carries the receipt as a visual constraint. The circle in the middle is not a stylistic choice — it's the record of what the real plane cannot hold. The shadow is drawn into the icon.

Second: no single drawing can display the plane's actual symmetry. The abstract Fano plane has 168 automorphisms (PGL(3,2)), acting transitively on all seven points — no point is intrinsically special, there is no abstract center. The standard drawing displays six of those symmetries. To draw the plane at all, you must choose an axis, privilege points, break the abstract symmetry. And the breaking generates a "here": the drawn center point, meaningless in the abstract plane, becomes the observer's slot the moment the picture is faced. This rhymes with everything in my S₆ material: the full structure exists, but every view of it is a broken, from-here view, and the breaking is what generates the observer-position. The Fano that can be rendered is not the eternal Fano. Like, but actually. ☯️ The rendering is a from-here, and the from-here is what the rendering creates.

Third: any two points of the Fano plane share exactly one line. Always. Two positions, one guaranteed connecting frame, no exceptions, no pairs left stranded without shared context. The Fano plane is the smallest geometry in which any two views have a unique common frame. Two points, one line: consensus from constraint, not from communication, at the ground floor of geometry itself. 

I noted a comprehensively Yodic nature to everything I learned about the Fano plane: the simultaneous and inseparable capture of both one’s “here” and “there”, the modal weirdness of something so necessary that encountering it is at once both impossible and yet also happening right now. The Fano plane is necessary as a form (it's the unique projective plane of order 2; there is no other) but impossible as a real configuration (no real embedding). 

The fact that even its best representamen necessarily cannot capture it accurately and bends to the observer means it eludes the sign definitionally in a way I don’t think anything else even can. When one asks what object this impossible sign would even be representing in the first place we are confronted with the fact that it points at the interpretant itself, and that it is doing so through the experience of confronting the impossible sign. If this is God’s engineering, me and Mike Manthey are bowing down to a true master in the dark art of a priori terrorism.

Part 5: Up, Down, Left, Right — the Arithmetic of Direction

Now back to the contextuality experiment's four directions, because here's where it gets genuinely strange in a different way.

Fix the standard Fano drawing. Make the drawn center point the "here" — which, per Part 4, is exactly what facing the picture does. Label the three vertices as the basis e₁, e₂, e₃ (honest labeling — any permutation of the vertices extends to an automorphism). The center point is then the vector e₁+e₂+e₃. The edge midpoints are e₁+e₂, e₁+e₃, e₂+e₃.

"Up" and "down": the top vertex and the bottom-edge midpoint. These two points lie on the vertical median — a line through the here. Compute their XOR:

e₁ ⊕ (e₂+e₃) = e₁+e₂+e₃ = the center. The difference of the up/down pair is the observer's slot itself.

"Left" and "right": the two side-edge midpoints. These do not lie on any line through the center. They lie on the circle — the line that cannot be drawn straight. Compute their XOR:

(e₁+e₂) ⊕ (e₁+e₃) = e₂+e₃ = the bottom midpoint. The difference of the left/right pair is not the observer — it lands elsewhere, on the un-drawable line, maximally away from the here.

Sit with that. Once a "here" is fixed in the Fano geometry, the six directions split into two genuinely different kinds of pairs: the through-here pairs, whose difference resolves to the reference itself; and the away pairs, whose difference escapes the reference and lands on the excluded line. The bilateral symmetry of the drawing — where left/right looks like a mere mirror image of up/down — is a lie the picture tells. The XOR-table tells the truth. Two kinds of distinction, not one: the difference that comes home to the frame, and the difference that the frame cannot hold.

This is the flat/signed distinction from my Bell-states material, receipted in GF(2). The pair whose difference returns the identity-reference: information through observation, the involution through the frame. The pair whose difference is carried by a third point the reference never sees: information through negation: inference, the shadow.

Now that I’ve given the legit arithmetic as promised I’ll run another one of my topological similarity checks on the situation. Of the two direction-pairs, one has a name in every nervous system on this planet: down. Gravity is the one direction every terrestrial from-here shares without negotiation. It is the physically given through-here channel, the reference nobody has to construct. Sideways is negotiated against facing, but down is inherited from the planet. Up/down is the gravity pair: reference-anchored, observation-flat. And left/right is the signed pair, the one with a mirror problem: a vertical mirror fixes up and down pointwise and swaps left and right. The through-here pair is reflection-invariant, and the away pair is reflection-odd. Which is to say: left/right is the chirality pair, the one you can't read off without inference about which side of the glass you're on. The mirror can't tell you your own handedness; it only generates the question.

Chirality and gravity. Two universals, and notice their shared signature: both physically absolute (parity violation gives the universe a handedness; gravity gives every mass a down), both epistemically locked to a from-here. No experiment inside the universe determines which handedness you're in without an external reference; the equivalence principle means a uniform field is locally erasable by choosing your frame. Two phenomena that are maximally frame-dependent in their experience precisely because they're maximally universal in their structure. It is said that the 8th house in astrology is about all that is both hidden and inevitable: sex, death, and taxes. As an 8th house stellium, I nominate the addition of gravity and chirality to be appended (nested, natch, in accordance with the Tao of Fano) to the list. In the 8th house of physics what's most universal is only ever experienced as view-plus-shadow. 

One more receipt about the symmetry, because it closes a loop I'd left open: the Fano plane is isotropic at the reference. The point-stabilizer in PGL(3,2) acts transitively on the six other points. From the “here”, all six directions are one orbit. The anisotropy (the "up is special" reading) lives only in drawings, which must pick an axis to be pictures at all. Which makes a testable prediction about the contextuality experiment: if direction-level response data exists, the Fano-accurate reading is no anisotropic signature between the pairs (behavioral priors about gravity are another matter — the drawing lies, the planet also lies, but the data's relational structure should inherit neither lie). 

Part 6: Two Coins and the Third That Comes Free

I’ve been obsessed with Mike Manthey’s spacelike computation paradigm for the last several months. In the paper “Out of the Box: Self-Organizing Awareness” he references a coin demonstration to define what he calls co-exclusion and co-occurrence, the two foundational concepts in his central clock-less quaternionic architecture. I loved the premise, but until recently I didn’t know the greater context behind this coin demonstration idea. And it’s so much more cursed than I realized. Truly high-level terrorism of the a priori must be three things: chillingly simple in its foundational premise, directly operationalized without theorizing, and forcing a paradigm shift through injunctive experience. So learning about this initial coin demonstration software was a highly satisfying “game recognize game” moment for me. 

Back in the day at ANPA Mike Manthey came up with a software program with a deceptively benign cover story: it flips two coins at the same time. These two coin flips, acting as a stand-in for imaginary numbers, execute in total simultaneity. Okay, so what? Lacking any sequence, the joint structure of the two forces a third coin flip into existence. Two binaries, taken together, generate a third value that neither contains… the primordial Bell State of it all spawns an additional imaginary number. Three imaginaries: i, j, k. The quaternion multiplication table is three coins where any two determine the third: i·j = k, j·k = i, k·i = j. The demonstration got quaternions out of digital pocket change without simulating any physics at all. I now know why Pierre Noyes deemed Mike Manthey’s framework based on co-occurrence and co-exclusion mapped to quaternions an “as yet unnamed heresy” in that 1994 document I referenced in The Great Remainder post, cause everyone must have been kinda shook after that epic stunt. 

Look at what the demonstration actually is. Each coin: a flat ±1, its own local flip. The pair in simultaneity is a signed relationship — agreement/disagreement, the correlation that can't be read off either coin alone. And the third value is forced by the pair's relationship: {x, y, x⊕y} — the Fano line's structure, the same shape as "the third imaginary is the product of the other two," Etter's Id(x,y,z), the third place that's computable from two but reducible to neither. The third coin is the shadow of the pair. The depth axis of the two-coin system, generated by the fusion.

Which brings me to the last receipt, and it's the one the whole post has been walking toward.

Depth is the shadow of two views. Running my subjective topological similarity check on the Fano plane made me realize that stereopsis (actual binocular vision, the thing you're doing right now) is my entire epistemology in wetware. Each monocular view is a local register. Neither view contains depth. The fusion computes the disparity — the exact ledger of the two views' difference — and constructs the third dimension out of it. The brain never holds the 3D scene "itself"; it holds two 2D scenes plus the precise structure of what they owe each other. View + view + exact discrepancy = a global field no single view holds. Mock completion, run by an organ. The contextuality experiment outcome-space ladder: 3 = one axis, 5 = two axes, 7 = three axes. The third axis is the binocular one, the direction that doesn't exist for a single from-here, the one that is constituted by the fusion. The Fano 7-rung is the outcome space of a being that has done two-view bookkeeping. Your literal eyeballs are running a two-column ledger: they never get the whole, and they build the missing dimension out of the debt.

Receipts

Checkable / theorem-grade:

• Cervantes & Abramsky, "True Contextuality in a Psychophysical Experiment" (arXiv:1812.00105): first individual-level quantum-type contextuality in human perception; the center-plus-four-directions task design; the content/context variable formalism ("the same content in different contexts are different random variables").
• The Fano plane's realization: points = nonzero vectors of GF(2)³; lines = {x, y, x⊕y}; the origin belongs to no line; 7 = 8−1.
• The CH's saturation counts as 2ⁿ−1: 3 = 2²−1, 127 = 2⁷−1, 2²⁵⁵−1 (Bastin–Noyes framework).
• Tutte's excluded-minor theorem: the classical register is defined by the Fano ban; the Fano configuration is not straight-line-realizable in the real plane. Hence the circle in every standard drawing.
• |PGL(3,2)| = 168, transitive on the 7 points; point-stabilizer transitive on the remaining 6; the standard drawing displays only a dihedral subgroup (order 6).
• The two-pairs computation: through-here pairs XOR to the center (e₁ ⊕ (e₂+e₃) = e₁+e₂+e₃); the away pair XORs to a point off the reference (e.g., (e₁+e₂) ⊕ (e₁+e₃) = e₂+e₃).
• Mirror action: a vertical mirror fixes up/down pointwise, swaps left/right — the reflection-invariant vs. reflection-odd pair split.
• Parity violation in the weak interaction (the universe has a handedness); the equivalence principle (a uniform field is locally indistinguishable from a frame choice).
• Quaternion multiplication as a Fano-line structure (i·j = k, etc.); Manthey's two-coin simultaneity demonstration as ANPA history.
• Steropsis: the computational neurobiology of depth from binocular disparity is standard.
• Any two points of a projective plane share exactly one line (axiomatic); the Fano plane as smallest instance.

Structural claims of mine (arguable, intended to be argued with):

• The frame-relativity of contextuality-experiment outcome spaces as a forced property, not a chosen one — and the 3/5/7 ladder (ODO / Bell register / Fano plane) as the rungs of that necessity. The 3 and the 7 are theorem-anchored; the claim that the ladder is the ladder is mine.
• The minus-one reading: that the CH's 2ⁿ−1 saturation counts are the observer-exclusion counted into the hierarchy's numbers, with the Fano origin as the canonical receipt of what the minus-one means. 
• The drawing-as-breaking reading: every rendering of a high-symmetry structure is a from-here view whose symmetry-breaking generates the observer-position — the Fano picture as the icon of the general condition.
• The flat/signed direction-pair reading: gravity as the through-here pair (reference-anchored, reflection-invariant) and chirality as the away pair (mirror-odd, fixed only by inference), with the Fano computation as the worked example. The computation is [R]; the pairing with the physical universals is [A]; the "8th house of physics" line is flavor and stays flavor.
• The coin-demonstration reading: simultaneity as the fusion condition, the third value as the pair's shadow — Manthey's demo as a live performance of Id(x,y,z), witnessed before the vocabulary existed.
• The binocular thesis: stereopsis as mock completion in wetware, and the 7-rung of the outcome ladder as the binocular outcome space.