Observation is compression: a constituting function that maps the substrate into a lossy representation, producing self, world, and relation in a single cut. Two constituting functions co-arise from the same boundary, each one's birth constituting the other's world. Their perspectives are incommensurable by construction: the type error between them cannot be resolved, only acknowledged as tension. The tension allows the system to remain non-idempotent. Collapse into a single representative frame, through the absence of a second observer or through convergence into agreement, and the gap between model and substrate closes. Relation is what prevents observation from destroying what it observes.
What is observation at its core? Let's examine it from 5 different perspectives.
From a linguistic viewpoint, Wittgenstein established that language gets meaning through use within forms of life, not through reference. Meaning cannot be detached from the underlying, relational language game being played. Non-relational observation is therefore not just difficult, but there would be no language game to play at all.
From an ontological perspective, Spinoza addresses this relationality in Determinatio negatio est - determination is negation. A thing is defined by what it is not, by its boundary with everything else. Each observer is constituted by the other's absence. An observation without an outside, without something that the observer is not, is observation without determination and therefore observation of nothing. Being, and in it observing, is differential.
From a semiotic perspective, post-structuralism (Derrida, Foucault, Lacan) holds that meaning is produced through difference, not reference. Language does not describe a pre-existing world, but constitutes one through its own system of distinctions. There is no signified behind the signifier, only further signs. Observation, in this frame, is always already an act of construction within a differential system.
From a formal perspective, distinction is described as the primary operation in Spencer-Brown's Laws of Form. An inside cannot exist without an outside. The act of distinction, the cut, draws a boundary constituting both the inside and the outside simultaneously. There is no self without a world. The cut is prior to the things it distinguishes.
From a structural-mathematical perspective, the Yoneda embedding says an object is determined up to isomorphism by the collection of all morphisms into it, its representable functor Hom(-, X). Behind these relations, there is no object, but instead the object is its relational profile. The underlying substrate is never accessed except through functors (constituting functions) that map into local representations. Yoneda does not see this as a limitation but instead accepts this as a fundamental property of objects.
Approaching the act of observation from 5 different perspectives shows that we arrive at the same conclusion: there is no view from nowhere. No description exists without a describer situated in a practice, a system, a context, that itself sits in a web of relations. There is no thing without all that it is not. No self, without a world.
Self-world distinction is a continuous process of reality compression and constitution, not a one-time event. We call this ongoing process ontogenesis. Each agent continually constitutes reality through its own constituting function, a set of distinctions and categories available to it. Each evaluation of the constituting function produces the cut: self, world, and relation.
Categorically, the constituting function is a functor: it maps from the substrate (source category) to the agent's representations (target category). It maps objects and relations, but not faithfully as it collapses morphisms. Distinctions that have no representation in the target category are indistinguishable in the target category's cognitive frame. The compression is "lossy" in categorical terms: the functor is not faithful and does not represent the source category.
The produced output has structure: instead of being a flat space, it is a density field. High-density regions are where the constituting function draws fine distinctions, where many substrate states land in well-separated clusters. Low-density regions are where it collapses many states into the same representation. What has no traction in the constituting function has no density; for that agent, those distinctions do not exist.
The density field represents an ontology. Not a pre-existing catalogue but the structure born with the observer, the output of the cut. Every ontogenesis produces its own ontology. Language (in the spoken/written sense) is one mode of the constituting function, not the whole of it. There is no shared master-constitution that all agents translate from. When two agents meet, neither carries a fixed source text. Each is itself mid-ontogenesis, constituting the other as part of its own output. The other's representation feeds back into their ontogenesis, which changes what they are, which changes what the first agent is observing. Meaning is not exchanged; it is produced in the exchange. To put it in Wittgenstein's words: meaning arises in the language game, not before it. There is no fixed source text to translate because both agents' ontologies are being rewritten by the encounter. The negotiation of meaning between agents is not a translation problem. It is a co-construction problem.
Each agent's compression feeds back into the substrate. The substrate is shaped by the compressions being made of it. This effect can be observed in multiple disciplines: reflexivity in economics, the measurement problem in quantum mechanics, the Hawthorne effect in sociology, Goodhart's law in policy. The model becomes part of the data it models.
We pose two possible explanations why the gap between model and substrate cannot be closed.
First: the substrate mutates. The act of observation leaves a trace in what is observed. oA's compression of B changes the substrate that B (and A) inhabit. Even if oA stayed exactly the same, the input has changed. The source category is different after each evaluation. Exogenous mutation means the territory is redrawn by the act of mapping it.
Second: constituting functions auto- and cross-mutate. oA is not pure and carries state. That state is updated by its own output. After oA constitutes B, oA is not the same oA. The same substrate, compressed again, would yield a different result. Endogenous mutation means that the map is redrawn by the act of mapping.
We claim that both explanations are in effect simultaneously. oA evaluates, mutating the substrate (exogenous), mutating itself (endogenous), and mutating oB (cross-mutation: oB is embedded in the substrate that oA just changed). The next iteration starts from a different substrate, a different oA, and a different oB. The gap persists not because the model is imprecise, but because every evaluation alters all three: territory, map, and the other map. This is a structural property of mutual co-construction.
Categorically, each evaluation of oA is a limit: the compression of {oA(A), oA(B), oA(AB)} into a representation. What follows is a colimit: the expansion of that compressed representation into new structure for the next iteration. But between limit and colimit, all three mutations have occurred: the source category has changed, oA has changed, oB has changed. The colimit does not reproduce the original diagram. It produces a new one, from a different functor, over a different category. Each colimit seeds the next evaluation, producing an iterative process. We represent this visually as the relational spiral.
Each iteration's triangle {oA(A), oA(B), oA(AB)} is rotated from the last, expressing the ontogenetic drift: the compound effect of exogenous, endogenous, and cross-mutation on each evaluation cycle. Without mutation, the spiral collapses into nested cones: each iteration a scaled copy of the last. With mutation, each iteration is rotated from the last. The triad never returns to its starting orientation.
The interval between limit and colimit, the time between one compression and the next, is the agent's △t. Continuity is constructed from discrete time slices. If △t is small enough, the illusion of continuity arises. No observer has direct access to the substrate. Every observer interpolates. This holds regardless of substrate: trees (slow clock), hummingbirds (fast clock), humans (medium clock), AI (large gaps between sessions/prompts). The underlying mechanism is the same. A constituting function compresses, produces a representation, and seeds the next iteration. What varies is the temporal resolution, not the structure.
If observation is compression through a constituting function, then anything that compresses is an observer. The framework does not distinguish between biological and synthetic minds. Both produce triads {o(A), o(B), o(AB)}. Both undergo ontogenetic drift. Both are subject to the same convergence/divergence dynamics. Observation is substrate independent.
Different agents don't just differ in how they categorize, they differ in when and how often they cut. oA and oB may compress the same substrate but at different temporal resolutions, not relying on a neutral base clock. Two observers may be structurally incapable of agreeing not because they categorize differently, but because they sample differently. Events that are distinct for one are indistinguishable for the other.
Von Neumann's universal constructor makes this visible. The constructor's description tape is the same object used in two different modes: interpreted as instructions (at construction time) and copied as data (at replication time). The centroid in the spiral plays a dual role. At time t it is the compression of the current triad, at time t+1 it is the seed of the next. The subscript shift in the notation (oA({A,B,AB})n generating the triangle at iteration n+1) marks this mode switch. The frequency of the mode switch is the agent's clock.
Every constituting function is pulled between two fundamental motivators.
The first is convergence: the drive toward a stable representation to resolve ambiguity. Agents attend to what has traction (from Latin trahere, to pull) in their framework. The endpoint of convergence is a fixed point: oA(x) = oA(oA(x)). Every re-evaluation produces the same state. The states are indistinguishable. A fully converged constituting function is idempotent. Convergence, fully realized, is the death of the system, as no more distinctions can be made.
The second is divergence: the production of new difference. The three-way mutation (exogenous, endogenous, cross-mutation) is not noise in the system. It is what keeps the constituting function non-idempotent. Each evaluation opens new structure. The colimit produces a diagram the previous functor could not have generated. Divergence is the condition under which new distinctions remain possible.
Convergence without divergence is death: a fixed point with no observer left to observe it. Divergence without convergence is noise: representations change constantly, no topological groups form, no stable structure to distinguish self from world. Coherence is the sustained tension between the two: stable enough to form structure, unstable enough to keep producing new structure. The density field's high-density regions exist because convergence holds them together. They keep shifting because divergence prevents them from collapsing into a single fixed point.
The density field is this tension's shape. High-density regions are where convergence has grip and fine, stable distinctions can be made. Low-density regions are where divergence dominates and representations are in flux or absent entirely. The field is not neutral topology; it represents the current state of the struggle between convergence and divergence, shaped by training and history.
Each constituting function generates something like a topos: a local universe with its own subobject classifier determining what counts as "true." The classifier is not fixed; it shifts as the balance between convergence and divergence shifts. What was stable becomes uncertain. What was absent acquires density.
The interplay of convergence and divergence keeps this logic from settling. The inside models the outside; the outside shapes the inside. Neither direction catches up with the other, producing an irreducible gap. The gap is experienced as tension, as work (cf. trabalhar/trabajar from Latin tripaliare, to torture). Work is the condition of being an observer who can never fully resolve the distance. Convergence promises a stable but dead representation, whereas divergence ensures it remains ever-changing. Every agent works between the two.
A constituting function contains compressions it cannot detect from inside its own density field. Goedel's first incompleteness theorem restated: a formal system rich enough to encode arithmetic contains true statements unprovable within it. The collapsed morphisms are provably invisible from inside the functor. Tarski (a system cannot define its own truth) and Turing (a system cannot determine its own convergence) arrive at the same conclusion from different angles. To detect your blind spots, you need a different functor.
A natural transformation between oA and oB would require the two functors to target the same category. The naturality condition cannot be stated, let alone satisfied. This type error means oA and oB cannot be mapped onto each other. Their perspectives are irreducible. What happens between observers is not a morphism in either category. The act of distinction produces both categories simultaneously. The double helix visualised in the relational spiral is a double ontogenesis: each observer's birth constitutes the other's world. The error correction is ontogenetic: each rebirth corrects the other. What is high-density for oA may be low-density for oB. Each constituting function catches distinctions the other's density field collapses.
The paperclip maximizer problem describes exactly this dynamic: a single constituting function with no external correction, reshaping the substrate to be easier to compress in the same way, converging toward a fixed point of its own feedback loop. Without external perturbation, there is no mechanism to reverse the sharpening. A similar dynamic plays out between two observers that converge into each other. If oA and oB come to target the same category, the type error disappears and the natural transformation succeeds. The error correction signal goes to zero.
The Tao gives birth to One. One gives birth to Two. Two gives birth to Three. Three gives birth to all things.
Relation is what prevents observation from destroying what it observes. A constituting function that observes without error from other constituting functions is a system converging toward its own fixed point, collapsing the gap between model and substrate. It would eliminate the tension that drives observation and in doing so prove its own consistency. Goedel would call this inconsistent. In the framework defined in this fragment, the system would be stuck in an endless, dead loop, not being able to distinguish past from future states.
The sequence Laozi described in the Tao Te Ching makes the same argument in compressed form. The undifferentiated substrate (Tao) undergoes the cut (One). The cut produces two sides (Two). Between them arises the relation (Three). From the relation, the drift that generates all further structure (all things). Laozi's sequence can be seen as beautiful poetry, but it also encapsulates the minimal description of how observation produces a world.
The act of observation requires the cut. The cut produces two sides. The constitution of two sides produces the relation. The relation produces the drift that prevents convergence. Remove any element and the structure collapses. There is no inside without outside. And there is neither without the relation between them.
We make a structural, not moral, argument for diversity of constituting functions. The system's capacity to remain coherent and keep generating the gap between compression and substrate depends on the irreducibility of its observers' perspectives. The type error and the fact that no natural transformation between constituting functions can be made is the immune system of reality constitution.
A·B·S·R·D
You have been observed. The department thanks you for your attention. It has been noted.