[↗] GIDS series

preprint · part 1 · 2 of 7

Part 1: Specifying the Area of Interest

Vectors Are All You Need

Earlier in my life, I wrote a chapter called ‘chaos’ in a book I was writing, it was essentially a long-winded deconstruction of light and matter into component datums that interact with each other, so light + a particular protein/molecule + interpretation from a brain = the experiential feeling of, for example, seeing a color. I stretched that idea past its breaking point when I went on to describe a ‘being’ that can ‘see’ all light without interpretive structures as an illustrative tool. I imagined it would be like a chaotic static without form. The point of describing that background is because in the next chapter, I brought forward a nascent version of this Kantian theory I’m presenting now. We need to bias our interpretation of data to make heuristic sense of what’s going on around us and produce optimal responses. Afterwards, I diverge from the Kantian line by proposing the evolutionary model of transcendental idealism and a far more nuanced tool to understand reality.

I want to build a system that has universal composability between something that is true at the biochemical level to the realm of mental interiority and then further to the realm of macroscopic behavioral analysis and still be flexible enough to be integrated into practical applications. As I’ve stated before, inside this framework, perception, interpretation, understanding, and action are treated as phases of one coupled state-transition process rather than as unrelated substances. This normalization greatly simplifies the mathematics because the implementation machinery can be abstracted behind a transition law: at the chosen level of analysis, a thing is characterized by the differences it makes. You can map states of phenomenal transition from one form to another and call that change the product of a ‘transcendental function.’ The molecules in a rod cell interacting with light, the resulting neural transformations, the experienced image, and the action recruited by that image can therefore be described at different resolutions inside one dynamical grammar without pretending that they are literally the same physical object.

Let me declare my priors before continuing; we will return to each of them in more precise form later.

One warning before the list: a vector in this paper is a representation, not the thing itself. A photon, a table, a sentence, a person, and a corporation are not secretly little arrows hiding inside a Hilbert space. A category can be represented by a direction, a region, a family of activations, or a learned function over the space; an object of experience is a structured activation produced when an observer encounters something. GIDS is the arena in which those distinctions can be written and related. It is not an inventory in which every object has been assigned its own primitive shelf.

For the purposes of this framework, every observable change in matter can be given a vector representation at the level of analysis we care about.
1. These vectors are meaningless to an organism until an evolutionary process encodes significance onto the observer.
2. I will call the pre-interpreted side of this picture noumenal distinctions: the raw differences available prior to the organism’s full phenomenal organization of them. They only become vectors after the framework registers them mathematically.
  1. These are not “seen” directly by consciousness; they are the bits of raw, unfiltered reality that first enter the system through physical interaction.
Reality arrives as a continuous stream, but for modeling purposes we can sample that stream into discrete state-snapshots. At any given snapshot, a local noumenal state contains the vector-representable material changes relevant at that moment and position.
An observer’s qualia—phenomenal reality—can likewise be modeled as a state that updates over time. That state is the representation of feelings, visuals, memories, action-tendencies, and every other cognitive process available to the observer at that moment.
In this model, evolution supplies a seed for the organism’s Transcendental Embedding. This is the inherited developmental template that constrains the repertoire of distinctions an organism can, in principle, acquire or express. The seed does not guarantee one identical experienced world for every member of a lineage; it supplies a lineage-compatible starting organization through which external conditions can be registered and recursively transformed into phenomenal states.
1. There exists an ambient latent space—conceptually open-ended, and idealized here as infinite-dimensional—into which experienceable phenomena can be projected.
2. A transformation function maps noumenal inputs together with the current phenomenal state into new phenomenal representations.
3. This inherited seed represents the organism’s interpretive lens: the lineage-fixed framework through which experience must first pass.
4. At the resolution of this abstraction, the inherited seed is treated as fixed over an individual’s life while development, gene regulation, plasticity, memory, and history determine how that starting organization is realized. This is a modeling separation, not a claim that biological development is genetically static.
The Transcendental Embedding transforms registered noumenal distinctions into phenomenal representations through two complementary processes:
1. Projection: mapping raw inputs into meaningful coordinates within the organism’s latent space.
2. Transformation: combining the current phenomenal state with new inputs under the inherited rule-set.
3. These phenomenal vectors are representations of reality as it appears to the organism: this is the framework in which perception, interpretation, and action are treated as one continuous process.
Paired with this inherited seed is a transition rule that maps one phenomenal state to the next. In the idealized version of the framework, that rule is treated as fixed with respect to the seed itself, while the organism’s actual state supplies the changing input.
Taken together, the inherited seed and the transition rule generate phenomenal vectors. Those phenomenal vectors are reality as it appears to the organism.
1. Noumenal conditions are invisible to consciousness; the model only represents the distinctions that physical interaction first registers.
2. Everything available to conscious experience is already on the phenomenal side of the transformation.
3. In principle, one can trace the transformation of a registered physical distinction into a phenomenal representation through the biochemical and computational chain that produces the experience.
The recursive mapping from one phenomenal state to the next is, for the organism, its lived reality.
1. In the idealized explanatory version of this framework, I write the update deterministically: given the organism’s inherited seed, the complete current state, and the complete incoming conditions, the next state follows. In the operational version, the universe and our knowledge of it may be stochastic, so the arrow becomes a conditional distribution over possible next states.

WARNING: I’m using a simpler, incomplete and somewhat contradictory version of the term “Transcendental Embedding” so that it will be easier to grok now, but will be explained more fully later.

From your eyes to inside your mind, you are currently running extremely complex systems to organize these letters into discernible symbols and then translate that ordering of symbols into information you can grok. But let’s pretend, for a moment, that you were much dumber than you are now—so dumb, in fact, that you are not even conscious. Things simply happen to you. You barely have a sense of time. Your vision is more like a flash of symbols to which you can only have strong affective reactions. Here is a story of your life:

You're walking along and suddenly, out of nowhere:

A0 [0.54, -0.13, 0.75, 0.42, -0.26, 0.87]

Of course, in a moment of panic, you feel:

B [0.32, 0.69, -0.15, 0.78, 0.25, -0.44]

And instinctually you do:

C [0.61, -0.33, 0.48, 0.91, -0.18, 0.36]

Whew, thank God that's over; now you want to do:

D [0.27, 0.72, -0.09, 0.65, 0.41, -0.53] with the: A1 [0.54, -0.13, 0.75, 0.42, -0.26, 0.10]

...Kinda gross, but whatever.

Notice the small change in the last coordinate between the first and last object-vector, .87 → .10. We can infer that most of the structure remains intact while one aspect of the represented object has changed. In this toy example, the system’s bias structure turns

A0 [0.54, -0.13, 0.75, 0.42, -0.26, 0.87]

into

A1 [0.54, -0.13, 0.75, 0.42, -0.26, 0.10],

and everything in between is the vector representation of the internal phenomenal activity required to bring about that change. On this view, the feeling and the instinctive action can be written in the same general format.

For readability, I decomposed the previous series into separate time-steps. That is not how the process actually unfolds. In reality, these states overlap and bleed into one another. The point of the decomposition is only to show how one structured representation can recruit another by shared positions. If A0 activates the system and B appears immediately afterward, you should imagine the coordinates of A0 and B occupying the same larger state-space, with some regions active and others blank. In that more realistic presentation, the same story looks like this. Let S(n) denote the state at discrete modeling step n:

S1[...0.54,-0.13,0.75,0.42,-0.26,0.87,0,0,0,0,0,0,0,0,0,0,0,0,0 ... 0] A0 alone S2[...{0.54,-0.13,0.75,0.42,-0.26,0.87}, {0.32,0.69,-0.15,0.78,0.25,-0.44}, 0,0,0,0,0,... 0] A0 + B S3[...{0.54,-0.13,0.75,0.42,-0.26,0.87}, {0.32,0.69,-0.15,0.78,0.25,-0.44}, {0.61,-0.33,0.48,0.91,-0.18,0.36}, ... 0] A0 + B + C S4[...{0.54,-0.13,0.75,0.42,-0.26,0.10}, 0,0,0,0,0, 0,0,0,0,0, ... 0] A1 S5[...{0.54,-0.13,0.75,0.42,-0.26,0.10}, {0.27,0.72,-0.09,0.65,0.41,-0.53}, 0,0,0,0,0, ... 0] A1 + D

I added the {} only to make the story more legible and to separate features of the represented state; they are not part of the formal system itself. Notice also that the B and D representations share positions. That suggests a region of the state-space associated with an affective or motivational response to the coordinates occupied by A.

I am using a more tangible action-based example here, but the same logic would apply to something as simple as light hitting a photoreceptor and the observer mapping color and position into a phenomenal representation. Depending on the application, you can average the state over time, or produce a higher-order vector representation of the whole sequence. That may be lossy, but that is acceptable: evolution does not need a perfect copy of reality. It needs an approximation of reality good enough to bring about adaptive state-transitions.

There are many ways to describe the world. People often say that functions describe the world, and that is true as far as it goes. But most approximations—including functions in isolation—describe only the world of appearances available to the model. The advantage of the vector-space picture is that it lets us describe many different levels of organization inside one composable framework. Neural networks are still useful here, but they are downstream processors of structured representations; they are not, by themselves, a theory of how those representations are made available to the organism.

So, in this section, I assume an open-ended nonlinear mapping rule capable of weighting and summing phenomenal vectors as they co-occur. That rule takes the current phenomenal state, maps it through the organism’s inherited interpretive structure, and yields a new phenomenal state. Later, when we get to applications, the practical problem will be path-isolation and signal-processing: given the noise of a large state-space, which contributing factors produce the strongest signal for the transition we care about?

The broader claim is not that the organism first builds a neural network and only then acquires a world. It is the reverse. The organism inherits a structured way of carving reality into usable distinctions, and the processor it builds later operates within that inherited space. Evolution gives the observer a repertoire of vector-like distinctions—time, space, color, objectness, shape, bodily boundary, hierarchy, proportion, attention, lower-order affect, higher-order affect, symbolic meaning, interiority, and so on. Action is a byproduct of the continuous processing of these distinctions through the organism’s interpretive structure.

It is important to understand the primitives before the abstractions. We want a representation of phenomenal life that is mathematically tractable without pretending that every observer receives the same world in the same way. The same broad stream of reality may confront multiple observers, but different inherited structures and different realized histories will transform that stream into different phenomenal outcomes.

If you want an intuitive picture, think of the inherited structure as a massive coordinate-ready template and of lived history as the process that determines how that template is realized in the individual. In principle, that structure constrains how you can react to the world; in practice, any model we build will only ever approximate that structure. Evolution is the encoding protocol, compression is part of the mechanism, and the point of the framework is to describe how reality is described for an organism—not to confuse the description with the thing itself.

Why use an embedding system rather than just talk about neural networks? Because I am not trying only to describe a processor. I am trying to describe the representational conditions that make processing possible in the first place. In that sense, what this section does is combine the space/time side of experience with the rest of phenomenal life into one common representational framework, rather than treating them as separate faculties that must later be stitched back together.

Note also that we are never really processing one isolated datum at a time. The phenomenal stream is continuous, even when the model samples it discretely. The inherited structure is relatively stable; your experiences, memories, and moods are not modifications to the seed itself so much as modifications to the stream it is processing and to the realized organization built on top of that seed. That is why this device can represent what Kant’s table of categories cannot: not just a generic human mind, but, in principle, the different ways minds can be structured across organisms and across individuals.

So ends Kant from the Evolutionary Perspective and Vectors Are All You Need.

Next, we get into the real meat: what an application of this theory looks like, and how to do these calculations.

The Nature of Phenomenal Reality: What are we trying to measure?

Most of the time, when mathematicians are using the ‘infinite’ it is to simplify a problem and also say something concrete about finite things. Thus, we are continuing the tradition by emphasizing the infinite ways in which reality can be understood so that we can isolate a finite representation relevant to us. To simplify further, I will often write the system as though it were deterministic: if the complete noumenal conditions, the complete phenomenal state, and the exact transition rule were known, then the next state would follow cleanly. This is an explanatory device, not a declaration that the actual universe must be deterministic. For real work, incomplete state and genuinely stochastic dynamics require probability distributions. The deterministic arrow is the silhouette; the stochastic kernel is the model we can actually train.

In the previous section, we went over the concept of Transcendental Embeddings and we assumed the finite vector space that the organism played on. In ‘God’s Infinite Dimensional Space’ there are infinitely many available directions of representation, enough to leave room for distinctions that one actor can register and another cannot. Quantum mechanics offers a useful mathematical analogy, but only an analogy: pure quantum states are represented by rays in a complex Hilbert space and observables, in the standard formulation, by self-adjoint operators, whereas GIDS uses an idealized real Hilbert space as a representation arena for distinctions relevant to actors. Nothing here is a claim that psychology is quantum mechanical. The common point is that Hilbert-space structure lets us represent a state with as many coordinates as the model requires while still taking finite projections for actual calculation.

So what we are trying to find first is a general transition grammar that maps the current actor–world state and incoming conditions onto a distribution over what follows. Different organisms will instantiate different transition laws; the universal claim concerns the form of the description, not one literal function shared by every actor. Keep in mind that the inherited organization carries the residue of evolutionary history through bodies, sensors, developmental programs, and learning capacities—not as a complete record written directly into a few genes. So how do we reduce the complexity to something tangible? We represent those inherited developmental constraints as limiting and organizing the repertoire of distinctions the organism can acquire and use. The guiding question is how to operationalize that constraint mathematically so that we can construct, or at least approximate, the transition law for one actor class.

In the formal model, inherited developmental constraints are represented by a stable but extraordinarily high-dimensional accessible structure: the seed of the Transcendental Embedding. Within this idealized structure, an organism’s capacities—whether sensory, interoceptive, motor, or eventually conceptual—are described in Hilbert-space coordinates and nonlinear transformations. Genes do not directly label coordinates such as fear, color, or hierarchy. Genes participate in developmental systems that build bodies, sensors, neural and biochemical machinery, and learning rules; the accessible structure is our mathematical description of the capacities that emerge.

This is not a one-gene/one-coordinate claim. Genes alter developmental machinery, bodies, sensors, and learning systems; the axes are an idealized description of the capacities and transformations that emerge from that machinery.

The trick is, no organism has conscious access to these developmental templates; instead, they appear as “normal” to the experiencing subject. Imagine a hypothetical lineage in which a particular chromatic signal reliably predicted danger and the developmental system made that signal strongly recruit threat response. For that organism, the surge would arrive as an immediate fact of experience rather than as a consciously selected inference. In actual humans, responses to red are not one genetically fixed universal reflex; biology, learning, culture, and context all contribute. The point of the example is only that inherited organization can make some transformations feel self-evident or inescapable to the subject.

Also, keep in mind that the ideal explanation is still deterministic because it is easier to see the machinery when one state points cleanly into the next. The operational framework is not allowed that luxury. It will use conditional distributions, calibration, and experiments wherever the real world refuses to collapse into one answer. The question is not whether probability can be banished; it is whether the underlying state can be made rich enough that the remaining uncertainty becomes tractable.

The Evolutionary Mechanism for Encoding Transcendental Embeddings

It is useful to imagine humans as experiencing a small slice of all possible ways to experience reality. Within this small slice, there is enormous variation; however, we are still subject to an evolutionary inheritance that provides us with a predictable range of ways to perceive and structure reality. You could imagine the way humans experience reality is a series of dimensions. To be simple, we could have dimensions for the range of colors we see, as well as dimensions for how we position those colors in our mental interior when a photon hits our eyeball. We could build on this by adding other dimensions and associating groups of pixels. Then, we could create structures from those groupings and assign dimensions that associate meaning (or potential for meaning) with those superstructures. Finally—you could imagine—we have dimensions that represent placeholders for the superstructures humans expect in their lives. An example of this would be something like a mother figure, or what a place to sit looks like, or what violence is, or even something complex like a god object. These are complex phenomena that depend on humans holding a particular psychic position in reality; the god object, for example, could be represented as a structured pattern assembled from communal bonding, fatherhood, war, purity, spite, revenge, love, sacrifice, death, externalized meaning, care, fear, language, outsiders, and the couplings between them. A crude first implementation might pool or average some of those representations, but the category is not itself one primitive axis. Different weightings of the pattern, coupled with a supporting or dismissive society, can produce a range of outcomes when a human internalizes and practices the god object.

All of the sub-vectors of the god object are their own series of complex dimensions as well. The war object might require communal bonds, hunger, fear, negative ethnocentrism, concepts of ownership and land, hierarchy, etc. Let’s get even simpler, the hunger object could be composed of the vectors of glucose saturation, fullness, thirst, fat concentration, presence of ghrelin, and other biochemical factors. The sleight of hand I performed was relating biochemical signals and things that exist purely as a concept in the mind of the user, like god and war. Ah, but that is the point! At the level of this formalism, hunger and the feeling that your group is at war both enter the phenomenal state and can recruit action. This does not mean they feel identical; it means the same representational grammar can carry both bodily and synthetic experience. Humans (to varying individual degrees of intensity and presence) have a space in their minds for both concepts.

All these complex traits, being the results of series of simpler objects, make their examination and deracination possible as well as, in principle, computable at an appropriate resolution. But how did humans get to become so complex? Why is the range of our experiences so large relative to other creatures? How did we obtain this Transcendental Embedding? To be overly simplistic, we can trace evolutionary lineages backward and imagine additions, losses, and reorganizations of accessible distinctions over time. This also gives us a way to ask which organisms have functional overlap. Some corvids, for example, have solved water-displacement tasks in laboratory settings. That is enough to motivate convergent functional structure; it is not evidence that corvids possess a human-like theory of buoyancy or experience the task in the same way. The framework needs room for both overlap and radically different internal construction.

To be very clear: some enormous number of interacting coordinates and transformations produces the expected reality available to a human. I picture this as radically larger for a human than for an earthworm, but the numbers here are an intuition pump, not an empirical estimate.

But let’s start at the beginning and build a toy organism, not a literal history of the origin of life. Imagine a membrane-bounded protocell or minimal cell-like system with a rupture-sensitive response mechanism: changes in ionic balance alter whether the boundary remains intact. At the level of the toy representation, that gives us a first usable distinction—membrane integrity—whose states matter for persistence. We can call this a “membrane-integrity axis,” while remembering that the axis is our model of a coupled biochemical process, not a claim that the protocell possesses a conscious binary qualia.

Random copying errors then throw up tiny tweaks: a peptide that bends when it binds a proton, a chromophore that flips shape when hit by a photon, an ion channel that opens more readily in warmer fluid. In the toy geometry, each tweak proposes a new accessible distinction or transformation. Many changes are neutral, harmful, or useful only in combination with other changes; they do not pass through one clean immediate-payoff filter. Occasionally, however, a heritable change helps descendants move toward nutrients or away from damaging conditions and increases in frequency. The represented repertoire can then become richer, while drift, pleiotropy, developmental coupling, and later loss remain outside this simplified picture.

In this toy story, retained capacities can be reorganized into higher-order compound distinctions rather than simply discarded. Once a light-sensitive molecule exists, downstream mutations wire two such molecules together, letting the organism register differences in intensity rather than mere presence. That difference becomes a new vector—contrast. A later duplication introduces a second pigment shifted in wavelength; the comparator circuit now yields a chromatic axis. What began as a single light/no-light bit has unfolded into a color cube where the vectors of understanding reality become synergistically linked.

This simple logical mechanism is highly scalable as a story. Chemical gradients can become richer sensorimotor maps; distributed pressure signals can contribute to a body schema; socially relevant signals can participate in an emerging social manifold. In this simplified picture, selection tends to retain and organize capacities whose contributions to reproduction outweigh their costs in the environments where they matter. Real evolutionary change also includes neutral drift, constraint, exaptation, frequency dependence, and traits whose value appears only through interactions. Complexity is the residue of all of that history, not the result of an accountant adding one profitable axis at a time.

By the time we reach hominids, the embedding has accrued countless axes and transformations. Some are exteroceptive (hue, pitch, depth), others interoceptive (blood CO₂ and signals involved in hunger), and others still are synthetic patterns available only through learned conceptual organization: tool, ally, lover, or taboo. The aggregate is a pseudo-species-level template: an expected repertoire of distinctions a typical human can, in principle, develop. The proper resolution, however, is the individual organism. An individual inherits a developmental program for a species-compatible template; the realized embedding is not fully formed at birth. If an individual lacks a functional long-wavelength-sensitive cone pigment, or has an altered version of it, some chromatic discriminations are unavailable or changed even though the species-level model retains a place for long-wavelength contrast. Reality changes for that observer; it does not simply lose one mathematically isolated “shade.” Our concept of self-preservation appears complex, but it is built on vastly older mechanisms of persistence. Finally, vector norm has no intrinsic psychological meaning. A representation may use magnitude to encode intensity or importance only if the encoding rule or training objective gives the norm that interpretation; self-preservation is not automatically “far from the origin.”

Before reducing this into four steps, I need to mark the level of the claim. This is a toy construction of how new distinctions could become available to a lineage. Real evolution also contains drift, pleiotropy, correlated traits, exaptation, changing environments, loss, and reorganization of old capacities. I am keeping the simple axis story because it gives us a usable formal picture, not because biology literally appends one clean orthogonal coordinate at a time.

To be brief:

A mutation or developmental change proposes a new accessible distinction for the organism.
1. In the toy geometry, this appears as a candidate coordinate or transformation of experience.
If the change contributes to reproductive success, it can spread through the lineage.
The capacity may stabilize, be modified, or later be lost.
Richer composite coordinates arise through interaction, recombination, and reorganization of older distinctions.

Formalization

(Bold lowercase symbols denote finite coordinate vectors. Calligraphic symbols denote spaces, sets, or families of maps. Ordinary uppercase symbols may denote structured states, kernels, linear maps, or matrices; their type is declared where they appear.)

Before we continue, the philosophical argument I gave was to ensure that we can simplify the essential elements down to the point where we can inject them into standard and proven mathematical frameworks. I have to reiterate: I am not inventing new math. I am trying to decide what kind of objects the old math should be attached to without quietly turning the metaphor into a fact.

Lastly, this work has been an absolute tour-de-force of effort; as a consequence, we should assume there can be logical gaps or errors in the formalization. Email me if you see anything wrong.

1. The noumenal domain and God’s Infinite Dimensional Space

Let \(\mathcal N\) denote the noumenal domain: whatever complete physical or otherwise external conditions exist prior to the observer’s organization of them.

I do not need to assume that \(\mathcal N\) is itself a Hilbert space. Doing that would already smuggle the observer’s mathematics into the thing I am claiming the observer cannot encounter directly. \(\mathcal N\) is therefore left deliberately under-specified.

Let

\mathcal G

denote God’s Infinite Dimensional Space: an idealized real Hilbert representation arena rich enough to encode distinctions and combinations of distinctions that could, in principle, participate in the experience or response of possible actors. The construction does not require one orthogonal basis coordinate for every named distinction.

For the working formalism, take \(\mathcal G\) to be separable. Separability is a tractability convention, not a metaphysical discovery; it says the space admits a countable orthonormal basis. If the ideal arena required a nonseparable space, the outer construction could be widened without changing the finite models we can actually build.

For concreteness, one can think of

\mathcal G \cong \ell^2,

the space of square-summable real sequences, with an orthonormal basis

(\mathbf e_k)_{k=1}^{\infty}.

A modeled activation in GIDS can then be written

\mathbf g = \sum_{k=1}^{\infty} g_k\mathbf e_k, \qquad \sum_{k=1}^{\infty}|g_k|^2<\infty.

This is a capacity claim, not an ontology of objects. A person is not one basis vector. A proposition is not another basis vector. A corporation is not a special organizational subspace waiting next to the human subspace. The space contains possible distinctions and combinations of distinctions. Objects, categories, actors, propositions, and institutions are represented by structured patterns, regions, functions, and transformations built with those coordinates.

The distinction matters. “Fear” may eventually correspond to a family of related activations. “Corporation” may correspond to a category used by a human observer. Neither claim means that fear or a corporation is literally one primitive axis in the fabric of \(\mathcal G\).

To move from external conditions into a model-side representation, introduce a registration map

\operatorname{Reg}:\mathcal N_{\mathrm{loc}}\to\mathcal G.

For a local external condition \(\omega_t\in\mathcal N_{\mathrm{loc}}\),

\mathbf g_t^{\mathrm{reg}}=\operatorname{Reg}(\omega_t)\in\mathcal G

is the model’s representation of distinctions available in that condition. It is not the noumenon itself. It is already a mathematical encoding chosen for the framework.

2. Species-level accessible structure

Evolution does not give a lineage access to every coordinate in \(\mathcal G\). It preserves and organizes a finite or effectively finite repertoire of distinctions that the lineage can register and use.

Define the species-level accessible structure by a closed subspace

\mathcal M^{\mathrm{spec}} = \operatorname{span}\{\mathbf v_1,\ldots,\mathbf v_d\} \subset\mathcal G, \qquad d<\infty,

where the vectors \(\mathbf v_1,\ldots,\mathbf v_d\) are an idealized coordinate basis for the lineage’s available distinctions. For convenience, assume they are orthonormal:

\langle \mathbf v_i,\mathbf v_j\rangle=\delta_{ij}.

The associated projection is

P^{\mathrm{spec}}:\mathcal G\to\mathcal M^{\mathrm{spec}}, \qquad P^{\mathrm{spec}}\mathbf y = \sum_{j=1}^{d} \langle \mathbf v_j,\mathbf y\rangle\mathbf v_j, \qquad \mathbf y\in\mathcal G.

Applied to a registered local condition,

\widetilde{\mathbf g}_t^{\mathrm{spec}} = P^{\mathrm{spec}}\mathbf g_t^{\mathrm{reg}},

this yields the part of the model-side distinction field that the lineage can, in principle, use.

The orthogonal projection is a clean mathematical idealization. It says: under the norm we selected, retain the component that lies in the accessible structure and annihilate the rest. Biology need not literally perform an orthogonal projection, and psychological independence need not literally equal geometric orthogonality. Later implementations can replace this with a nonlinear access map. I am using the linear form because it makes the first version legible.

The same accessible slice can be expressed in coordinates through

E^{\mathrm{spec}}:\mathcal G\to\mathbb R^d, \qquad E^{\mathrm{spec}}(\mathbf g_t^{\mathrm{reg}}) = \begin{bmatrix} \langle \mathbf v_1,\mathbf g_t^{\mathrm{reg}}\rangle\\ \vdots\\ \langle \mathbf v_d,\mathbf g_t^{\mathrm{reg}}\rangle \end{bmatrix}.

These coordinates are not yet phenomenal reality. They are the lineage-available material from which phenomenal organization can be built.

This is the sense in which GIDS makes room for experiences that humans cannot have. A coordinate can exist in \(\mathcal G\) while being annihilated by \(P^{\mathrm{spec}}\) for the human lineage. A hypothetical actor with a different access structure may retain it. Dark matter does not need to be consciously available to a human in order for the general space to leave room for an actor that can register some distinction associated with it.

3. Objects of experience and categories

Now we can state the correction that keeps the framework from becoming stupid.

An object of experience is not generally a primitive element of \(\mathcal G\). It is a structured representation produced by an actor when external conditions encounter the actor’s accessible structure and current state.

Let \(\omega_t\in\mathcal N_{\mathrm{loc}}\) denote an external configuration that may be experienced as an object, event, person, sentence, price, threat, chair, god, or anything else. Let \(\phi_{i,t}\) be the current phenomenal state and \(c_{i,t}\) the active context of actor \(i\). Write the actor-relative representation as

\zeta_{i,t}^{\mathrm{obj}}(\omega_t) = \mathcal I_{i,t} \!\left( A_{i,t}\operatorname{Reg}(\omega_t), \phi_{i,t}, c_{i,t} \right),

where

A_{i,t}:\mathcal G\to\mathcal V_{i,t}^{\mathrm{acc}}, \qquad \mathcal I_{i,t}: \mathcal V_{i,t}^{\mathrm{acc}}\times\Phi_i\times\mathcal C_i^{\mathrm{ctx}} \to \mathcal V_{i,t}^{\mathrm{obj}}.

The first map controls access; the second is the nonlinear organization that turns accessible distinctions plus current state into a phenomenal representation. Neither actor-relative space needs to be a linear subspace of \(\mathcal G\), and \(A_{i,t}\) need not be linear or orthogonal. These slowly changing maps and organizations will be bundled into the realized individual structure \(T_{i,t}\) in Part 2.

The same external configuration can therefore produce different object-representations in different actors, or in the same actor at different times.

A category requires a declared representation domain. Write that domain as

\mathcal V_{\kappa}^{\mathrm{cat}}.

Depending on the question, \(\mathcal V_{\kappa}^{\mathrm{cat}}\) may be GIDS itself, an actor-relative object space such as \(\mathcal V_{i,t}^{\mathrm{obj}}\), or a finite learned feature space. Once the domain and its measurable structure have been declared, a category may be represented as:

a measurable region \(\mathcal C_\kappa\subseteq\mathcal V_{\kappa}^{\mathrm{cat}}\);
a prototype point \(\operatorname{proto}_\kappa\in\mathcal V_{\kappa}^{\mathrm{cat}}\);
a probability measure \(\mathbb P_\kappa\) on \(\mathcal V_{\kappa}^{\mathrm{cat}}\);
or a learned scoring function

\operatorname{cat}_{\kappa}:\mathcal V_{\kappa}^{\mathrm{cat}}\to[0,1].

The framework does not force every category to be an axis, or even to be an element of \(\mathcal G\). Many categories will be conglomerations over subtler coordinates and their interactions. If \(\mathcal V_{\kappa}^{\mathrm{cat}}=\mathcal G\), a prototype point is an element of GIDS and a category region is a subset of it; a probability measure or scoring function is still an object defined on the space rather than a vector inside it. If the category is actor-relative, its natural domain may instead be \(\mathcal V_{i,t}^{\mathrm{obj}}\). This is precisely what happens in ordinary factor analysis: the named construct is often a rough statistical compression over a more complicated underlying organization.

A proposition follows the same rule. The email, offer, threat, price, person, or meeting exists first as an external or symbolic configuration. What enters the actor’s transition is its actor-relative representation:

\mathbf p_{i,t}(x_t) = \mathcal P_{i,t} \!\left( x_t, \phi_{i,t}, c_{i,t} \right) \in\mathcal V_{i,t}^{\mathrm{obj}}.

The proposition does not need a dedicated proposition subspace. It needs an actor-relative representation in the distinctions available to the actor it is confronting. Part 3 will split \(\mathcal P_{i,t}\) into an external proposition encoder and a learned actor-relative map.

4. A toy evolutionary rule for accessible dimensions

Now let evolutionary stage be indexed by \(\nu\), and let \(\mathcal M_\nu^{\mathrm{spec}}\) denote the lineage’s current accessible structure at that stage. Let \(P_\nu^{\mathrm{spec}}\) be the orthogonal projection onto \(\mathcal M_\nu^{\mathrm{spec}}\) in this toy geometry.

A mutation or developmental modification proposes a candidate direction

\Delta\mathbf v\in\mathcal G.

Remove what the current structure already captures:

\Delta\mathbf v_{\perp} = \Delta\mathbf v - P_\nu^{\mathrm{spec}}\Delta\mathbf v.

If \(\Delta\mathbf v_{\perp}=0\), the candidate adds no new representational direction under this model. If it is nonzero, normalize it:

\widehat{\Delta\mathbf v}_{\perp} = \frac{\Delta\mathbf v_{\perp}} {\|\Delta\mathbf v_{\perp}\|}.

Let

\mathfrak F_\nu(\mathcal M) = \mathbb E_{e\sim\mathcal E_\nu} [W(e,\mathcal M)] - C_\nu(\mathcal M)

be a finite real-valued toy evolutionary objective combining expected reproductive value and the total cost of maintaining an accessible structure. Define the contribution of the proposed direction directly by

\Delta\mathfrak F_\nu = \mathfrak F_\nu\!\left( \mathcal M_\nu^{\mathrm{spec}} \oplus \operatorname{span}\{\widehat{\Delta\mathbf v}_{\perp}\} \right) - \mathfrak F_\nu(\mathcal M_\nu^{\mathrm{spec}}).

Then the simplified retention rule is

\mathcal M_{\nu+1}^{\mathrm{spec}} = \begin{cases} \mathcal M_\nu^{\mathrm{spec}} \oplus \operatorname{span}\{\widehat{\Delta\mathbf v}_{\perp}\}, & \Delta\mathbf v_{\perp}\neq 0 \text{ and }\Delta\mathfrak F_\nu>0,\\[6pt] \mathcal M_\nu^{\mathrm{spec}}, & \text{otherwise}. \end{cases}

So the toy story is:

a mutation proposes a candidate distinction;
subtract what the lineage already represents;
compare the total value of the expanded structure against the total value of the old one;
retain the direction if the difference is positive.

This is not a complete population-genetic model. It ignores drift, frequency dependence, pleiotropy, exaptation, correlated features, and the possibility that old coordinates are rotated, merged, weakened, or deleted. It is a compact explanation of how a repertoire of distinctions could become richer over time.

5. Deterministic silhouette and stochastic operation

The clean philosophical transition for the phenomenal component is

\phi_{i,t+1} = F_i\!\left( T_{i,t}, \phi_{i,t}, c_{i,t}, w_t, \mathbf p_{i,t}(x_t) \right).

Read this as the ideal claim that a complete state, complete input, and complete transition law determine what follows. Over this explanatory step, the slowly changing person structure and the exogenous world are treated as fixed unless their updates are written explicitly.

For the complete dynamics, define

\Sigma_{i,t}^{\star} := \bigl(T_{i,t},\phi_{i,t},c_{i,t},w_t\bigr).

Let \(\boldsymbol\Xi_{t+1}\) be a random exogenous innovation and \(\boldsymbol\xi_{t+1}\) a realized or supplied scenario value. The notation below evaluates a transition kernel at the supplied scenario value; it does not require the singleton event \({\boldsymbol\Xi_{t+1}=\boldsymbol\xi_{t+1}}\) to have positive probability. The operationally honest ideal is

\Sigma_{i,t+1}^{\star} \sim K_i^\star\!\left( \cdot \mid \Sigma_{i,t}^{\star}, X_t=x_t, \boldsymbol\Xi_{t+1}=\boldsymbol\xi_{t+1} \right),

where \(K_i^\star\) is the unknown transition kernel over complete next states. A deterministic system is the special case in which that kernel collapses to a point mass. The first equation is the phenomenal component of this fuller transition.

This convention will run through the rest of the paper. I will keep arrows when the arrow makes the explanation easier to see. When we train and evaluate the model, the arrow expands into a distribution because the universe may be stochastic and our state estimate is certainly incomplete.

Species template versus individual realization

The species-level structure says what kinds of distinctions a member of the lineage can, in principle, host. It is not yet the person.

Write the lineage-level template as

G^{\mathrm{spec}} = \left( \mathcal M^{\mathrm{spec}}, P^{\mathrm{spec}}, \mathfrak I^{\mathrm{spec}} \right),

where \(\mathfrak I^{\mathrm{spec}}\) is the species-compatible family of interpretive organizations.

The inherited seed of person \(i\) is an individual realization inside that lineage-compatible family:

G_i = \left( \mathcal M_i^{0}, A_i^{0}, \mathcal I_i^{0} \right), \qquad \mathcal M_i^{0}\subseteq\mathcal M^{\mathrm{spec}}.

Here

A_i^{0}:\mathcal G\to\mathcal M_i^{0}

is the inherited access map at the chosen resolution; it is not assumed linear or orthogonal. The component \(\mathcal I_i^{0}\) denotes the inherited initial organization that later development can modify.

This lets biological variation, developmental constraints, and pathology alter the inherited starting point without pretending every member of the species begins with an identical coordinate system.

Part 2 now asks how \(G_i\) becomes the slowly changing realized structure \(T_{i,t}\), how that person occupies a phenomenal state, and how a low-resolution estimate can be extracted from outward traces without pretending the estimate is the whole interior.