God as the invariance of projectability

The most naive way to imagine what lies beyond our universe is to treat it as the room next door. On one side there would be the physical world, with space, time, fields, particles, laws and observers; on the other, a metaphysical elsewhere, perhaps vaster, subtler or more mysterious, but still imagined as a place. The image feels natural because the mind spatializes almost everything: if something can be distinguished from nothing, it must be somewhere; if it is somewhere, it inhabits a space; if it inhabits a space, that space is already a kind of universe.

The fragile point is exactly there. To be determinate does not necessarily mean to be localized. A law, a mathematical structure, a global condition, an equivalence class, a consistency constraint or a symmetry principle can be something without occupying a place. They are not nothing, because they make a difference in thought and, in some cases, in the description of reality. But they are not things resting inside a geometry. Metaphysics becomes interesting when it stops looking for a spatial outside and starts asking what structure must remain for a universe to be a universe.

The thesis I want to defend is this: if we call the ultimate principle of reality “God”, then it is not useful to identify it with the undifferentiated totality of possibilities. A totality that contains everything in the same way does not explain why order exists, why some forms are worlds and others are only incoherent combinations, why physics finds invariants instead of noise. The stronger version of the idea is different: God, if the term still helps, is the intensive principle of projectability, what lets a loss of absoluteness become a world without dissolving into arbitrariness.

This is not a proof of God’s existence in the ordinary religious sense. It does not produce a personal being, found a cult, authorize a local miracle or turn a scientific gap into theological evidence. It is a more austere metaphysical claim: if reality is not pure chaos, then the foundational level cannot be pure undifferentiated abundance. It must have form. And if it has form, it is not omnipotence as caprice, but power as coherence.

The false outside

The first error to remove is the idea that every being must be inside a space. The phrase “something outside the universe” is almost always ambiguous, because it uses outside spatially while trying to speak about what should not be spatial. If the universe includes physical spacetime, then an outside understood as an external region of space is no longer an outside: it is an enlargement of the geometry. If, instead, the outside is not spatial, then it should not be imagined as a distance, a wall or a border.

Cosmology helps avoid a caricature. The Big Bang should not be imagined as a fragment of matter exploding inside a surrounding void. In the standard description, the point is the evolution of spacetime itself from an early hot, dense state, not a detonation in a preexisting room. The cosmic microwave background is an observational trace of the early universe; the image of an explosion in static space is misleading because, in standard cosmology, space itself is what expands, not material thrown into an external void. This physical clarification does not solve metaphysics, but it prevents us from building it on the wrong picture.

If something interacts causally with our universe, the situation changes at once. A local, measurable and temporally ordered influence enters the enlarged physical domain: it may be an unknown field, a source, a boundary condition, a hidden degree of freedom, but it no longer remains simply “beyond”. The act of causing an event within spacetime makes it part of the network of dependencies that physics must describe. The mystery may be enormous, but the conceptual form becomes physical.

Another possibility remains: not a local cause, but a global foundation. An atemporal principle does not push events one after another. It does not wake up before the first instant, choose a parameter, wait, correct, intervene. It grounds the whole block of relations in which time, causality and observers appear from within. To fix the distinction, the first line treats $G$ as if it produced three separate local events, $e_1,e_2,e_3$:

$$G \to e_1,\quad G \to e_2,\quad G \to e_3$$

This notation describes an agent producing separate events. The second line instead uses $U$ to name the whole totality of the world:

$$G \Rightarrow U$$

describes a principle that makes a totality possible. In the first case God is a cause among causes, only more powerful. In the second, God is the foundation of the scene in which causes can exist.

From the possible to the projectable

The next temptation is to identify God with the absolute possible. Let us call:

$$\Omega = \text{abstract totality of combinable possibilities}$$

This definition is seductive because it seems maximal. If God is everything that can be, nothing remains outside. But precisely this greatness makes it weak. A totality that contains every combination does not yet have an internal criterion for distinguishing world and noise, law and accident, structure and heap, real possibility and grammatically constructible sentence.

The problem is not that $\Omega$ is too small. It is that it is too permissive. If everything is admitted on the same terms, then “possible” loses contrast, because possibility makes sense only if something remains impossible. A modal system is not an infinite landfill of alternatives; it is a regulated distinction between what can hold together and what cannot. Where there is no exclusion, there is no form.

David Lewis is a useful comparison because he takes possible worlds radically seriously. In his modal realism, possible worlds are concrete and isolated from one another: no part of one world is spatiotemporally related to parts of another. It is an extremely powerful theory for modal logic, but it is not the point I am after here. It is not enough to say that many worlds exist. We must explain what makes a possibility a coherent totality rather than a shapeless combination.

So we need a less extensional symbol. From here on, $G$ does not mean a large archive of possibilities, but the principle that establishes which forms remain coherent when projected:

The difference between $\Omega$ and $G$ is the heart of the theory. $\Omega$ is extensional: an ideal sum of configurations. $G$ is intensive: it is not a list, but what establishes which representations are still coherent forms. The question is no longer:

$$\Omega \to G \text{ chooses } U$$

because this form would make $\Omega$ more fundamental than $G$. The right form is:

where $U_G$ is the domain of the legitimate projections of $G$. All possibilities do not exist first, followed by a divine selector. There is a principle of projectability from which only the forms capable of preserving its invariants emerge.

A projection is a loss

The word “projection” has to be taken seriously. A projection does not reproduce everything. It reduces, compresses, identifies, makes some differences invisible. If a universe were identical to $G$, it would not be a projected universe; it would be $G$ itself. Worldliness begins when the absolute becomes finite without becoming arbitrary.

Write the projection with the Greek letter $\pi$:

$$\pi: G \to U$$

The map $\pi$ sends the original principle into a projected totality. But if $\pi$ loses information, then distinct elements of $G$ may become indistinguishable inside $U$:

$$x \sim_\pi y \Longleftrightarrow \pi(x)=\pi(y)$$

Here $x$ and $y$ are not two already given physical objects. They are two determinations of the foundational level. The symbol $\sim_\pi$ means “equivalent with respect to the projection”: $x$ and $y$ may be different in $G$, but they become the same element when seen through $\pi$. This equivalence relation is the scar of finitude. The world does not see all the distinctions of its foundation. It groups them into classes, treats them as the same thing, builds objects, events, properties and laws from this compression.

In compact form:

A universe is a quotient. It is not a piece cut away from God like a slice from a metaphysical cake. It is a structure obtained by identifying deeper differences so that a legible totality remains. Locality, time, opacity, phenomenal irreversibility and the separation between observer and observed may be effects of this loss.

Loss alone, however, is not enough. If you collapse too many distinctions, you do not get a world but indistinction. If you identify elements without respecting structure, you get noise. The point is not to lose information; the point is to lose it well.

The quotient must respect structure

Mathematics offers a precise model. Here $\star$ is not a specific physical operation: it is a placeholder for any structural operation at the starting level, such as composing two relations, applying a transformation or combining states. If you want to build a quotient, you cannot identify elements at random. You want the operation to remain well defined on classes. The notation $[x]$ means the class of all elements that the projection makes indistinguishable from $x$:

$$[x] \star [y] = [x \star y]$$

This formula is legitimate only if the result does not depend on the chosen representatives. Saying $x \sim x'$ means that $x$ and $x'$ belong to the same projected class; the same holds for $y$ and $y'$. If you change representative within the same class, the result must remain in the same class:

$$x \star y \sim x' \star y'$$

If this condition fails, the quotient becomes ambiguous. The same projected state would have different consequences depending on a hidden difference that the projection claimed to erase. Such a structure is not a world: it is a machine using one word for things it later wants to treat differently.

This is the most important criterion of the whole theory:

Call this compatibility congruence. An equivalence relation is a congruence when it respects the structural operations and relations. Then:

This formula removes the need for an external filter. We are not saying: first there are all possibilities, then God applies criteria such as stability, observers or computability. We are saying: a world is what derives from $G$ through a quotient that does not destroy the well-definedness of invariants.

Computability, in particular, is too internal to a metaphysics of process. To compute means to execute a procedure, and a procedure introduces an operational order. If $G$ is atemporal, it does not generate universes by running an algorithm. The structure may be mathematically representable without being the result of a program in execution.

The invariants of worldliness

An invariant is not a property identical in every universe. We should not imagine that every world has particles, space, energy, entropy, an electromagnetic field or the speed of light. These are candidates for local traits of our projection. Metaphysical invariants are more abstract: conditions without which a projection would not be a world. The formulas that follow are not physical calculations; they are a minimal notation for introducing those conditions one at a time.

The first is determination. For there to be something rather than indistinction, a difference must be able to hold. The letters $a$ and $b$ name any two elements of the projected world $U$:

$$\exists a,b \in U \quad a \neq b$$

This is not schoolbook logic yet. It is minimal ontology. If every difference collapses, no object, event, property, relation or perspective remains. What remains is a phenomenally null point. A world begins when at least one distinction can be maintained.

The second is relationality. Two isolated differences do not make a totality. A universe is not an inventory of elements lined up in a row; it is a domain in which determinations imply, limit, order and transform one another. The letters $R$, $S$ and $T$ do not name three particular forces; they indicate three possible relations among elements of the world:

$$R(a,b),\quad S(b,c),\quad T(a,c)$$

Here structural realism becomes a conceptual ally. In philosophy of science, this position also arose to explain why some mathematical structures survive theory change even when the ontology of objects changes. Structural continuity can be more reliable than the description of the world’s ultimate furniture. In this metaphysics, the point is radicalized: to be a world is first of all to be a structure of relations.

The third is global compossibility. It is not enough for the pieces to be thinkable one by one. They must be able to cohabit the same totality. Many configurations are locally describable but globally cannot be glued together. A universe requires its parts to fit together without producing contradictions at the seams.

The geometric image is useful. $U_\alpha$ and $U_\beta$ indicate two local descriptions of the same world, like two partial maps of a city. They must agree where they overlap:

$$U_\alpha|_{\alpha \cap \beta} = U_\beta|_{\alpha \cap \beta}$$

Compossibility is stronger than immediate non-contradiction. A sentence may not contradict itself and still be unable to enter a world. What is needed is coherence of totality, not just the absence of local sparks.

The fourth is regulated transformability. A world need not be motionless. It must be able to change without losing every identity. This requires a set of admissible transformations, which I call $A_U$ because they belong to the world $U$:

$$A_U = \text{legitimate transformations of } U$$

If $I_U$ indicates an internal invariant and $T$ a legitimate transformation chosen from $A_U$, then for every state or element $u$ of the world:

$$I_U(T(u)) = I_U(u)$$

The deep meaning of law is here: not a sentence describing surface regularities, but a constraint on what may vary while leaving form intact. A world is change under conservation. If nothing can change, you have immobility; if everything can change into everything, you have noise; if change takes place within a field of invariants, you have history.

The fifth is internal closure. Once projected, a world must be readable from within as an autonomous totality. This does not mean absolute independence from $G$. It means that the world’s events do not require a punctual external act for each transition. Causality belongs to the projection; foundation belongs to the principle.

This distinction makes the miracle understood as a local rupture of laws suspicious. Not because metaphysics must impose a flat naturalism, but because a punctual intervention reintroduces God as an acting object on the very plane God is supposed to found. If $G$ is the principle of worldliness, it does not enter the world as an additional event. It is that by which events can be events.

The sixth is modality. A universe is not only the list of what happens; it is also a domain of what can or cannot happen according to its structure. I call $M_U$ this space of possibilities internal to the world $U$:

$$M_U = \text{possibilities internal to } U$$

Laws of nature are philosophically difficult precisely because they do not seem to be mere summaries of what has happened. In many interpretations, they also delimit what would be physically possible: the law helps define the space of physical possibilities. This metaphysics needs the same idea at a higher level: $G$ is not a law of our universe, it is what makes possible a regulated distinction between possible and impossible.

The seventh is representability. This does not mean that every world must contain minds. That would be an anthropic condition, and therefore too narrow. It means that a projection must have differences and relations determinate enough to be, in principle, modelable. Consciousness appears only when an internal substructure manages to build representations of its own world. It is not the criterion of a universe’s existence; it is a special case of internal resonance.

Noise, world, principle

Now we can distinguish three levels:

Noise is not simply complexity. A theory can be complex and still ordered; a string can be short and still meaningless for the system that receives it. Noise is a multiplicity that does not preserve form through transformations, does not allow well-defined quotients, does not distinguish legitimate variation from destruction of structure.

This distinction corrects a formula that is too quick: “God is noise.” If this means that the absolute contains all combinations without hierarchy, then the definition is almost inert. Such a God does not explain why we observe regularities, why some constants are measurable, why mathematics intercepts physical structures, why an arrow of time exists or why a consciousness can reconstruct laws. To contain everything is to select nothing. And without internal selection there is no explanatory information.

Kolmogorov complexity helps, provided it is not used too casually. Informally, it measures how short the shortest description capable of generating an object can be. An infinite set can have a short description. “All finite strings” is a short rule, even though it generates an enormous totality. Conversely, a single finite sequence may be hard to compress. So it is not enough to say that God has infinite cardinality or infinite information. The decisive question is whether its description generates structure or only indifference.

So I will use two provisional labels. The noise-God is extensional:

$$God = \Omega$$

The act-God is intensive:

$$God = G$$

In the first case God is the sum of possibilities. In the second, God is the form that makes some possibilities projectable. The first is maximal but mute. The second is not arbitrarily maximal, but it is intelligible.

Omnipotence must be rewritten

The classic question “Can God do anything?” already contains a hidden metaphysics. It presupposes a set of options available before God, like an ontological menu:

$$\Omega = \{\text{all doable things}\}$$

Then it asks whether God can choose every item on that menu. But if $G$ is more fundamental than $\Omega$, this image is inverted. There are not already constituted options first. There is the principle by which something can be an option, a law, a difference, a world.

The correct question becomes:

$$G \text{ can produce a projection that preserves nothing of } G?$$

The answer is no, but this no does not indicate an external force limiting God. It indicates identity. A representation that represents nothing of its principle is not a representation of that principle. A projection that destroys all the invariants of $G$ is not another world generated by $G$; it is a non-projection.

Therefore:

$$G \text{ can do everything that is a coherent expression of } G$$

and:

$$G \text{ cannot make a non-expression into its expression}$$

This is not an accidental limitation. It is ontological grammar. Asking $G$ to violate its own essence is like asking a structure to be itself by ceasing to be itself. It is not a missing power; it is a sentence without an object.

This idea is close to Spinoza, but it does not coincide with him. Spinoza thinks God or Nature as the single, necessary, immanent substance from which things follow according to the necessity of divine nature. The distinction between Natura naturans and Natura naturata, that is, between productive nature and produced nature, separates the productive aspect from what is produced. This theory inherits the critique of a capricious, interventionist God, but preserves a strong distinction between $G$ and every single projected universe:

$$G \neq U_i$$

and also:

$$U_i = \pi_i(G)$$

No world exhausts $G$. Every world, if it is a world, preserves a structural trace of it. This is not simple pantheism; it is closer to a structural panentheism, meaning a position in which the worlds are in the principle without coinciding with the whole principle.

Our universe is not the measure of the absolute

Our universe is a particular projection:

$$U_{\text{our}} = \pi_{\text{our}}(G)$$

From this follows an essential caution: the invariants of our physics are not automatically invariants of $G$. The value of the speed of light, Planck’s constant, the gravitational constant, particle masses, the gauge group of the Standard Model, meaning the symmetry structure of known interactions, $3+1$ dimensionality, meaning three spatial dimensions plus one temporal dimension, local cosmological history and low initial entropy are profound for us, but they may be properties of the specific quotient in which we find ourselves.

The numerical detail may be local while the abstract form is deeper. The value of $c$ belongs to our physics; the presence of a causal structure that distinguishes admissible and inadmissible dependencies is a more general candidate. The group $SU(3)\times SU(2)\times U(1)$ belongs to our theory of known interactions; the fact that physical content is often identified with what survives redundancies of description seems a more abstract trace.

The many-worlds interpretation of quantum mechanics must be handled with the same caution. The Everettian interpretation does not arise to add a collapse that creates universes, but to take the unitary evolution of the wave function seriously and explain the appearance of branches through quantum structure and decoherence. The language of “worlds” is already delicate within physics; for our metaphysics, the point is not to import many-worlds cosmology literally, but to use one lesson: apparent branching can be the internal effect of a more global structure.

So we should not say that $G$ contains worlds as a box contains objects:

$$G \neq \{U_1,U_2,U_3,\dots\}$$

Better:

$$U_i = \pi_i(G)$$

or:

$$U_i \vDash G$$

The symbol $\vDash$ should be read in a weak logical sense: $U_i$ satisfies or manifests the structure of $G$, without exhausting it. In the first case God is a warehouse of universes. In the second, God is a grammar of manifestations. A grammar does not contain sentences as already printed sheets; it contains them as possibilities generable by its own form.

Symmetries and gauge as local clues

Modern physics offers a grammar very well suited to this metaphysics: invariance. In relativity, physical content must not depend on arbitrary choices of coordinates. In gauge theories, different mathematical descriptions can represent the same physical situation. In mechanics and field theory, symmetries play a constructive role in the form of laws.

Noether’s theorem is the clearest symbol of this idea. In physical terms, it connects continuous symmetries and conserved quantities. Metaphysically, the lesson must be abstracted:

$$\text{regulated transformation} \Rightarrow \text{persistence of structure}$$

A world is not a fixed thing. It is a domain in which something can vary without everything becoming indistinguishable. Change is not the enemy of invariance; it is the place where invariance reveals itself. What remains through variation is more informative than what appears in a single frame.

Gauge theories make the point even more radical, because they place the problem of representational redundancy at the center. The philosophical point is simple: not everything that appears in the mathematics of a theory is an autonomous piece of reality. Some differences are differences of description, not physical differences.

This suggests a stratification:

$$G \to U \to \text{physical equivalence classes}$$

First there is the metaphysical projection: $G$ loses information in $U$. Then, inside $U$, physics discovers further redundancies: coordinates, gauge, representations, choices of formalism. Reality appears less and less as a list of raw objects and more and more as what remains when we remove arbitrary dependencies.

From this comes an epistemic rule:

$$\text{the more a content survives changes of representation, the more ontological weight it deserves}$$

It is not an automatic proof. It is an orientation. The fact that physics seeks invariants does not prove $G$, but it indicates that knowledge of the world already works as an archaeology of what resists transformations.

To know is to remove perspective without removing content

Scientific objectivity is not a magical view from nowhere. It is the work of removing what depends too much on an observer, an instrument, a scale, a coordinate, a community or an interest. Scientific objectivity is precisely this aspiration: to obtain statements, methods and results not dominated by particular perspectives, biases or personal interests.

In our language:

$$\text{objective} \approx \text{what remains stable across legitimate representations}$$

This formula holds at three levels. In science, a quantity is more physical if it does not depend on arbitrary coordinates. In epistemology, knowledge is more objective if it survives changes of observer and procedure. In metaphysics, a world is more than noise if it preserves invariants of the principle from which it derives.

The full chain introduces the last technical object needed here. $M_O(U)$ means “the model of the world $U$ built by observer $O$”; $\eta_O$ is not a new substance, but the perspectival filter through which that observer receives the world:

$$G \xrightarrow{\pi} U \xrightarrow{\eta_O} M_O(U)$$

$G$ is the principle of projectability. $U$ is the projected world. $M_O(U)$ is the model built by an internal observer $O$. We do not touch $G$ from outside. We do not even observe $U$ in its totality. We work with internal models and look for what remains when we improve measurements, change formalism, correct biases, compare scales and translate between languages.

Physics does this:

$$M_O(U) \to I_U$$

It seeks the internal invariants of our universe. Metaphysics attempts a riskier move:

$$I_U \to I_G?$$

It asks which physical or epistemic invariants are only local and which are traces of more general conditions of worldliness.

The useful formula separates two contributions. $\rho_\pi$ indicates the way the projection $\pi$ translates deeper invariants into our world; $L_\pi$ indicates what arises locally from this particular projection:

$\rho_\pi(I_G)$ indicates the invariants of $G$ translated into our projection. $L_\pi$ indicates the local invariants generated by the specific way $G$ is quotiented in our universe. Much of the difficulty lies in distinguishing these two terms.

Why mathematics works so well

Wigner’s old problem of the effectiveness of mathematics in the natural sciences returns with a new meaning. Wigner was not simply saying that everything is mathematics; he focused on the surprising fact that abstract mathematical structures seem to describe the physical world with a precision that was not obvious to expect.

In the framework of invariants, the mystery does not disappear, but it changes shape. Mathematics is so effective because it is the most powerful language we have for speaking about relations, symmetries, equivalence classes, transformations, conservations, global structures and limits. If a world is a coherent quotient of a deeper principle, then what we can know robustly is not its naively imagined substance, but the architecture of its invariances.

Mathematics does not describe the world because the world is a notebook of formulas. It describes the world where the world is stable structure. Elsewhere, indeed, mathematics is not enough: we need physical hypotheses, measurements, model choices, data, approximations, interpretations, experiments. Its success does not authorize a total Platonism in which every mathematical structure exists as a universe. On the contrary, this theory rejects precisely the identification between existence and mere formal consistency.

The difference is subtle but decisive:

$$\text{not every mathematical structure is a world}$$

but:

$$\text{every world must have enough structure to be modelable}$$

The first statement rejects $\Omega$. The second brings $G$ close to representability.

Consciousness as an internal reflection of projection

Consciousness should not be placed at the beginning as an anthropic filter. A world without observers can be perfectly projectable. It would lack internal self-representation, not coherence. Consciousness enters later, when substructures capable of modeling the world they inhabit emerge within a projection.

Write:

$$O \subset U$$

where $O$ is an observer or, more generally, a cognitive structure. This observer builds:

$$M_O(U)$$

an internal model of the world. When it does science, it seeks invariants of $U$. When it does metaphysics, it tries to understand whether some invariants of $U$ point back to the form of projectability itself.

Consciousness, in this theory, is not a psychological fragment of God that God must perceive in time. That would still be an anthropomorphic projection. Better to say: some projections become capable of partially representing the conditions of their own representability. Not God thinking us as a temporal mind; us, from inside a projection, thinking the principle from which the projection can be thought.

Here metaphysics becomes self-reflective without becoming mystical in the easy sense. The subject does not leave the world. It does not leap beyond its own finitude. It does something more modest and stranger: it uses the internal invariants of the projection to infer the possibility of deeper invariants.

Kant, pushed beyond Kant

The form of the argument is transcendental. Transcendental arguments move from a fact of our experience, thought or knowledge to necessary conditions that make it possible. In this sense they start from a convincing premise and seek a necessary, non-obvious presupposition.

Here, however, the ambition changes. We are not only asking:

$$\text{what conditions make our experience possible?}$$

We are asking:

$$\text{what conditions make a projected world possible?}$$

Kant imposes a discipline: do not turn the foundation into an object inside experience. This theory accepts the discipline but attempts an ontological extension. It does not limit itself to the conditions of human knowledge; it seeks the conditions of worldliness in general. It is a riskier move, and for that very reason it must remain aware of its own status: it is not physics, not revealed theology, not pure mathematics. It is formalized metaphysics.

The risk is mistaking a condition of our thinking for a condition of being. The defense cannot be absolute certainty. It can only be a strategy: push the argument toward invariants that are less and less anthropic and more and more formal. Determination, relation, compossibility, closure, modality and the well-definedness of the quotient do not depend on the existence of human beings. They seem to be conditions for any totality that wants to be more than noise.

Necessary objections

The first objection is that $G$ may be only an elegant name for the laws of logic. If by logic we mean a small list of formal principles, the answer is no. $G$ includes at least the possibility of determination and non-contradiction, but it is not exhausted by them. A projection also requires relationality, global compossibility, regulated transformability, modality, internal closure and structural representability. Logic is a minimal layer, not the whole architecture.

The second objection is that $G$ may be a mathematical structure. Perhaps, but the formulation does not require it. Saying that $G$ can be treated with symbols does not mean that it is identical to an already given mathematical object. Mathematics may be our best way of describing invariants without exhausting their ontological status. Confusing model and foundation would repeat, at a more refined level, the error of mistaking a coordinate for a thing.

The third objection is that the theory predicts nothing. In the strict physical sense, true. It does not give the value of the cosmological constant or the mass of the neutrino. But not all theories have the same task. A metaphysics does not compete with a cosmological model; it clarifies which forms of explanation make sense. Its criterion is not numerical prediction, but the ability to order concepts that otherwise collapse into one another: possible, world, law, foundation, causality, consciousness, noise.

The fourth objection is that $G$ reintroduces God under technical vocabulary. The danger exists. The answer is not to use “God” as an emotional shortcut. The word remains useful only if it is purified of three images: external object, temporal agent, capricious will. If these images return, the theory gets worse. If they are excluded, “God” becomes a historical name for the foundation of the world’s intelligibility.

The fifth objection is that such a theory may be merely aesthetic. This is also serious. Elegance is not enough. To avoid rhetoric, explicit constraints must be kept: distinguish $G$ from $\Omega$, do not call every physical law a “divine trace”, separate local and ontological invariants, do not confuse observers with criteria of existence, do not use physics as metaphorical decoration.

A minimal formulation

We can compress the whole theory into one structure. The notation in parentheses does not add new entities: it gathers in one place what has already been introduced. $D_G$ is the domain of determinations, $R_G$ the relations, $A_G$ the admissible transformations and $I_G$ the essential invariants:

$$G = (D_G, R_G, A_G, I_G)$$

where:

$$D_G = \text{domain of determinations}$$ $$R_G = \text{fundamental relations}$$ $$A_G = \text{admissible transformations}$$ $$I_G = \text{essential invariants}$$

A single projection, marked by the index $i$, is:

$$\pi_i: G \to U_i$$

and induces in the corresponding world a structure of the same kind:

$$U_i = (D_i, R_i, A_i, I_i)$$

Here $D_i,R_i,A_i,I_i$ are not new principles: they are determinations, relations, transformations and invariants as they appear inside the projected world $U_i$. The projection is legitimate if these elements are well defined through $\pi_i$. That is: if two elements are indistinguishable in the projection, the operations and relations concerning them must not generate incompatible ambiguities.

More tersely:

$$U_i \in U_G \Longleftrightarrow U_i = G/{\sim_i}$$

with:

$${\sim_i} \text{ a congruence compatible with the invariants of } G$$

This is the technical definition of a world:

“Non-trivial” means that not everything is collapsed into one undifferentiated point. “Loss of information” means that the world does not coincide with the principle. “Preserves invariants” means that it is not noise. “Relational” means that beings are stabilized by relations. “Modal” means that it distinguishes possible and impossible. “Internally closed” means that it does not need punctual external interventions to be a totality.

The invariant test

Given an invariant $J$ discovered in our universe, where $J$ is any candidate for “what remains stable” in a theory or description, we can evaluate it with a sequence of questions:

1. Does $J$ change if I change coordinates?
2. Does $J$ change if I use an equivalent physical formulation?
3. Does $J$ depend on specific details of our local law?
4. Does $J$ seem necessary for something to be describable as a world?

If it falls at the first question, it is probably an artifact of description. If it falls at the second, it may be a residue of the formalism. If it falls at the third, it is a local invariant of $U_{\text{our}}$. If it survives the fourth, it becomes a candidate trace of $G$.

Likely local:

$$c,\quad \hbar,\quad G_N$$

particle masses, coupling constants, observed dimensionality, the specific group of the Standard Model, concrete cosmological history.

Deeper candidates:

$$\text{determination}$$ $$\text{relation}$$ $$\text{global compossibility}$$ $$\text{invariance under legitimate transformations}$$ $$\text{modal structure}$$ $$\text{internal closure}$$ $$\text{representability}$$

These are not things observed next to particles. They are conditions for a set of particles, fields, events or any other local ontology to constitute a universe.

The concluding formula

The absolute should not be sought as a gigantic object outside the cosmos, nor as an infinite collection of ready-made worlds, nor as a will that chooses events in time. All these images import into the foundation categories that are valid only after projection: space, succession, choice, local causality, container, inventory.

The most coherent form is more sober. A world exists when a loss of information preserves form. Law exists when change does not erase the invariant. Knowledge exists when an internal substructure manages to remove perspective without removing content. Metaphysics exists when this search for invariants is pushed beyond the single universe and becomes a question about the conditions of projectability.

Then “God” does not name the total possible. The total possible is too easy: it includes everything and therefore does not explain the world. If the term still has force, it names the invariance that lets the finite not be chaos.

The most compact formula is:

It is not the noise from which order happens to emerge by chance. It is the principle by which order can be a projection and not an accident without reason.