1. Preamble—The Fundamental Problem of Being
Classical metaphysics has often been located in idea (Plato), substance (Aristotle), necessity (Spinoza), and the condition of possibility (Kant). But one question remains: what does it mean to belong to a world? Answering with “resemblance” or “passive inclusion” is not enough, since resembling does not imply stable coexistence or the absence of predation. Naïve inclusion confuses presence and participation: we can be there, but we cannot hold together.
The central intuition of the TUB is to reverse the presupposition: belonging is not an automatic consequence of existence; it is an active relationship. To exist in a universe is to resist the forces that decompose, without in turn decomposing. In this way, the TUB converts the language of inclusion into the grammar of balance.
2. Methodology: Arithmetic Derivation and Ontological Transposition
The method used to establish the Law of Universal Belonging (TUB) is based on a new two-step process, starting from an arithmetic truth to arrive at a metaphysical law.
2.1. The Model of Numerical Irreducibility
The analysis begins with the study of prime numbers which, by definition, are the most stable and fundamental entities in arithmetic. Their primality condition is based on the irreducibility to previous factors (the gcd (n, k#) = 1 of the Kadouno Test) (Kadouno, 2025).
2.2. Philosophical Transposition
We formalize the divisibility relationship as a relationship of destruction or structural weakening in a general system U. The concept of coprimality (gcd(x, u) =1) then becomes the principle of reciprocal irreducibility, a condition for stable coexistence.
It is this transposition that allows us to move from the mathematical fact \“number is prime\” to the ontological imperative \“being must be non-reducible and non-reductive\”.
3. Formal Definition of Universal Belonging
To belong is to be unalterable in otherness.
This sentence condenses the entire TUB.
To belong means “to coexist without destroying”, or “it is to resist the destruction of a predator without being one”.
We will therefore say:
A being x belongs to a universe U if, for any u ∈ U, the relation x/u does not produce dissolution or reduction.
It is a logic of co-stability. Belonging is not a relationship of inclusion (as in topology), but a relationship of balance: each being is the bearer of a coefficient of irreducibility that measures its capacity for peace or structural stability.
Spinoza said: “Every thing, as far as it is in it, strives to persevere in its being” (Ethics, III, prop. VI).
The TUB extends this conatus: “to persevere in one’s being is also not to dissolve that of others”.
Being is therefore not a simple persistence, but a co-persistence.
4. The Arithmetic Model: Prime Numbers and the TUB
The universe of prime numbers is a perfect illustration of this law. Each prime number exists in an infinite field of numbers, but no other prime number divides it, and it does not divide any other prime. It belongs to the world of the former, not by resemblance, but by reciprocal irreducibility. Compound numbers, on the other hand, do not really belong to this world: they can be broken down by others. Thus, it amounts to saying that the condition of belonging manifests itself in the form of an equilibrium of rank: each being, each number, is situated at a level of stability proportional to its resistance to decomposition.
Symbolic formulation of the TUB
5. Rank Balance: The Foundation of All Stability
In every universe, every being occupies a rank of equilibrium.
This rank, moreover, is not hierarchical but rather ontological: it expresses the force of stability that a being confers on the system.
The universal order does not come from a fatality, but emerges from the mutual stability between beings.
Typology of beings: unstable, unbalanced, balancing
Three types of beings can be distinguished:
1) The unstable—liable to be destroyed;
Characteristic: they disintegrate under the action of the surrounding forces.
2) The unbalancers—capable of destroying others;
Characteristic: they break the fabric of the universe’s coherence.
3) Balancing agents—neither destructive nor destructible.
Characteristic: they are the basis of universal cohesion by their inviolability alone.
Balancing beings form the stable core of the universe.
They are analogous to prime numbers: irreducible, but fertile in their associations.
Their “rank of equilibrium” does not express a moral hierarchy, but a cosmic function, that of stabilizing without dominating.
Heidegger, in Being and Time, wrote:
“To be-in-the-world is always to be-with-others.”
But being with is not enough. The TUB adds: it is still necessary that this being-with does not imply mutual destruction.
Ontological authenticity is not only the recognition of the other, but the reciprocal stability of being-with.
The TUB is not limited to the recognition of the other, but embraces the ethical principle that applies to all, for the balance of the whole in a U universe.
Concept Table
Arithmetic concept |
Ontological Interpretation |
Division |
Structural destruction |
Coprimality |
Reciprocal irreducibility |
Primality |
Ontological autonomy |
K (Universe of the product of the first) |
Element Interaction Field |
Definition of terms:
1-a) Define Ontological “Decomposition”:
Ontological decomposition (or destruction) is the act by which one being denies the constitutive irreducibility of another being. It occurs when the identity of one being is reduced to its mere function or usefulness to another, transforming it from an end in itself into a means.
1-b) Irreducibility:
Irreducibility is the condition of a being that resists the forces that decompose it or that resists the destruction of a predator. It is the ability of a being to maintain its structural identity.
Reciprocal irreducibility is the principle of stable coexistence, formalized by coprimality (gcd(x, u) = 1), meaning that no logic of reduction operates between beings x and u.
1-c) Ontological Autonomy:
Ontological autonomy is the state of a being that is its own foundation. It is a condition of being that is analogous to primality in arithmetic.
A being that possesses ontological autonomy is indivisible and does not depend on the destruction of another to survive.
1-d) Interaction field of the elements:
The Field of Interaction of the Elements (the Universe U) is the system where beings coexist.
It is the ontological analogue of the K# (Universe of the product of the first) in arithmetic.
Analogy analysis
a) The Ontological Analogy of the pgcd Function:
In the symbolic formulation of TAU, gcd (x, u) = 1 is a mathematical metaphor to signify the absence of common dissolution factors.
The ontological analogue of the pgcd (Greatest Common Divisor) is the “force of shared dissolution” or the “factor of mutual reducibility”.
A number N is reducible if it shares a factor with another, because that factor decomposes N into smaller units.
Two beings x and u have a gcd > 1 when one possesses a destructive power (dx) that the other cannot absorb or counter (du), or when both are vulnerable to the same external factor, creating a bond of mutually destructive dependence.
The gcd = 1 means that no reduction logic operates between x and u. They are “co-prime” in that their coexistence does not lead to mutual decomposition.
For example, for two political systems, a gcd > 1 could represent a common principle of domination (such as militarism or imperialism) that allows one to reduce the autonomy of the other.
b) Arithmetic-metaphysical analogy:
The philosophical justification for this transposition is based on the concept of Structural Primacy or Formal Transposition.
The Model of Stability: Mathematical structures, especially arithmetic, are considered the purest and most stable expression of the relationship. The principle of primacy (irreducibility) is chosen because it embodies the very idea of stable autonomy in the formal realm.
Form is Law: The approach is not to say that beings are numbers, but that the condition of irreducibility in the formal world is the condition of stability in the real world. The choice of coprimality is therefore an operation of formalization that raises a truth about the stability of numbers (they are maintained without dividing) to the rank of a law of equilibrium for beings.
Platonic heritage: This recourse to formal structures to found ontology is part of an ancient philosophical heritage, where the stability of reality is often sought in Ideas or mathematical relations (Pythagoras, Plato), considered to be purer and more permanent than the phenomenal world. The TAU only actualizes this principle by identifying the arithmetic law of non-division as the metaphysical principle of co-stability.
6. The Ethics of Co-Stability
The Theory of Universal Belonging (TUB) is the basis of a new ethic:
The good is no longer what an authority commands, nor what custom dictates, but what reinforces the stability of the whole, that is to say, what sustains the coherence of the universe without ever destroying its constituent elements.
Evil, on the other hand, is that which dissolves the bonds of stability, which weakens coexistence by transforming difference into the right of predation or domination.
The TUB replaces the morality of duty with a morality of non-reduction.
Kantian duty commands the universality of the moral law; the TUB commands the universality of stability.
The good cannot be a categorical imperative, but an ontological invariant: the right act is the one that does not destroy the other.
To act morally is therefore to preserve the balance of irreducibility.
This conception finds a major echo in Jean-Jacques Rousseau, in On the Social Contract (I, 6):
“To find a form of association which defends and protects with all the common force the person and property of each partner, and by which each one, uniting with all, nevertheless obeys only himself.”
Rousseau was looking for the foundation of a political equilibrium in which the union of wills does not jeopardize individual freedom.
The TUB takes up this principle but transposes it to the ontological level:
Just as the Rousseauist citizen remains free while participating in the general will, so each being in the universe remains irreducible while contributing to the cohesion of the whole.
The social contract is shown here as “an ontological contract of non-destruction”.
The ethics of co-stability thus broaden the scope of the contract: it binds not only individuals to each other, but also forms of being in their diversity (human, natural, technological, spiritual, etc.).
It affirms that there is a universal contract, tacit but real, the first clause of which is the following: No being must maintain itself at the price of the dissolution of another.
This is the fundamental law of co-stability, the principle of balance between the powers of existence.
Here we can invoke Spinoza, for whom “desire is the very essence of man, insofar as it is conceived as determined to do what increases his power to act” (Ethics, III, prop. IX).
But where Spinoza defined virtue as the power to exist, the TUB extends its meaning: true power is not to grow alone, but to subsist with others.
Virtue is therefore the power to exist without destroying the power of existence of others.
Finally, Heidegger sheds a new ontological light on this ethics.
In Being and Time, he writes that human existence is always a being-with (Mitsein).
The TUB redefines this idea: being-with is not enough; you have to be-with-without-destroying.
In other words, true being is being-with-stability, and morality consists in maintaining this stability in the presence of the other.
By extension, an ethics without transcendence, but universal, is emerging, an ethics immanent in the very structure of being.
The good is not an imposed value, but the necessary consequence of co-belonging.
To live according to the TUB is to recognize that the survival of the whole depends on the non-dissolution of the parties.
It is to be understood that freedom, justice, and peace are not contingent social contracts, but substantial conditions of ontological existence.
The ethics of co-stability then become the matrix of all morality, all law, and all true politics:
It makes reciprocal irreducibility not only a rule of coexistence, but the constitutive principle of reality.
7. The Political Dimension: From the Social Contract to Structural Peace
The Social Contract establishes the balance between individual freedom and the general will.
Similarly, the TUB seeks a balance between individual irreducibility and universal stability.
It provides a metaphysics of politics: the right system is the one in which the elements (individuals, institutions, peoples) are co-primary: autonomous, but compatible.
Thus:
True sovereignty is not expressed by domination, but by coexistence without destructive dependence.
And a stable state is not one that absorbs its parts, but one that allows them to be irreducible while participating in the general equilibrium.
Rousseau writes: “Sovereignty, being only the exercise of the general will, can never be alienated” (Social Contract, II, 1).
Similarly, through the TUB, stability cannot be transferred: it is exercised collectively or disappears.
8. Discussions
a) “Everything interacts: how can we speak of non-destruction?”
If everything is in permanent interaction, how can we demand “non-destruction”? Is it not naïve, even contradictory, to speak of coexistence without alteration when all becoming is transformation?
The TUB responds that interaction is not the problem; the problem is predation.
A clear distinction must be made between:
1-interaction, which is movement, the exchange of energy, influence, and information;
2-predation, which is the disintegration or reduction of one being by another, i.e. its reduction (the reduction of being) to a function, or its dissolution into a “resource” (the means, in the Kantian concept).
To say that two beings can coexist without destruction does not mean that they do not change each other: it means that neither denies the ontological identity of the other.
It is possible to transform the other without abolishing it.
This point is already in germ in Spinoza, for whom every being “strives to persevere in its being” (Ethics, III, prop. VI)—which implies that the encounter with the other must not annihilate this perseverance.
In terms of the TUB:
Two beings can touch, respond to each other, and shape each other, without ever reciprocally reducing each other to a simple means.
What is rejected is not mutual interaction (inevitable), but ontological absorption: the fact of using the other as material without acknowledging their own irreducibility.
On this point, the TUB is more demanding than the simple morality of tolerance. Tolerance accepts others. The TUB demands better: it requires that the other remain other after the interaction.
The real ontological crime is not to influence; it is to dissolve.
This is why co-stability is not a dead or immobile peace, nor peace by absence of contact; it is peace by recognized reciprocal resistance.
b) “What about vulnerability?”
What do we do with vulnerable, fragile, unstable beings?
If only those beings who cannot be destroyed belong fully to the stable universe, does this mean that the weak, the suffering, the dominated, and the crushed peoples would be excluded?
Answer: No.
The TUB introduces a notion that is intended to be essential: gradual belonging.
An unstable being is not denied in its existence; it is simply in a precarious state of belonging.
He is in the universe, but his link to the common fabric is fragile. His mode of existence is all the more threatened, either because it can be destroyed, or because it risks, through imbalance, distress, or reaction, destroying in turn.
As a result, the ethical task is not to reject it, nor to dominate it, nor to neutralize it.
The ethical task is to raise it to co-stability, that is, to allow it to reach the state in which it can exist without being destroyed and without becoming destructive.
It is here that Rousseau’s approach becomes, to weigh the words carefully, decisive and takes on its full meaning.
In On the Social Contract, Book I, Chapter 8, Rousseau reminds us that the man who is to be made a citizen must be formed, transformed:
He does not exist spontaneously as the bearer of the general will; he becomes so (Rousseau, 1762).
In the same way, the vulnerable being is not born stable. He becomes one.
Therefore:
Solidarity, in the TUB, is not reduced to assistance that keeps the other in his or her dependence; it is an ontological stabilization operation.
This is a major difference from the classic forms of charity or even humanitarian aid:
Charity keeps us alive.
This point also has political significance:
A just society is not one that eternally protects the weak as well as the weak, but one that actively works to make them balanced beings, beings capable of not being destroyed, but also of not destroying.
In this sense, we are shaping bearers of stability, and not perpetual or permanent welfare recipients.
c) “Spinoza and Monism”
Spinoza affirms the absolute unity of being: “All that is, is in God” (Spinoza, 1677). There is no possible “outside”.
So how can the TUB say that some beings “belong” to the universe, and others do not? Is this not a betrayal of Spinozist monism?
Answer: The TUB does not negatively monism; it refines it.
In Spinoza, everything that exists proceeds from the same infinite substance—Deus sive Natura.
But Spinoza also recognizes that not all modes (i.e., particular beings) have the same power to exist, nor the same capacity to persevere.
They do not have the same degree of reality. (Ethics, V, prop. XL).
The TUB takes hold of this difference in degree and raises it to the rank of a criterion of membership:
Yes, everything is “in” the universe in Spinoza’s sense (nothing is outside Nature).
But no, not everything contributes equally to the stabilization of this universe.
Membership is therefore not given in advance; it is not an automatic right; it is won through co-stability.
One only fully belongs by becoming irreducible without being predatory.
Spinoza’s God, in this reading, is not reduced to the unity of substance; it is the name of the system of cooperating irreducibilities.
That is to say: what Spinoza called \“God\”, the TUB interprets as the structure of a universe where each stable being supports the others without destroying them.
The divine becomes the balance of the powers of being.
d) “Heidegger and being-with”
Heidegger has already said that the human being is “being-with” others (Mitsein). The TUB thus only repeats Heidegger’s ontology of co-presence.
Answer: The TUB is more extensive than Heidegger’s.
Heidegger, in Being and Time, describes the fundamental structure of Dasein:
“To be-in-the-world is always to be-with-others.” (Heidegger, 1927)
In other words, human existence is never purely individual; it is always already exposed to the other, immersed in a shared world.
More simply: The human being is never alone in his being; he always exists within a network of co-presence.
The TUB recognizes this analysis, but identifies a universal principle from it.
Heidegger’s “being-with” is no longer limited to human Dasein: it is a cosmic law applicable to any form of being.
The TUB is therefore not limited to describing being-with; it norms it.
Being-with is not enough if it is not also being-with-stable.
Shared existence must be governed by mutual non-destruction.
In other words, being-with is right when it guarantees the possibility of the being of the other—when it ensures co-stability in presence.
Consequently, Heidegger’s Mitsein finds its ethical significance in the TUB: co-presence becomes an ontological obligation.
Living with others, in this context, means preserving one’s existence, not consuming or dominating it.
This leads us to conclude that:
The universe is not only a set of beings present together, but a system of stabilizing presences.
Development of “Progressive Belonging”:
The notion of progressive belonging is compatible with the formal definition by distinguishing the condition of the system from the property of being. Coprimality is an ideal condition of stability, while progressive belonging describes the dynamic reality of being in the system.
The formula gcd (x, u) = 1 describes the maximum and stable state of belonging—the ethical and ontological goal. This state defines stabilizing beings.
Real Progression: A vulnerable or unstable being is not excluded from the system, but their relationship with others does not yet respect the ideal gcd = 1. It is in a state where:
Either another being can decompose it (it is reducible), meaning gcd (x, u) > 1 for a certain u.
Either it risks becoming destructive in reaction to its own instability.
The Ethics of TUB: Progressive belonging is the path by which the vulnerable being is raised to the state where its pgcd with all other stabilizing beings approaches 1.
Ethical action (solidarity) is therefore the operation aimed at reducing the factors of dissolution that x shares with U, until it reaches reciprocal irreducibility.
9. The TUB in the Contemporary Context
To situate the TUB, we can confront it with the currents of Relational Ontology and the New Political Pluralism.
Relational ontology (e.g., Levinas, Ricoeur): These philosophies place the relationship with the Other at the center of being.
Distinction: Although TUB is relational, it is more prescriptive. Levinas and Ricoeur emphasize the recognition of otherness and responsibility. The TUB goes further: the responsibility is not only to recognize the Other, but to guarantee its structural stability (its non-reduction). The ethics of the TUB is an imperative of co-stability, not just a face-to-face duty (Lévinas, 1961; Ricœur, 1990).
Contemporary political pluralism (e.g., Chantal Mouffe, post-structuralist theories): These theories seek to manage agonism and heterogeneity in the public sphere.
Distinction: They deal with coexistence through conflict management or procedural frameworks. The TUB provides a metaphysical foundation for this pluralism: the autonomy of the parties (irreducibility) is not a political contingency to be managed, but the ontological condition for the stability of the system (Mouffe, 2014).
10. Application of the TUB in the Real World
Example 1: A multicultural society
Let us take the example of a Multicultural Society (U).
1) Stabilizing Beings (Primes): These are the strong, self-sustaining cultural identities that, while interacting, do not break down or try to break down others. They maintain their irreducible structures of meaning. (Ex: a language, a religion, a tradition that cannot be assimilated, but not dominating.)
2) Decomposition (gcd > 1): Interaction becomes decomposition when the state or a dominant culture imposes a single norm (the common factor) that reduces other cultures to folklore (a function) or forces them to assimilate (dissolution).
Progressive Belonging: Fragile minority cultures (vulnerable beings) are in a precarious state of belonging.
Ethical action (the policy of the TUB) is not mere tolerance, but active investment (education, law) to strengthen their internal structures, so that they themselves become irreducible entities that contribute to global cohesion without being destroyed or becoming destructive by reaction.
The goal is to achieve a structural peace based on the co-primality of identities.
Example 2:
Let us think of the international system as the universe U, where the beings are nation-states, non-governmental organizations (NGOs), and fundamental treaties. The goal is to achieve a stable global peace.
1) The Stabilizing Being (The Irreducible State)
A stabilizing state is one that possesses irreducible sovereignty and contributes to global stability without seeking to break up others.
Irreducibility: Its sovereignty is non-negotiable (it resists division or absorption by superpowers) and its internal political regime is stable enough not to collapse and create a destructive vacuum or chaos. The state is a “prime” because it cannot be reduced to a protectorate or a simple functional ally without upsetting the balance of the region.
Co-stability: It is in a relationship of gcd = 1 with its neighbors because it respects the territorial and political integrity of the latter, not imposing a common factor of domination.
2) Ontological decomposition (imperialism): GCD > 1
Decomposition in international relations is the act of denying the sovereignty of the other, of reducing them to a zone of influence or a simple source of resources.
Forced decomposition: A weak state sees its sovereignty decomposed when it is reduced to an economic colony or a client state. The international agreement or the aid it receives contains a common factor of dissolution (gcd > 1), allowing the dominant power to control it (“divide” it) and deny its essential political autonomy.
Active Decomposition (Predation): Imperialism or annexation is the most direct form of decomposition. The aggressor state (the “destroyer”) seeks to abolish the structural identity of the neighboring state, reducing it to a mere province.
Fragile peace occurs when common factors (such as a mutual defense treaty with a subordination clause) bind states, but do not achieve the reciprocal irreducibility of ideal co-stability.
Progressive Membership and the Law of Co-Stability:
Progressive Membership applies to new states or states in crisis (fragile or post-conflict states).
The Vulnerable Being: A state emerging from a civil war has not yet reached the state of co-stability; it is vulnerable to interference and internal dissolution. Its membership in the international system is precarious.
The Ethical Objective (Ontological Stabilization): The intervention of International Law based on the TUB must not be limited to humanitarian aid (simple assistance), but must be an ontological stabilization operation.
Action: Aid should be aimed at strengthening the internal institutions (justice, administration, economy) of the vulnerable state, helping it to develop its resilience against internal factors of division and external factors of domination.
Outcome: The goal is to elevate the vulnerable state to the status of a stabilizing Being that, by its mere autonomous presence, contributes to the non-reduction of the entire international system.
The TUB thus offers a metaphysical foundation for International Law: its ultimate function is to be the guarantor of the co-stability of nations.
Explanatory summary:
The four main objections show that the TUB ignores neither the dynamics of reality, nor vulnerability, nor the metaphysical tradition; it integrates them all into a higher conception of being as an active equilibrium.
1) Interaction is allowed, but predation is prohibited.
2) Vulnerability is recognized, but it calls for an elevation toward stability.
3) Spinoza’s monism is preserved but enriched by a functional hierarchy of belongings.
4) Heidegger’s being-with is accepted, but extended into stable-being-with.
In short, the TUB does not oppose the great philosophical systems; it extends them and confers on them a common law, that of reciprocal non-destruction, which becomes the condition of possibility of any lasting universe, whether physical, ethical, or political.
11. Conclusion
The Theory of Universal Belonging (TUB) redefines the metaphysics of being on a principle of active equilibrium.
It shows that the substance of the real is not an immobile given, but a harmonic tension between forces that maintain each other without canceling each other out.
By deriving the condition of existence from the concept of arithmetic irreducibility, the TUB elevates coprimality to the rank of an ontological law: true being is not confused, does not divide, does not destroy itself; it persists in harmonic resistance with all that is.
So, presence is not a simple spatial fact of “being there”, as in classical ontology.
It is an act of balance: “to be with without destroying.”
Existence is now defined by three fundamental requirements, which constitute the Ontological Trinity of Belonging:
1) Be your own foundation:
To exist is to be indivisible, like a prime number in the universe of numbers.
The being that merges within itself possesses the internal stability necessary for duration. It does not depend on the destruction of another to survive.
It is, according to the Spinozist formula, the cause of itself (causa sui), and not the product of a reductive otherness.
2) Allow the being of others:
True power is not to overcome, but to let be.
Each stable entity must guarantee the stability of the others by not breaking them down.
Where force seeks to impose itself, stability seeks to coexist.
Thus, freedom no longer consists in acting without hindrance, but in existing without destroying, in becoming the guardian of the existence of others.
3) Contributes to the permanence of the universe:
The universe, from the perspective of the TUB, is not a homogeneous whole, but a symphony of irreducibilities in balance.
Each being that reaches co-stability strengthens the overall structure.
To exist fully is to participate in this structural harmony, where diversity does not lead to dislocation, but to coherence.
The universe is not maintained by inertia, but by the active harmony of its irreducibilities.
Thus, every stable being becomes the first of the universe to which it belongs: it is not just an element of the whole, but a pillar of balance that sustains the whole.
Primality, in this expanded sense, is no longer a mathematical property, but a spiritual and metaphysical state: it designates the capacity of a being to persist without dominating and to last without devouring.
From this conception flows an ethics and a policy of universal stability.
What arithmetic expresses by co-primality, the TUB raises it to the level of the right to be:
“It is just that which does not destroy. It is stable which does not reduce. It is great what allows permanence.”
The history of ideas had hitherto sought being in substance, consciousness, or will.
The TUB teaches that it is now necessary to look for it in the relationship of equilibrium.
And it is only in the harmony of irreducibility that a truth is discovered:
“Being does not triumph over nothingness by force, but by the stability it shares.”