This research is based on an alternative ontological hypothesis compared to the traditional view of numbers: I hypothesize that prime numbers constitute the only fundamental numerical entities, which I define as Promoters, and that composite numbers are structural effects that emerge from the combinatorial interaction of the Promoters themselves. This perspective suggests that the nature of numbers is intrinsically dynamic. They do not "exist" in a fixed sequence, but "emerge" according to an order of constructive necessity. A direct consequence of this hypothesis is that the linear order with which we are accustomed to counting (n → n+1) represents only one of the possible readings of this structure, but not necessarily the fundamental order of its generation. To investigate this dynamic vision, I developed the P System, a theoretical framework and symbolic language whose purpose is to make explicit the grammatical rules that govern the generation of numbers.
The P Symbolic Language
A New Alphabet for a New Grammar
To overcome a purely quantitative view, I developed a formal language to represent the structure and genealogy of numbers. The basic elements of this language are:
Symbols (πk): Each prime Promoter is represented by a symbol. π₁ is the first promoter of the system, π₂ the second, and so on. They are the irreducible elements of the alphabet.
Operators (⊗, ↑): To describe interactions, I defined two fundamental grammatical operators. Composition (⊗) describes the interaction between different genealogical families (e.g., the structure of 6 is ⦅π₁ ⊗ π₂⦆). Self-Interaction (↑) describes a promoter interacting with its own family (e.g., the structure of 4 is ⦅π₁↑2⦆).
The Morphogenetic Signature: The Metric of Structure
To measure and classify these symbolic structures objectively, I developed the Morphogenetic Signature, a function φ that maps each symbol to a vector of 5 qualitative parameters:
This signature reveals non-obvious structural connections. For example, the numbers 6 and 8:
6 (symbol ⦅π₁ ⊗ π₂⦆) has signature (δ=2, ν=2, φ=1, ρ=-1, ω=2). It is a "flat" structure (δ=2) but "wide" (ν=2), born from the interaction of two distinct families.
8 (symbol ⦅π₁↑3⦆) has signature (δ=3, ν=1, φ=3, ρ=2, ω=1). It is a "deep" structure (δ=3) but "narrow" (ν=1), generated entirely by the χ₁ family.
Although numerically close, they are qualitatively very different entities.
A New Definition of Prime Number
The P System offers a new definition of primality, not based on divisibility, but on grammatical constructibility.
πₙ ∉ CLOSURE(π₁, ..., πₙ₋₁) ∧ φ₅(πₙ) = (1, 1, 1, 0, 1)
Where CLOSURE represents the set of all symbols constructible through the operators of the P System from the preceding promoters. A prime, therefore, is a necessary emergence, a symbol that the system is forced to introduce when it encounters a constructive gap that it cannot fill with the elements at its disposal.