u/Inevitable-Sea6754

I am studying a deterministic structure based on the gaps between consecutive prime numbers.

The observation that started this research is the following sequence:

89 -> 97 -> 101 -> 103

with gaps:

8, 4, 2.

For each prime p and its following gap g, I define S as the cumulative sum of the previous prime gaps. In this construction, S = p + g - 2. I then define V = S + g.

The interesting phenomenon is not simply that an arithmetic operation sometimes produces another prime. The interesting point is that different positions in the prime-gap construction can produce exactly the same future value V before that value itself appears in the sequence of primes.

The clearest example is 103.

When we reach the prime 97, the following gap is 4.

So:

S = 97 + 4 - 2 = 99

and:

V = 99 + 4 = 103

Therefore, while the current prime is still 97, the future value 103 has already been determined by the construction, even though 103 has not yet appeared as a prime.

The previous row produces exactly the same value:

S = 89 + 8 - 2 = 95

and:

V = 95 + 8 = 103

So we have two different rows producing the same value:

95 + 8 = 103

99 + 4 = 103

The prime sequence then continues:

97 -> 101 -> 103

and the third representation is:

101 + 2 = 103

The same mechanism can also produce a composite value. For example:

395 + 8 = 403

399 + 4 = 403

but:

403 = 13 x 31

So the phenomenon is not simply a way of generating primes. The construction can determine a future integer before that integer appears in the prime sequence, whether it eventually turns out to be prime or composite.

For consecutive rows producing the same value V, a simple relation appears: the next prime gap is half the previous one.

This gives structures such as:

8, 4, 2

16, 8, 4, 2

32, 16, 8, 4, 2

My question is:

Is there a known number-theoretic explanation for this phenomenon?

In particular, is there a known result explaining why different trajectories constructed from cumulative prime gaps can converge to exactly the same future value, and why consecutive convergences produce this repeated halving of the gaps?

I am mainly interested in the mathematical mechanism behind the convergence and the advance determination of the future value, rather than in claiming that every announced value must be prime.

Estoy investigando una estructura determinista construida a partir de los huecos entre números primos consecutivos.

El ejemplo que originó la investigación es:

89 → 97 → 101 → 103

con huecos 8, 4, 2.

Para cada primo p y su hueco g, construyo:

S = p + g − 2

y después:

V = S + g.

En el caso de 103, cuando todavía estamos en el primo 97:

S = 97 + 4 − 2 = 99

V = 99 + 4 = 103

Por tanto, el valor 103 queda determinado cuando todavía estamos en 97, antes de que 103 aparezca en la sucesión de primos.

La fila anterior produce el mismo valor:

89 + 8 − 2 = 95

95 + 8 = 103

Después la trayectoria continúa:

97 → 101 → 103

El mismo mecanismo también puede anunciar valores compuestos. Por ejemplo:

395 + 8 = 403

399 + 4 = 403

pero 403 = 13 × 31.

Lo que me interesa no es simplemente que una operación produzca a veces un primo. Lo interesante es que distintas posiciones construidas a partir de los huecos convergen en el mismo valor futuro antes de que ese valor aparezca.

En las cadenas consecutivas aparece además una relación:

8, 4, 2

16, 8, 4, 2

32, 16, 8, 4, 2

y algebraicamente, cuando el mismo V se mantiene entre posiciones consecutivas, el siguiente hueco es la mitad del anterior.

¿Alguien conoce una explicación o resultado de teoría de números relacionado con este tipo de convergencias de trayectorias construidas a partir de huecos primos?