Why can a prime-gap trajectory converge to a future value before that value appears as a prime?

I am studying a deterministic construction based on the consecutive gaps between primes.

The phenomenon I am interested in is NOT that arithmetic on primes sometimes produces another prime. The interesting point is that the same future value can be generated by different rows of a cumulative prime-gap construction before that value itself appears in the prime sequence.

Let $p_n$ be the $n$-th prime, and let

$$ g_n=p_n-p_{n-1}. $$

Define the cumulative gap sum

$$ S_n=\sum_{i=2}^{n} g_i=p_n-2, $$

and then define

$$ V_n=S_n+g_n=p_n+2g_n-2. $$

A convergence occurs when two different rows produce the same value $V$.

For example, consider $V=103$. The relevant prime-gap sequence is

$$ 89\xrightarrow{8}97\xrightarrow{4}101\xrightarrow{2}103. $$

The corresponding rows give

$$ 95+8=103 $$

and

$$ 99+4=103. $$

The important chronological point is that the second occurrence of $V=103$ happens when the current prime is $97$.

At that moment, $103$ has not yet appeared in the prime sequence. Nevertheless, the construction has already determined the future value $V=103$. The actual sequence then continues

$$ 97\rightarrow101\rightarrow103. $$

So the value $103$ is determined before the prime $103$ itself appears.

The same construction can also announce a composite value. For example,

$$ 395+8=403, \qquad 399+4=403, $$

while

$$ 403=13\cdot31. $$

Thus the phenomenon is not simply that an arithmetic operation happens to produce a prime. The construction can determine a future integer $V$ before $V$ appears in the prime sequence, whether $V$ eventually turns out to be prime or composite.

There is also a simple algebraic relation behind consecutive convergences. If

$$ V_n=V_{n+1}, $$

then

$$ p_n+2g_n-2=p_{n+1}+2g_{n+1}-2. $$

Since $p_{n+1}=p_n+g_{n+1}$, this gives

$$ g_n=2g_{n+1}. $$

This explains the halving structure $8,4,2$ in the $103$ example.

My question is:

Is there a known number-theoretic interpretation or theorem describing this phenomenon?

In particular, when the same value $V$ is produced on consecutive rows, is there a known result explaining why the corresponding prime gaps must form a halving chain such as

$$ 8,4,2, $$

$$ 16,8,4,2, $$

or

$$ 32,16,8,4,2? $$

More specifically, what mathematical structure determines the fact that the repeated value $V$ is already fixed before the prime $V$ itself appears?