On the dynamic invariant of $6n \pm 1$ twin-track arithmetic lattice and its consecutive prime structures

I am an independent researcher investigating the arithmetic and structural properties of prime distributions formulated within the twin-track lattice of $6n \pm 1$.

I would like to inquire about a potential algebraic and geometric invariant regarding Goldbach pairs. Consider the following model representing the relationship between a base prime, a directional displacement variable, and the intervals of the resulting prime pairs:

$$2P_{\text{base}} + C = K - (6 + 2n) + K = S_{\text{prime\_even}}$$

Where:

• $P_{\text{base}}$ is a defined base prime.
• $C$ is the shift invariant, formulated as $C = 2m$, where $m \in \mathbb{Z}$ and $2P_{\text{base}} + C \neq 0$. This constraint allows $C$ to be any even integer (including zero and negative values), ensuring the directional symmetry within the twin-track lattice remains fully flexible while guaranteeing that the resulting even sum never collapses to zero.
• $K$ represents the larger prime in the targeted pair.
• $K - (6 + 2n)$ represents the smaller prime in the pair, where $n \in \mathbb{Z}$.
• $S_{\text{prime\_even}}$ is the resulting sum of the prime pair.

Geometric Vantage Point & Prime Progressions

The core insight of this framework is that the shift invariant $C$ acts as a dynamic geometric vantage point (a translational camera view). By shifting the coordinate reference point via $C$ to a chosen base prime ($P_{\text{base}}$), any valid prime pair assigned to that even sum structurally aligns under two distinct geometric configurations relative to the local prime sequence:

1. Symmetric Median Structure (The $\{-1, 0, 1\}$ Alignment): The base prime $P_{\text{base}}$ acts as the exact local prime median ($0$), where the resulting prime pair radiates symmetrically outward, occupying the equidistant adjacent lattice positions (e.g., $P_{\text{median}-1}$ and $P_{\text{median}+1}$).
2. Consecutive Boundary Structure (The $\{0, 1, 2\}$ Alignment): The base prime $P_{\text{base}}$ acts as the absolute boundary zero ($0$), causing the target prime pair to align precisely with the subsequent consecutive prime positions (e.g., $P_{\text{base}+1}$ and $P_{\text{base}+2}$), mapping a localized triad of consecutive primes.

Is this particular lattice-shift invariance, median symmetry, or consecutive structural property already explored within modern sieve theory or additive number theory? I would highly appreciate any guidance toward relevant literature, existing frameworks, or formal methods that address such dynamic structural intervals.