MODULAR OBSTRUCTIONS AND CONSTANT DENSITY FAILURE PHENOMENA IN ARITHMETIC PROGRESSIONS OF SHIFTED GOLDBACH PRIMES.

We investigate the behavior of additive transformations applied to primes arising in representations associated with the Goldbach Conjecture. For an even integer define the Goldbach set where denotes the set of prime numbers. Each corresponds to a Goldbach decomposition with Fix the shift transformation which induces a mapping on even integers. This transformation defines a structural propagation mechanism between Goldbach decompositions of consecutive even numbers. An even integer is said to satisfy the structural propagation property if there exists such that both and are prime, yielding a valid Goldbach decomposition of Conversely, is defined as a failure case if no such propagation occurs, that is We restrict attention to arithmetic progressions of even integers of the form with and analyze each progression independently. A comprehensive computational study over the first even integers reveals that structural propagation holds in a substantial proportion of cases but systematic and reproducible failure occurs for specific values of The number of observed failure cases within this range is given by We establish that all observed failures are governed by a modular obstruction modulo . Since , the shift transform induces If all primes satisfy then forcing to be composite. This modular obstruction condition implies that no Goldbach decomposition of propagates to and hence is necessarily a failure case. Empirical analysis confirms that every observed failure satisfies this condition with no counterexamples in the tested range. In the case the failure set exhibits stronger structural property. All failures lie in a single arithmetic progression yielding a constant spacing of This reflects a simultaneous alignment of modular constraints modulo demonstrating that the failure set collapses into a single residue class modulo For other values of , the failure sets remain modularly constrained but decompose into multiple classes resulting in irregular spacing. We identify a structural persistence phenomenon associated with failure cases in the progression. For every failure value tested up to , the intersection is non empty, indicating that at least one prime is shared between the Goldbach sets of consecutive even integers separated by the shift. Frequency analysis of these intersections reveals recurrent primes including small primes such as and as well as larger primes appearing with high multiplicity. A further striking phenomenon is observed in the case A large scale computational analysis up to reveals that the proportion of failure cases stabilizes at a constant density of approximately across all tested intervals. This density remains invariant under scaling and does not exhibit decay, fluctuation or drift. The stability of this ratio indicates that the failure set is not governed by probabilistic fluctuations in the distribution of primes but instead from an underlying deterministic residue class mechanism. This provides strong evidence that the observed failure structure is a consequence of modular synchronization rather than random sparsity. The results demonstrate that additive transformations of Goldbach primes are governed by a layered modular structure in which residue class constraints determine the success or failure of structural propagation. The emergence of arithmetic regularity in specific cases together with persistent overlap in Goldbach sets suggests deeper structural interdependencies in the distribution of Goldbach representations.