Diophantine Stability and Adelic Obstructions in Reciprocal Sums of Arbitrary Subsets: A Generalized Solution to Erdös Problem #257

This paper establishes the unconditional irrationality of the series $S_A = \sum_{n \in A} (b^n - 1)^{-1}$ for any infinite subset $A \subseteq \mathbb{N}$, resolving a generalized variation of Erd\H{o}s Problem \#257. The proof is structured in two fundamental stages, addressing first the arithmetic stability of the digital expansion and second the structural impossibility of periodicity in the general setting. First, for sets of pairwise coprime integers, we formalize the asymptotic expansion via a \textbf{recursive period construction}. We control the propagation of arithmetic carries by mapping structural carry interference into a specific \textbf{Diophantine Region}. We derive a \textbf{Deterministic Density Bound} based on the distribution of solution lattices, proving that the density of structurally compromised blocks vanishes asymptotically. This ensures the \textbf{Diophantine Stability} of the integer block representation, guaranteeing that the vast majority of the digital sequence remains free from arithmetic interference. Second, we extend the result to arbitrary sets via a \textbf{Smoothness Shell Filtration} (a stratification based on arithmetic friability). To overcome the absence of global coprimality, we construct an \textbf{Adelic Obstruction} by identifying a ``Witness'' term $n^*$ with maximal $p$-adic valuation in each shell. Utilizing a \textbf{CRT-variant projection} (Chinese Remainder Theorem) onto the local component $\mathbb{Z}/p_k^e\mathbb{Z}$, we demonstrate that the witness induces a strict \textbf{Phase Shift} relative to the background period. This shift forces the \textbf{existential distinctness} of the digital blocks, creating a topological obstruction in the profinite limit $\hat{\mathbb{Z}}$ that precludes rationality.