Within the domain of high-performance interconnection architectures, ensuring reliability emerges as a critical priority, particularly given the escalating susceptibility of components in expanding networks. Resilience, in this context, stands as a pivotal consideration, with conditional connectivity under distinct fault patterns serving as vital evaluation criteria. The present study develops a foundational theoretical scheme for dual-attribute fault tolerance within the augmented Half Hypercube architecture. Furthermore, we establish rigorous bounds for characterizations on the 1-extra r-component connectivity through combinatorial proofs: (1) For r = 3, we prove \(ECC^{1}_{3}(HH_{n})=4\lceil {\frac{n}{2}} \rceil -4\) where \(n\ge 7\) ; (2) For r = 4, we derive \(ECC^{1}_{4}(HH_{n})=6\lceil {\frac{n}{2}} \rceil -10\) where \(n\ge 11\) , revealing \(HH_n\) ’s resilience scales linearly with dimension n.