We present three algorithms with formal correctness guarantees and complexity bounds for the problem of selecting a diverse, multi-locale set of sources from ranked search results. First, we formulate weighted locale allocation as a constrained integer partition problem and give an $O(n \log n)$ algorithm that simultaneously satisfies minimum-representation, budget-exhaustion, and proportionality-bound constraints; we prove all three hold with a tight deviation bound of $< 1$. Second, we define a cascaded country-code inference function as a deterministic priority chain over heterogeneous signals (TLD structure, model-inferred metadata, language fallback) and prove it satisfies both determinism and graceful degradation. Third, we introduce a $κ$-domain diversity constraint for source selection and give an $O(|K| \cdot R)$ algorithm that maintains the invariant via hash-map lookup, eliminating the aggregator monopolization pathology present in URL-level deduplication. We further formalize Latent Objective Induction (LOI), an environment-shaping operator over prompt spaces that steers downstream model behavior without restricting the feasible output set, and prove its convergence under mild assumptions. Applied to a multi-locale retrieval pipeline, these algorithms yield 62% improvement in first-party source ratio and 89% reduction in same-domain duplication across 120 multilingual queries.

翻译：我们针对从排序搜索结果中选择多样化多区域来源集的问题，提出了三种具有形式化正确性保证与复杂度界限的算法。首先，我们将加权区域分配建模为约束整数划分问题，并给出一种$O(n \log n)$算法，该算法同时满足最小表示、预算耗尽与比例边界约束；我们证明所有三个约束均以$< 1$的紧偏差界成立。其次，我们将级联国家代码推断函数定义为异构信号（顶级域名结构、模型推断元数据、语言回退机制）上的确定性优先级链，并证明其同时满足确定性与优雅降级特性。第三，我们为来源选择引入$κ$域多样性约束，并提出一种通过哈希映射查找保持不变量的$O(|K| \cdot R)$算法，从而消除了URL级去重中存在的聚合器垄断问题。我们进一步形式化潜在目标诱导——一种作用于提示空间的环塑形算子，其能在不限制可行输出集的前提下引导下游模型行为，并在温和假设下证明其收敛性。将这些算法应用于多区域检索管道后，在120个多语言查询中，第一方来源比例提升了62%，同域重复率降低了89%。