The Solvability Complexity Index (SCI) provides an extensional limit-height formalism for recovering a target map $Ξ$ from finite samples of an evaluation interface $Λ\subseteq\mathbb C^Ω$ by finite-height towers of pointwise limits. We first give a foundational analysis of what this extensional framework does and does not determine. We show that the SCI separation axiom is equivalent to a factorization of $Ξ$ through the full evaluation table, and we isolate the minimal logical role of $Λ$ as an information interface. To connect the SCI to Type-2 computability and Weihrauch reducibility, we give an effective enrichment for countable $Λ$ by viewing the evaluation table image $I_Λ\subseteq\mathbb{C}^{\mathbb{N}}$ as a represented space and factoring $Ξ$ as $\widehatΞ$. We then define the Weihrauch-SCI rank of a problem as the least number of iterated limit-oracles needed to compute it in the Weihrauch sense, i.e. the least $k$ such that $\widehatΞ\le_{W}\lim^{(k)}$, and prove well-posedness and representation invariance of this rank. A central negative result is that the unrestricted raw type-G SCI model (arbitrary post-processing of finite oracle transcripts) is generally not a computability model in the Type-2/Weihrauch sense: finite-query factorizations collapse raw type-G height, and analytic non-Borel decision problems yield examples with raw $\mathrm{SCI}_G=0$ but infinite Weihrauch-SCI rank. We therefore distinguish the raw extensional SCI from implemented SCI variants, where the indexed approximation table is required to be realized uniformly by a chosen class of operations. To recover a robust bridge, we introduce an intermediate SCI hierarchy by restricting the admissible deepest-level post-processing to regularity classes (continuous/Borel/Baire).

翻译：可解性复杂度指标（SCI）为通过有限高度点态极限塔从评价接口$\Lambda\subseteq\mathbb C^\Omega$的有限样本中恢复目标映射$\Xi$提供了外延式极限高度形式体系。本文首先对该外延框架所能与不能确定的内容进行基础分析，证明SCI分离公理等价于$\Xi$通过全评价表的因子分解，并厘清$\Lambda$作为信息接口的最小逻辑作用。为将SCI与第二型可计算性和魏劳赫可归约性建立关联，我们通过将评价表像$I_\Lambda\subseteq\mathbb{C}^{\mathbb{N}}$视为表示空间并将$\Xi$分解为$\hat\Xi$，对可数$\Lambda$给出有效强化。进而定义问题的魏劳赫-SCI秩为在魏劳赫意义下计算该问题所需最小迭代极限神谕次数，即最小的$k$使得$\hat\Xi\le_{W}\lim^{(k)}$，并证明该秩的适定性与表示不变性。核心否定性结论是：无限制的原始类型-G SCI模型（对有限神谕转录的任意后处理）通常不构成第二型/魏劳赫意义下的可计算性模型——有限查询因子化会导致原始类型-G高度坍塌，且解析非波莱尔决策问题能产生原始$\mathrm{SCI}_G=0$但魏劳赫-SCI秩无穷的反例。因此我们区分原始外延SCI与可实现SCI变体（要求索引逼近表通过选定的操作类别均匀实现）。为重建稳健桥梁，我们通过将容许的最深层后处理限制为（连续/波莱尔/贝尔）正则类，引入中间SCI层级结构。