We show that no total functional can uniformly transform $Π_1$ primality into explicit $Σ_1$ witnesses without violating normalization in $\mathsf{HA}$. The argument proceeds through three complementary translations: a geometric interpretation in which compositeness and primality correspond to local and global packing configurations; a proof-theoretic analysis demonstrating the impossibility of uniform $Σ_1$ extraction; and a recursion-theoretic formulation linking these constraints to the absence of total Skolem functions in $\mathsf{PA}$. The formal analysis in constructive logic is followed by heuristic remarks interpreting the results in informational terms.

翻译：我们证明，在$\mathsf{HA}$中，任何全函数都无法在不违反正规化条件的情况下，将$Π_1$型素数性质一致地转化为显式的$Σ_1$型见证。论证通过三种互补的翻译展开：一种几何解释，其中合数与素数分别对应局部与全局的填充构型；一种证明论分析，展示一致提取$Σ_1$型见证的不可能性；以及一种递归论表述，将这些约束与$\mathsf{PA}$中全斯科伦函数的缺失相联系。构造性逻辑的形式分析之后，附有从信息论角度解读结果的启发性评注。