On the Realizability of Prime Conjectures in Heyting Arithmetic

We show that no total functional can uniformly transform ${\mathrm{\Pi }}_1$ primality into explicit ${\mathrm{\Sigma }}_1$ witnesses without violating normalization in $\mathsf{H}\mathsf{A}$. 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 ${\mathrm{\Sigma }}_1$ extraction; and a recursion-theoretic formulation linking these constraints to the absence of total Skolem functions in $\mathsf{P}\mathsf{A}$. The formal analysis in constructive logic is followed by heuristic remarks interpreting the results in informational terms.