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On the Realizability of Prime Conjectures in Heyting Arithmetic
arXiv:2511.07774v2 Announce Type: replace-cross
Abstract: We show that no total functional can uniformly transform $\Pi_1$ primality into explicit $\Sigma_1$ 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 $\Sigma_1$ 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.
Abstract: We show that no total functional can uniformly transform $\Pi_1$ primality into explicit $\Sigma_1$ 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 $\Sigma_1$ 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.
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