The equal-norm case: norm-separation is not a hypothesis but a characterization, and the gauge-equivariant repair

Abstract

When distinct primes of S share the same norm, the naive reconstruction arguments of the series degenerate. This paper shows that norm-separation is exactly the condition under which the dual gauge action separates the prime coordinates, characterizes the equal-norm degeneracy, and provides the gauge-equivariant repair: the full gauge action of Definition 4.1, which restores the separation equivariantly and is the hypothesis used by the reconstruction theorems of the later papers.