Hurwitz Retypes RH as Two-Sector Strict Completion
Every finite completed Euler section is holomorphic and nowhere zero in the open critical strip. If source-derived unit renormalizations make these sections converge locally uniformly to separately on
then Hurwitz's theorem forces to be nowhere zero in both sectors. This proves RH. Conversely, under RH the backward ratios are sectorwise units, so existence of such unit completions is equivalent to RH.
The explanatory target is noncircular: derive the renormalization units and their normal-family compactness from the graded Euler and theta boundary data. An isolated zero is exactly where strict invertible descent fails through nonuniform convergence, loss of a unit, escaping inverses, determinant discontinuity, or a nonexact boundary operation.
This also gives a precise reason for two sectors. The critical seam may remain a singular common boundary while completion stays strict in each open chamber.
Artifact:
- research/grothendieck/hurwitz-retypes-rh-as-two-sector-strict-completion.md