Toroidal Temperedness Criterion

Abstract

The strip-native Riemann hypothesis is equivalent to temperedness of every nontrivial toroidal Eisenstein parameter.

Theorem 1.1 (RH is equivalent to toroidal Eisenstein temperedness).

$$\forall Index\in \mathrm{Type}\left(\right),T\in Index\to \left(\mathrm{Complex}\left(\right)\to \mathrm{Complex}\left(\right)\right),\left(\forall s\in \mathrm{Complex}\left(\right),\exists i\in Index,T\left(i\right)\left(s\right)\ne 0\right)\Rightarrow \left(\left(\forall s\in \mathrm{Complex}\left(\right),\mathrm{IsNontrivialZero}\left(s\right)\Rightarrow \mathrm{Re}\left(s\right)=\frac{1}{2}\right)\iff \left(\forall s\in \mathrm{Complex}\left(\right),\left(\forall i\in Index,\mathrm{completedRiemannZeta}\left(s\right)\times T\left(i\right)\left(s\right)=0\right)\Rightarrow \mathrm{Re}\left(s-\frac{1}{2}\right)=0\right)\right)$$

Proof. Machine-checked in Lean as D5/S3/Analytic/Adelic/ToroidalTemperednessCriterion.rh_iff_all_toroidal_eisenstein_tempered (✓ std3). ∎

Source. Repository-derived.

Commentary.

The left side is the canonical nontrivial strip-zero formulation. On the right, toroidal invisibility is stated directly by vanishing of every completed-zeta-times-twist period.

Pointwise twist nonvanishing makes simultaneous period vanishing equivalent to a completed-zeta zero. The frozen completed-zeta zero-locus theorem then identifies exactly the strip zeros.

The normalized principal-series parameter is s minus one half. Its real part vanishes exactly on the critical line, which is the displayed temperedness condition.

References

• Truth anchor: D5/S3/Analytic/Adelic/ToroidalTemperednessCriterion.rh_iff_all_toroidal_eisenstein_tempered