Dual Characterization of the Critical Midline

Abstract

Mirror fixed points and unitary half-density parameters define the same midline.

Theorem 1.1 (Mirror fixed points and unitary parameters define the critical midline).

$$\forall A[\mathrm{AddMonoid}(A)],\forall \ell :A{\to }_{+}\mathbb{R},(\exists a,\ell (a)\ne 0)\Rightarrow (\{s\in \mathbb{C}:\mathrm{mirror}(s)=s\}=\{s\in \mathbb{C}:\forall a,\Vert \mathrm{halfDensityReading}(\ell ,s,a)\Vert =1\}\wedge \{s\in \mathbb{C}:\mathrm{mirror}(s)=s\}=\{s\in \mathbb{C}:\Re (s)=\frac{1}{2}\})$$

Proof. Machine-checked in Lean as D5/S3/Midline/DualCharacterization.midline_dual_characterization (✓ std3). ∎

Source. Repository-derived.

Commentary.

For any additive ledger with at least one nonzero length, the set of conjugate-reflection fixed points equals both the set of parameters whose half-density readings all have unit norm and the line of parameters with real part one half. This set-level theorem is derived from the existing pointwise critical-line characterizations. It locates no zeta zero and asserts no Riemann-hypothesis conclusion.

References

• Truth anchor: D5/S3/Midline/DualCharacterization.midline_dual_characterization
• Dependency: D5/S3/Weil/SpectralDynamics