Full Symmetry Does Not Force Localization

Abstract

An explicit entire quartic has every zeta symmetry while all four zeros remain off line.

Theorem 1.1 (A fully symmetric entire function with four off-line zeros).

$$\begin{gathered} \begin{array}{r}\forall delta,gamma\in \mathbb{R},delta\ne 0\wedge gamma\ne 0\Rightarrow \\ \exists F:\mathbb{C}\to \mathbb{C}, \\ F=\mathrm{offCriticalQuartic}\left(delta,gamma\right)\wedge \\ \mathrm{Differentiable}\left(\mathbb{C},F\right)\wedge \\ \forall s:\mathbb{C},F(1-s)=F(s)\wedge \forall s:\mathbb{C},F(\mathrm{conj}\left(s\right))=\mathrm{conj}\left(F(s)\right)\wedge \\ \forall s:\mathbb{C},F(s)=0\iff s\in \mathrm{sourceZeros}\left(delta,gamma\right)\wedge \\ \mathrm{card}\left(\mathrm{sourceZeros}\left(delta,gamma\right)\right)=4\wedge \\ \forall s:\mathbb{C},F(s)=0\Rightarrow \mathrm{re}\left(s\right)\ne criticalAbscissa\wedge \neg \forall F:\mathbb{C}\to \mathbb{C}, \\ \mathrm{Differentiable}\left(\mathbb{C},F\right)\Rightarrow \forall s:\mathbb{C},F(1-s)=F(s)\wedge \forall s:\mathbb{C},F(\mathrm{conj}\left(s\right))=\mathrm{conj}\left(F(s)\right)\Rightarrow \forall s:\mathbb{C},F(s)=0\Rightarrow \mathrm{re}\left(s\right)=criticalAbscissa.\end{array} \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/Zeros/SymmetricPolynomial/FullSymmetryNonlocalization.full_symmetry_not_fixed_line_localization (✓ std3). ∎

Source. Repository-derived.

Commentary.

For nonzero real delta and gamma, the witness is exactly the source quartic P_delta,gamma(s), formed from z = s - 1/2. It is complex differentiable everywhere and satisfies both generators of the source Klein-four symmetry: reflection s maps to 1-s, and complex conjugation commutes with evaluation.

The zero condition is an equivalence, not a one-way inclusion: a point is a zero exactly when it belongs to sourceZeros(delta,gamma). That named finite set consists of 1/2 plus or minus delta plus or minus i gamma, has cardinality four, and every zero has real part different from the repository critical abscissa.

The second top-level conjunct is the boxed consequence. It negates the universal implication from entire full-zeta symmetry to fixed-line localization, using the same explicit quartic as counterexample. No Riemann-hypothesis assumption or unformalized zero data enters the declaration.

References

• Truth anchor: D5/S3/Zeros/SymmetricPolynomial/FullSymmetryNonlocalization.full_symmetry_not_fixed_line_localization
• Dependency: D5/S3/Weil/ReflectionLedger