Discriminant Determinant Product

Abstract

Conditional Lerch data identify the two mod-five holonomy determinants.

Theorem 1.1 (The mod-five sector determinants recover the golden constants).

$$\left(\mathrm{HasReflectedHurwitzDerivativeAtZeroFormula}(\frac{1}{5})\wedge \mathrm{HasReflectedHurwitzDerivativeAtZeroFormula}(\frac{2}{5})\right)\Rightarrow \left(\mathrm{masslessHolonomyDeterminant}(\frac{1}{5})\times \mathrm{masslessHolonomyDeterminant}(\frac{2}{5})=\mathrm{sqrt}(5)\wedge \frac{\mathrm{masslessHolonomyDeterminant}(\frac{2}{5})}{\mathrm{masslessHolonomyDeterminant}(\frac{1}{5})}=goldenRatio\right)$$

Proof. Machine-checked in Lean as D5/S3/Analytic/HolonomyDeterminant/DiscriminantDeterminantProduct.discriminant_determinant_product (✓ std3). ∎

Source. Repository-derived.

Commentary.

Assuming the reflected Hurwitz derivative formula at both source-fixed mod-five representatives, the frozen determinant bridge evaluates the two zeta-regularized massless holonomy determinants.

Their product is the square root of five, while ordering the second sector over the first gives the golden ratio.

References

• Truth anchor: D5/S3/Analytic/HolonomyDeterminant/DiscriminantDeterminantProduct.discriminant_determinant_product
• Dependency: D5/S3/Analytic/HolonomyDeterminant/MasslessHolonomyDeterminant