Four-Point Power Defect

Abstract

Four symmetric exponential points have a hyperbolic-trigonometric power defect.

Theorem 1.1 (The four power defects collapse to a real product).

$$\mathrm{Defect}(q,\theta ,k)=4\cdot (1-\cosh (kq)\cdot \cos (k\theta )).$$

Proof. Machine-checked in Lean as D5/S3/Zeros/ToySpectrum/FourPointPowerDefect.four_point_power_defect_eq (✓ std3). ∎

Source. Repository-derived.

Commentary.

For real q and theta, the four points are exp(q + i theta), exp(q - i theta), exp(-q + i theta), and exp(-q - i theta). Defect(q, theta, k) is the sum of 1 - z^k over these points.

The power-of-exponential identity moves k into each exponent. The two angular signs cancel the sine terms, while the two radial signs combine into twice the hyperbolic cosine.

This records only the four-point algebraic identity from the source atom. Its detection estimate, asymptotic formula, reciprocal-power expansion, measure interpretation, and numerical certificates are not claimed.

References

• Truth anchor: D5/S3/Zeros/ToySpectrum/FourPointPowerDefect.four_point_power_defect_eq