2. Follow the proof🔗

1. Relate critical-line zeta zeros to the zeros of a continuous real function.

2. Prove the theta identities and the decay bound at the cusp.

3. Prove the Fourier transform identity, the contour shift, and absolute convergence of the weighted integrals.

4. Apply finite differences. These cancel the cosine term and give a nonnegative kernel.

5. Prove that the normalized full integrals tend to zero. Pass to limits on compact intervals.

6. Assume an eventual fixed sign. A positive tail interval then grows faster than the initial interval can bound.

7. Derive a contradiction. Thus, a zero exists above every height, and there are infinitely many zeros.

Every step is proved. The argument does not require estimates for all higher derivatives at the boundary.