Move a small patch away from the equator
Compare local scale factors for ideal spherical Mercator and Lambert cylindrical equal-area. Both are normalized to true scale at the equator.
Inspect the calculation
Let c = cos(latitude). Mercator: east-west scale =1/c, north-south scale=1/c, area factor=1/c². Lambert cylindrical equal-area: east-west scale=1/c, north-south scale=c, area factor=1. These are local factors for an infinitesimal patch on a sphere. The equal-area map keeps area while stretching shape.
No coastlines, national borders, or country-size estimates are modeled. This is not an ellipsoidal survey calculation. Latitude 80° is the control limit because Mercator diverges at the poles.