Resonance Frequency of the Antenna.

Show me Smith chart plot of S11.

This is the smith chart.

"Resonance" frequency (im(S11) = 0) is about 5 GHz, but radiation resistance is > Z0, not matched. In practice, the matching may be sufficient for the intended antenna purpose.,

Double resonance.

How did you know that the radiation resistance is more than Zo? Thanks.

Zin=Z0*(1+S11)/(1-S11).

For example, if S11=1/100, Zin=Z0*(101/99).
If S11=-1/100, Zin=Z0*(99/101).

Radiation resistance is something special, what you discuss here is input resistance.

If we can ignore copper loss, dielectric loss, matching element loss, guided loss, etc., real(Zin) is equal to radiation resistance.

Even if loss is not zero, real(Zin) is close to radiation resistance in many cases.
Here I treat object including matching elements as antenna, although I don?t think matching should be included in antenna.

Yes, in an ideal world things are ideal, which makes it simple.

In my work on antenna simulation, I can separate radiation resistance from losses, and losses are often a major influence.
For patch antennas on FR4, we have large dielectric loss.
For electrically small antennas, as used in many devices today, we often have radiation efficiencies of 50% or less, so that radiation resistance is much smaller than input resistance.

Even if we use low loss substrate such as Teflon, loss due to surface wave is fairly large for patch antenna.

Surface wave does not contribute to radiantion but it causes mutual coupling of patch array antenna.

Maybe you notice that you just disproved your own statement.