I would like to know when do Carnations (Dianthus caryophyllus) become responsive to their photoperiod for flowering. That is, at what leaf stage e.g. two leaf, four leaf. Thank you a lot.
Adequate photoperiod is not enough for carnations (Dianthus caryophyllus) to start flowering. They need a few months of cold environment like I said in your other question, about 5°C (41F°) for 10 weeks.
I live in a temperate climate, zone 6b, and my experience with carnations is as follows:
They were started by seed in spring 2016, experienced increasing hours of daylight in spring and summer, then decreased hours of daylight in autumn and winter. I have kept them outdoors for the winter because they needed the low temperatures. During winter they did not grow and they started to grow again in spring 2017. In June 2017 they have flowered. It took them one year to reach maturity and another 2-3 months to flower after reaching maturity.
I didn't count the leaves, there may be 30 leaves or so (I can't give a good estimate), but I will post some pictures for you to see how the leaves look when the plants start flowering. Also, I'm not a good photographer, sorry. In my opinion, you need at least 12 weeks of primary growth (autumn) + at least 10 weeks of cold (winter) + at least 12 weeks of gradually increasing light again (spring). This means that it's unlikely to have flowers before the plant is 34 weeks old.