= Solution
Add a stratum indicator that is “First” for the first episode and “Later” for every subsequent episode, including the final censored episode:
|| Patient
|| Next-event episode
|| Gap start
|| Gap stop
|| Event
|| $z$
|| Stratum
| 001
| 1
| 0
| 24.8
| 1
| 0
| First
| 001
| 2
| 3
| 8.3
| 1
| 0
| Later
| 001
| 3
| 3
| 7.1
| 1
| 0
| Later
| 001
| 4
| 3
| 11.7
| 1
| 0
| Later
| 001
| 5
| 3
| 8.1
| 0
| 0
| Later
A <Stratified Cox model> then uses distinct <baseline hazards> $h_{0,\mathrm{first}}$ and $h_{0,\mathrm{later}}$, with a common treatment coefficient $\beta$ unless an interaction is explicitly desired. Form separate <risk sets> within the two strata and multiply their partial <likelihoods>. There is no need to force every later event number to have its own baseline when the stated aim is only first versus subsequent headaches. This is a <first-event versus recurrent-event baseline stratification>.
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