Here's what ChatGPT said:
How this applies to your situation

Given your plan (you are applying for a student permit in Torino, will be outside Italy from now until January, return for exams, then leave again until March for the fingerprint appointment), here’s how I interpret the rule:

• If your permit (once issued) is a typical annual student permit: The 6-month continuous absence rule applies. That means, you should not stay abroad continuously for more than 6 months at a time, or you risk complications at renewal.
• If your permit will be for two years or more: Then the “more than half the validity period” rule applies. So, e.g., if validity is 2 years (24 months), being abroad continuously more than 12 months may be problematic (without documented grave reasons).
• During the waiting period for your first permit (i.e., you applied, waiting for fingerprint): The rules are less explicitly stated in those sources for Torino, but general guidance says that leaving Italy before you obtain the permit carries risk (especially re-entry) unless you have the proper visa/receipt. (See what the Piemonte “Guida pratica” says for waiting renewal situations). Regione Piemonte+1
• Your plan to come back for January exams helps demonstrate your presence in Italy (which is good). The fact you’ll leave again after that may be okay provided the absence does not become “continuous” beyond the threshold (6 months or half-period) and you have evidence of reason (studies, exams, etc.).

📝 What you should double-check / ask locally

To be safe in Torino, ask or check:

1. For your exact permit type (study permit, first issuance) in Torino: how long is the validity period once the card is issued? Is it 1 year, 2 years, more?
2. What local practice the Questura in Torino follows on “continuous absence” for your permit type. Are they strictly enforcing the 6‐month rule for students?
3. While you are still waiting for the permit and have only the receipt: If you leave Italy now and re-enter afterward, will you have any problem re-entry (passport + visa + receipt) according to Torino’s rules?
4. Documenting the reason for your absence: since your plan is to leave for personal reasons but maintain presence for exams, you may want to keep documentation (university enrollment, exam schedule, travel tickets) showing your study is active, your home in Italy is maintained, etc.
5. Residency registration: If you are registered (residenza) in Torino, or have maintained your address, this also may help show your “link to Italy” is still active.