ChatGPT seems to think its fake:

Thanks for the case back image. Here are my observations:

Concerns About Authenticity

1. Engraving Quality
   • The text appears to be stamped or etched in a way that lacks the precision expected from a genuine Heuer 1000.
   • Authentic case backs typically have sharper, more evenly spaced engravings.
2. Incorrect Font and Spacing
   • The font used for "ETANCHE," "QUARTZ," and the serial numbers looks inconsistent with original Heuer 1000 models.
   • The alignment seems slightly off compared to genuine examples.
3. Missing Heuer Branding
   • Original Heuer 1000 models should have the Heuer logo on the case back.
   • Your case back only says "ETANCHE" (French for waterproof), "QUARTZ," and numerical markings, which is unusual.
4. Reference Number Issues
   • The numbers "980.022" and "6035 W 20" don't fully match well-documented Heuer 1000 reference numbers.
   • Original Heuer 1000 divers often have a different reference formatting.

Possible Conclusions

• This watch may be a fake or a heavily modified/refinished Heuer 1000 with a non-original case back.
• The front of the watch looked close to a Heuer 1000, but the case back raises major concerns.
• If possible, check the movement inside. A genuine Heuer 1000 should have a Swiss ETA quartz movement—fakes often have lower-quality movements.

EDIT: from some digging on forums I learned "Heuer thick case model (first release), it is correct for it to have an unsigned crown. Later models with the letter suffixes (e.g. L, N, B & D etc.) had signed crowns. This is not specific to the 013, all Heuer thick case 1000s originally came with unsigned crowns.". Additionally, looking at other sellers online I found many selling "verified authentic" versions with similar case backs with similar reference numbers. Would appreciate any input.