Carbon fixation converts inorganic carbon dioxide into carbon that is covalently incorporated into an organic molecule.

In the Calvin cycle, the enzyme ribulose-1,5-bisphosphate carboxylase/oxygenase (RuBisCO) catalyzes fixation of $CO_2$ onto the five-carbon acceptor ribulose-1,5-bisphosphate (RuBP).

For one fixation event,

$$\mathrm{RuBP;(5C)+CO_2;(1C)\rightarrow unstable;6C;intermediate}.$$

The six-carbon intermediate rapidly splits into two molecules of 3-phosphoglycerate (3-PGA), each containing three carbons:

$$\boxed{\mathrm{RuBP+CO_2\rightarrow2,3!!!-PGA}}.$$

Carbon accounting makes the reaction transparent

The carbon count is

$$5+1=6=3+3.$$

Nothing about carbon fixation creates carbon atoms. The key event is that carbon previously present as gaseous $CO_2$ becomes part of an organic carbon skeleton.

For three $CO_2$ molecules,

$$3\times CO_2+3\times RuBP\rightarrow6\times3\text{-PGA},$$

so

$$3+15=18\ \text{carbons on both sides}.$$

Fixation is only the first stage

3-PGA is more oxidized than the carbohydrate products ultimately exported from the Calvin cycle. Fixing $CO_2$ therefore does not by itself make sugar.

Subsequent reactions use ATP and NADPH to reduce carbon and rearrange intermediates. Most of the resulting carbon is also needed to regenerate RuBP so fixation can continue.

The full name RuBisCO includes oxygenase because the enzyme can also catalyze a competing reaction in which $O_2$, rather than $CO_2$, reacts with RuBP. The consequences of that competition belong to photorespiration. Its central role here is carboxylation: inserting carbon from $CO_2$ into the Calvin-cycle carbon pool.