Let us say there's a level 5 fighter that has a 10 as his constitution score and 50 hit points. The fighter character gets into a fight and takes 40 points of damage, for 10 hp remaining.
Then let's say the fighter somehow loses 4 points of Constitution (for the sake of the question, imagine a DM's homebrewed poison that deals 4 points of Con damage/drain).
Now that the fighter's constitution score has dropped from 10 (+0 mod) to 6 (−2 mod), he would retroactively lose hit points based on his modifier times the current level: 2 × 5. His maximum hit point total would drop by 10 hit points and he only had 10 hit points left.
Would this situation drop the fighter to zero hit points because of the negative modifier to his constitution, or is there a rule that a character would have a minimum 1 hit point no matter what the constitution decrease is, until the character hits zero constitution and dies?