spoiler : A theory of how L could have had definitive proof against Light only by the time of episode 7-8
We all know that Naomi's disappearance was the thing that led L to lock into 2 families, Yagami and Kitamura family. now, L didnt really put much thought into Naomi's disappearance but what if he did. i just want to expand on that here
L gets the call about naomi's disappearance about 4-5 days after she met light and got killed. lets see what information L already has. He already knows that Kira is probably a student and has connections to the police. after he gets called about naomi's disappearance he deduces that naomi's disappearance or suicide doesnt actually make any sense. she would actually be actively trying to catch kira and might have uncovered some crutial evidence against kira and if she did, did kira manage to get to her first? this was where L's thought about naomi ends. what if we expand it.
if we assume that she got killed by kira, then kira must have known about her and that she has crutial evidence against him but, how did he know? its not like she would be shouting on the streets that i have information against kira, she must have gone somewhere where she could actually report this thing, which is the police station where she went. the police station is the only place where she would actually tell this. L already knew that kira has connections to the police, so its possible that kira was present at the police station when she told them. thats the only way kira would have even known about naomi's existence. but the thing is, she used an alias, so her name wouldnt be on the police logs, so there is no definite time where he would be able to check security footage, so to get to know who listened to the information about kira, L would have to check days of footage just to find naomi and who talked to her. but since naomi was the one who uncovered the information, the suspects list is only 2 families, so while there is a possibility that she used alias, there is no reason for the people of those families to use alias, so just checking the footage of the entries of the people from those families we should be able to find naomi, and if we dont, then the people who were present at the time of the report of naomi would become suspects ( this point covers that if kira was not one of the members of the two families and naomi still got killed as only those people knew that naomi had information about kira), we would have to check days of footage but there is a strong possibility that we would find kira, so its worth the time as L went over the footage of light's house multiple times for days.
also this thing depends on the fact that naomi did try to catch kira and uncovered some evidence and got killed by him, this would be an absolute waste of time if she was actually just griefing about her husband's death and committed genuine suicide, but since L already knew about her and her talents, we can give this thing a little whoosh.
please analyse this and give your counter arguments, it would be highly appreciated