Why do I have mysterious cellular data usage (Verizon) every 6 hours on all 3 of my iphone 5's?

Question

I have recently uncovered mysterious cellular usage on three different iPhones. I am a Verizon customer and discovered this by examining the cellular data use logs. What I found are a long series of mysterious data usage logs. I have visited the Genius Bar at my local Apple Store 3 times now to log notes and discuss the issue. It is being escalated.
The characterstics are as follows:
All my family phones have the appropriate IOS and hardware updates (verified by the GeniusBar at my local Apple Store).
This is occuring across three phones, which happen to all be iphone 5. Two are 5 and the other a new 5s. We do have one iphone 4 in the family but the issue (so far as I can tell), is not happening on it.
One iphone 5 has IOS 7, the other IOS 6. The new 5s has of course IOS 7.
Mysterious data use happens even while connected to wifi.
Each mysterious data use log entry is exactly 6 hours apart. For example: 2:18 AM, 8:18 AM, 2:18 PM, 8:18 PM, 2:18 AM ... etc. It cycles AM and PM, same times, for a day to many days until some condition causes it to change (evolve).
The times evolve. One day it could be cycling through at one time series, then it changes to another time sequence and continues until the next condition.
The data usage is anywhere from a few K to many MB. The largest I've seen is over 100 MB.
The logs clearly show these usages are not due to human interaction. It is a program.
If cellular connection is used frequently (by the owner), the pattern is hard to pick out. Luckily, my family member is very good about only using wifi whenever possible, so these mysterious use patterns are easy to pick out.
Verizon allows access to 90 days worth of data logs, so I downloaded and analyzed them. This has been happening for at least 90 days. I have found 298 instances of mysterious use out of 500 total connections to cellular. A total of 3.5 GB of mysterious cellular data has been recorded as used in that 90 days by this phone alone. Only .6 GB of the total cellular data use is legitimate, meaning by the person.
This issue is occuring across three different phones. Two are iPhone 5, and the third is a recently purchased iPhone 5s. The 5s I have not touched beyone the basic startup. I have left it alone on a desk for 3 days, and looking at the logs, the mysterious data use in the same pattern is occuring.
So ... I am speaking to both sides, Verizon and Apple to get answers. Verizon puts their wall up at data usage. It doesn't matter how it is being used, you simply need to pay for it. Yes, the evidence I have gathered is getting closer to someting on Verizon's end.
Before pressing in that direction, I am hoping someone reading this may recognize this issue as a possible iPhone 5 issue ... OR ... you, right now, go look at your data usage logs available through your carrier's web account, and see if you can pick out a pattern of mysterious use. Look especially at the early morning instances when you are most likely sleeping.
I am hoping this is a simple issue that has a quick resolution. I'm looking for the "ooohhh, I see that now ..." But I am starting to think this might be much bigger, but the fact is, most customers rarely or never look at their data usage details, much less discover mysterious use patterns.
The final interesting (maybe frightening part) thing about all this is that I discovered while talking to Verizon ... they do not divulge which direction the data is going. This goes for any use, mysterious or legitimate. Is cellular data coming to your phone, or leaving it? Is something pulling data from your phone? We know that it is possible to build malware apps, but the catch with my issue is that it is also happening on a brand new iphone 5s with nothing downloaded to it.
Thanks for your time

Answer 1

Thanks for the replies. It took a while not hearing anything so thought I was alone. I have done many of the suggestions already. The key here is that it occurs on both phones with apps, and phones still packaged in a box.
A Genius Bar supervisor also checked his Verizon data usage log and found the same 6 hour incremental use. Suprisingly, he did not express much intrigue over that. Maybe he did, but did not show it.
I think the 6 hour incremental usage is the main issue here. I spoke with Verizon (again) and they confirmed that all they do is log exactly when the phone connected to the tower and used data. The time it records is when the usage started. I also found out that the time recorded is GMT.
What is using data, unsolicited, every 6 hours?
Why does it change?
Why does it only happen on the iPhone 5 series and not the 4?
Since no one from Apple seems to be chiming in on this, and I have not received the promised calls from Apple tech support that the Genius Bar staff said I was suppose to receive, it is starting to feel like something is being swept under the rug.
I woke up the other day with another thought ... What application would use such large amounts of data? Well ... music, video, sound and pictures of course. Well ... what would someone set automatically that is of any use to them? hmmm ... video, pictures, sound. Is the iPhone 5 succeptible to snooping? Can an app be buried in the IOS that automatically turns on video and sound recording, and send it somewhere ... every 6 hours? Chilling. I noted that the smallest data usage is during the night when nothing is going on, then it peaks during the day. The Genius Bar tech and I looked at each other when I drew this sine wave graph on the log print outs during an appointment ...