Jtdylktuy

I was recently a passenger on a commercial aircraft at cruise altitude when I noticed another aircraft just reaching its cruising altitude while crossing our flightpath from below, at what looked like a very close distance.

I looked up the two flights (LH491, AC172 on Nov 10, 2018) on Flightradar 24, linearly interpolated the following values in Excel to roughly 2-3 seconds resolution, and calculated the great circle distance between the coordinates:

AC 172 was heading 106° at the very end of its climb to FL 350. LH 491 was heading 51°, cruising at FL 350.

The smallest horizontal separation by my calculation was about 3.67 nm (at 1300 ft vertically). When we reached 1000 ft vertical separation, that distance was about 4.7 nm.

To my understanding, aircraft at these altitudes should always have a separation of least 1000 ft or 5 nm.

Not being an aviation professional myself, could someone please help me understand what was happening here?

I'm sure there must be a duplicate question, but I couldn't find one that involves both the vertical and horizontal components.

I think you may have answered your own question note the use of the word OR

To my understanding, aircraft at these altitudes should always have a
separation of least 1000 ft or 5 nm.

If you take a look at your data the aircraft are always separated by 1000 feet OR 5nm. The singular exception to this is when you have the separation quoted at 4.81 miles and 982ft. This can to an extent be explained away by the inaccuracies in systems like FlightRadar24