How do you tell a genuinely sharp edge from one that only feels sharp?

The tests I use, roughly in order of how much they predict real kitchen behaviour. Slicing a ripe tomato with no downward pressure, using only the weight of the knife, is the best cheap test I know because tomato skin is unforgiving and it tests the edge along its whole length rather than at one spot. Cutting a sheet of paper held vertically tells you the apex is clean but only at the point you cut. Shaving tells you almost nothing useful because a burr shaves.

The one that changed how I sharpen was the onion. If the first cut into an onion skin skates instead of biting, the edge is not there regardless of what the paper said.

An edge that shaves and then dies in three days is almost always a deburring problem rather than a sharpness problem. What you have is a wire edge still hanging on, which is thin enough to shave and weak enough to fold over the first time it hits a board. It feels spectacular for two days and then feels blunt, and the steel underneath was never actually apexed properly.

If you want an actual number rather than a feeling, there are testers that measure the force required to cut a standardised medium, reported on a scale where lower is sharper. They exist, hobbyists use them, and they turn arguments into measurements. I borrowed one for a weekend and the most useful thing it told me was not which of my knives was sharpest but how much sharpness each of them lost after a week of normal use, which is exactly the retention question you are asking. That is a lot of money for a home cook, but if you ever get the chance to borrow one, do.

Politely disagreeing with the whole framing. You are testing at the stone and asking it to predict a week away, and it cannot, because retention is mostly about the steel, the angle and what you cut on rather than about how the edge left the stone. Two knives that test identically will diverge completely if one lives on a wooden board and the other on glass or in a drawer full of other metal. Before you buy any test, check what your board is made of and whether the knives touch each other in storage. That changed my retention more than any sharpening change ever has.

I used to test by shaving my arm and I once opened up a decent cut doing it on an edge that was better than I expected. Testing on your own skin is a bad habit with a genuinely unpleasant failure mode, and it does not even measure the thing you care about. Cork, paper, tomato, onion. Nothing attached to me.

Deburr properly, then test on a tomato with no pressure. If that works and it still fails in a week, the problem is your board or your storage, not your stones.