[quote name='mykevermin']cumulative.[/quote]
Partly correct. For most items, yes, once you sell the required total of items, the reward item will become available, regardless of when you sold them, or whether you sold them singly or all at once. If you need, say, 3 Gemsteels for a Tournesol and sell them at 3 totally different points in the game, the sword will pop up in the bazaar after the 3rd one (assuming you've gathered the other goods). However, if you sell 2 Gemsteels and then accidentally toss in a couple Mallets and another item that i can't recall, a Masamune will pop up in the bazaar. Once you've triggered Masamune, you'll need ANOTHER 3 Gemsteels (god help you) before the Tournesol will be available.
Another point is that if you sell a whole mess of one item (say, 99 arcanas) and trigger a high arcana that requires 10 arcanas and a few other goodies, your arcana total will reset to zero, rather than to 89.
In a nutshell, if you have items that need to be used multiple times, or on multiple items, hang onto them until you need them. When you sell them, only sell the exact number you'll need unless you're sure you won't need any more. I could've saved myself a few hours of collecting had i known this early in the game.