Why do you need one so small? I think you may have a real problem finding such a thing, because the electrical box in the ceiling, where connections are made between the light fixture's wiring and the house's wiring, is likely to be bigger than 4" across, and the canopy has to be big enough to cover it with room to spare. I don't have an encyclopedic knowledge of available components, but my intuition tells me you're looking for a part not likely to exist.

I highly recommend LED bulbs; we have about 100 3.6W MR16-style LED bulbs in our house in 3" mini-can housings, all at about 4500K (neither warm nor cool white). Each gives about as much light as a 35-40W halogen bulb. Our 19 x 21' kitchen has 15 for general lighting, plus the following task lighting: 4 over our 13' counter with main sink; 8 over our 5 x 9' main island; and 4 over our 3'3" x 9' small island. The 72" wide range hood came with ten 40W PAR20 bulbs but light a 9' wide area including range and landings; we'll be replacing the 40W bulbs with 5W dimmable LED bulbs. This is a good amount of light, not too much, not too little. Well, in the range hood area 50W of LED bulbs (equivalent to about 500W of halogen lights) is actually too much light but it's dimmable, not that my wife will use any setting but MAX. (There are also 10W LED bulbs available, but my wife has no self control when it comes to light and WILL burn out her retinas if I let her.) The standard medium base "Edison" bulb is not a form factor well suited to LEDs because while filaments (tungsten, halogen) are point sources and radiate in all directions, LEDs are VERY directional and have limited dispersion (15-60 degrees). Most LED medium base bulbs have more LEDs pointing outward with very few pointing up, some nearly exclusively point upward (downward in a pendant application). The distribution and "point" of the individual LEDs will affect whether the ambience (outward glow) or the task function (downward lumens) will work. Most pendant shades do have an inner white lining to even out the light distribution, and that will improve your downward light output for task lighting. LED color is tricky as well, as many are very cool white and won't give you that warm glow you expect from the pendant shade. A warm white (3000-3500K) bulb will give you the best ambience, whereas a pure white (4000-4500K) will give you the best task lighting. Another caution: pendants spaced too far apart look odd, but too close looks worse. I would recommend a spacing of about every 2-3'. Also, there should be ideally 3, or 4 max. Slimmer/smaller ones can be spaced a little closer and you can have more of them and still look fine. There are metallic shades (if modern works for your kitchen) that will give you a lot more task lighting. Here is a link that might be useful: [Houzz Pendant Light Ideabook[(https://www.houzz.com/magazine/kitchen-islands-pendant-lights-done-right-stsetivw-vs~158689)

So far everyone has been in agreement. I'm also not a fan of track lighting, but that is what came with the house and it at least works. Can lights over the sitting area would do the job, but that would be pricy and challenging DIY, for another solution that isn't really attractive and we don't often use for more than a 15-30 minutes per day. It is a tricky problem, as we have a lot of furniture already packed in there, and I don't really want to obstruct the sight lines to the garden (we have a 10' wide slider where this house's French doors are located). Maybe I will experiment with finding a much brighter floor light for behind the seating, and hiding the torchiere floor lamp behind the entertainment center, for when we just need a little room illumination.

I don't have pendents or recessed lighting - so no help there, but if you don't have under cabinet lighting, add that to your list. My under cabinet lighting is the most helpful lighting I have. At a guess, I would say 2 pendants over the island, but your lighting store should also be able to help you. It depends on what you use and if your recessed lights light up the island - then pendants aren't necessary. Change out the track light over the sink for sure.