The push-off / push-on cap
The all-the-time (inseparable) attached bottle cap already exist, and the problems here are (depending on their type): are expensive (special because there are two pieces, produced separately and then combined together), not so easy to use, unsafe (for sparkling sodas or on handling could pop-up), they could produce another small protective cap or tear-off waste and – in some cases – the bottles cannot be refilled ( reused).
There are too many types of bottles and contents, so it will not be an universal solution, but maybe ( and hopefully ) we’ll have more small ones.
Something like a collar on the bottle neck with an attachment to the cap for sure is better then the existing free disposal cap model, as long as the cap move free on the attachment ( like a small bump on the cap with a small lasso, or stretched circle, on the collar’s attachment). It already exist, special in the multi-use bottles, but the screw-type cap combined with the collar is not quite functionally.
One of the solutions could be the push-on / push-off cap. The bottle (plastic or glass) have to have a rim on top (fig. 1) (the glass bottles already have that rim), and another rim a few millimeters down from the top (like on the plastic ones). On the top rim the tight cap could be stretched, passing after the rim, by pressing in the middle or all around ( depending on the diameter)(fig.2). The cap have an attachment with the collar (collar on the bottle neck , passed down of the bottom rim), and is only one piece of seal, non-disposal, opposite with this attachment and easy to tear it (same width like the attachment but shorter, for example, but for the handling is better the attachment to be thinner, as thin as the collar). The broken seal will serve like handle for taking out the cap (the push-off movement). Is not exactly necessary to have an (special) elastic material for the cap, is enough a thinner material on top with a thicker collar (rim) all around.(fig.3)
For certain types of bottles, or bottle mouth diameters, or for certain drinks could be two seals, on opposites sides of the cap, to keep it safe.
That kind of bottle caps are much easier to handle, even with only one hand, so that could mean that they could be welcomed by the customers!
Another (very) good thing, the bottle mouth diameter could be bigger than the actually standards, making the refilling much easier ( that means the reusing will be more probably!).
Another good thing is the plastic economy, for (almost) the same kind of bottle we have (calculated) up to 20 % less plastic! (or, for the same amount of plastic, an diameter increase of up to 15%!). Great!
In that way the plastic cap waste problem will be (partially) solved, the cap will stay together with the bottle!
ADD could be used for the glass bottles, too!