We could do an unmanned mission today, for not that much.
Nuclear pulse + slingshot and assembly in space. Cost per tonne, around 15 bucks, not including the cost of the slingshot.
Total cost would be around a couple trillion dollars, for a 50 year one way shot at AC. Of course there is plenty that could go wrong, but it could be done today with existing technology.
You could probably do it manned as well, if you were willing to spend 10x the cost to bring aboard the essentials for life aboard a spaceship for 50 years + the increase in nuclear fuel.
The only question is the will to do something rather silly like trusting 10 trillian dollars to 2 18 year old kids on a one way shot to AC, without any way to know if it’s been done thanks to relativistic effects.
Too bad about TAU,never heard of itIF only we would send a Hubble clone or similar "up" and one more "down" we might get a huge amount of positional data.But better save the money to give bonuses to government slugs and Wall Street money-mismanagers .
Look at our society: it is considered normal for a kid to spend hundreds of dollars and hours avidly playing some video game extolling car theft and street violence but spending a similar amount of effort in amateur astronomy,amateur radio,woodworking, or something productive is seen as "odd"!!!!