I must reiterate that I do not wish to use a negative regulator in this design.
And that's perfectly fine. It's not different than two positive lab supplies hooked together to make +/-. You've just put them in one box and freaked some people out
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The problem is that your schematic shows a line under your diodes (D1 and D4) , those dots are showing a connection to that green line under the diode symbols. You may not have intended that, so you don't see it, but other people will point it out but you will now be talking at cross purposes. If you ever make a netlist from this schematic to go to a PCB, the short circuit WILL end up on the PCB unless you notice it. Then you've got problems.
The diodes are there to protect the regulators because usually when Vout>Vin+.7V, they fry. The diodes prevent that, they are cheap.
I would also reduce that 1500uF output cap. Maybe check the transformer specs at max current, and the filter cap.
The only thing is I'd get rid of is the connection between the two supplies, and make that connection externally, for future flexibility. Banana jacks and cables are cheap. And if you are going to put this thing in a box, maybe some input protection on the AC side like a fuse and a common mode filter, with a connection for earth to chassis.
You've basically built my student project from college, a dual linear supply with a Hammond transfo, and a sheet metal box. The thing is robust and I still use it today.
Of course I am old and lazy and if I wanted a new supply, I would buy an already made supply, if only for the transformer and input is already taken care of.
https://www.ebay.ca/itm/International-Power-IHBAA-40W-100-240V-Input-5VDC-Output-Power-Supply/274523214819?epid=1933579616&hash=item3feadbd3e3:g:qP0AAOSwewdffNlLThese are cheap and great. I have one of those, and if you were in Montreal we could arrange a pick up since I rarely use it. 20$?
You could then use the chassis to hack away to your heart's content, or just use it as-is... 20$ doesn't even get you a decent transformer IMO.