What is the maximum tooth sprocket you could fit on the back using a Deore DX rear derailleur?

That is the main thing I took from it yes because you appear to have checked the vertical alignment of the hanger and tweaked it (and well done for doing so) but then ruled out the alignment being the issue for your continued indexing problems.

I'm pointing out that you haven't completed the job fully and there is more to do before you can rule out alignment.

From the pictures you can clearly see that there are massive alignment issues on both axis so until you correct that no amount of hanger extensions, friction shifter or anything else is going to help.

I'm also out at this point.
 
pics - hoorah!

its kinda hard to tell cos most of the photos from the rear aren't quite in line with the cage
good you got something sorted on the hanger measuring side - would concur that you should try and be a little more patient and sort that out more accurately - front to back as well as bottom to top. Taking the wheel out and reversing it so the cassette isn't in the way would be a good idea too.

The derailleur cage does look a little bent out of alignment on some of these images - which would obviously cause issues - but I'm not 100% sure of this

The other thing I noticed is that this appears to be the GS or so called 'medium cage' version of this RD which has a rated capacity of 33T - NOT 45T - the SGS or "super long cage" version that DOES have a 45T capacity looks like this:

1727294067248.png

With you're setup your significantly exceeding the rated capacity (11-40T at the back is 29T and then I'm presuming at least a 10T difference at the front between the 2 chainrings? so over 39T for sure. Even if its a 36T out back, you're still over, but admittedly not by much and it would probably be ok)

You could probably buy a longer cage to swap out (if its indeed bent that might not be a bad idea anyway)

This is the parts diagram for this derailleur - the only difference between GS and SGS is the inner and outer cage parts which can be purchased separately

1727294836185.png

SJScycles would the place I would go to see - they stock lots of older shimano spares - looks like the M770 parts are the same as M771:

inner plate: https://www.sjscycles.co.uk/gear-spares/shimano-deore-xt-rdm770-inner-plate-sgs-type-y5w510000/

I couldn't find the corresponding outer plate assembly on SJS's site

it might be cheaper to buy another derailleur TBH - but make sure you get an SGS one - you can find loads of suitable models for about 20 quid

While it looks like your chain is sized correctly for big/big (ie the angle isn't too bad in the photos and I'd agree that its' not stretching the RD too much) if I'm correct the issues will be more the higher gears in general. Especially with that tiny granny ring, the cage probably isn't long enough to provide adequate chain tension on the smaller sprockets which would cause a lot of chain slap and possibly the jockey wheels essentially clattering against one another. Would be good to get an image of small/small combination as well

The derailleurs guide pulley also looks uncomfortably close to that 40T sprocket at the back (again if it is a 40T, its too close if its a 36T as well!) - I think you're trying to push too big a sprocket for this derailleur without an extension or at least more B screw adjustment
 
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note "capacity" = the max range of teeth difference across front and back - NOT the maximum sprocket size at the back
I think you know this, but this does cause a lot of confusion with people mistaking one for the other
 
In the vintage Shimnao derailleur numbering scheme 770 would have been Low Normal aka Rapid Rise. and 771 would have been the conventional High Normal part. 10 speed and up they stopped offering that option.
 
note "capacity" = the max range of teeth difference across front and back - NOT the maximum sprocket size at the back
I think you know this, but this does cause a lot of confusion with people mistaking one for the other
Sometimes you can go a bit beyond the listed spec. with a longer B screw. It's been suggested that Shimano is conservative here on parts like the Zee FR derailleur because on long travel suspenaion bikes there can be some chain growth with suspension movement. The guys at MTBR which is a huge offroad site usually know what works and what doesn't. T least for MTB parts, which I like for street anyway.
Another way this limit has been breached is to only use the largest cogs in the smallest chain ring.
 
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Didn't have a bolt that was just the right fit for hanger thread so I got a close one and found two nuts, one for either side to hold it fast and a third to hold the wood fast. Took a couple of pictures but not uploaded

I would also be very careful that you're not damaging the hanger threads with this arrangement - getting a properly sized 10mm bolt that screws in correctly would be advisable :)
 
Sometimes you can go a bit beyond the listed spec

this is true of course but the question is always by how much :)
older RD's rated for 32 max sprockets can usually be pushed fine for 34T and sometimes 36T

capacity wise i've found that you can usually squeeze another few teeth without too many issues over the rated max but the more you push that the more you are compromising either one end (too stretched out on big/big) or the other (too much chain slap, jockey wheels rubbing on small/small)
 

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