The message
from The Natural Philosopher contains these words:

And 3" dip is nothing for a bit of timber of that size. May have been
cut from a not particularly good bit of timber when green and it may
have simply shrunk to that shape under drying out, and possibly central
heating.

If the main span is not rotten, then there is little to worry about I'd say.

I agree that if the movement is ancient there shouldn't be much to worry
about.

I don't have the original message to look at (on a different computer
240 miles away) but I thought the op said he thought the 3" sag was
recent in which case it is a totally different ball game. Timber gets
stiffer as it seasons and more brittle with age.

If it is some peculiar effect caused by recently installed central
heating why was only this beam effected and why when shrinkage along the
grain is so limited would the beam bow. The op could still differentiate
between bowing and breaking though. If the beam looks like 2 straight
lengths with a hinge in the middle then it is likely to cracked at the
hinge if it is merely bowed then the chances are it hasn't broken.

If the ends of the beam are merely built in to the wall there probably
will not be sufficient rigidity for the beam to behave as a 'built in'
beam in the engineering sense of the term and the bow should extend from
one wall to the other. If however there is sufficient mass at each end
(or the beams continue through the wall to provide a jetty for the upper
storey) the shape would be rather different with 2 convex sections
joined by a concave section. With only 3" deflection that might be
difficult to distinguish.

--
Roger