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So is there a way to force a particular video mode? Or not?
I just looked up the standard way to set a particular mode to a KMS kernel at boot time, and one needs DRM for that. So no, you cannot do that for now.
Currently the HDMI driver's logic appears to be:
1. unless asked not to, try the highest-refresh mode among some low-refresh (<= 30Hz) 1080p modes (subject to aforementioned filtering)
2. unless asked not to, try the closest to a 'default' 720p mode (ditto)
3. pick the mode that is closest to the monitor's preferred mode (as declared by the monitor itself) - i.e. could be the preferred mode itself, if that was not filtered out (but chances are the preferred mode gets filtered out in your case, if that's a native 1080p set). Preferred mode is marked with an asterisk in the list of accepted modes, or with a captialized rejection code, in the rejected modes list.
4. if everything else fails, set the most vanilla 720p mode known (not subject to filtering).
From your description of the problem, the driver most likely gets 'stuck' somewhere on step 1 - it picks a low-refresh 1080p mode which, when attempted to be actually fed to the display, is met by a refusal.
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I also tried the demonstrational Android image, and it did get a display mode going with the Sony.
That just indicates that Android is not trying to feed your monitor the exact same mode the stock HDMI driver is. Another mode, by the same display/HDMI driver combo, could be absolutely fine. Apropos, what does the display say about the mode supplied by the Andorid image? There should be a source-signal info box somewhere in the display's on-board menus.
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Anyway, there was one full HD mode listed as supported, the one with 24 frames per second. What was wrong with that then? I don't know about the level of "sophistication". How sophisticated is it that after doing all this so-called logic, it finally decides to set a mode that the Sony TV cannot display?
Normally that should not be the case. Bar bugs in the kernel driver and/or monitor's firmware (yes, some monitors/tvsets report modes the former cannot actually reproduce). Definitely give the teneighty=0 option a try.