Newby from Delaware...

[echo1]

Welcome from Kali, don't cut chit, PAX

[Phosphorus32]

Given how far that unfired cartridge inserts, it appears that you just have the neck or a bit more than that stuck in the chamber.

If nothing above is working (always best to punch out from the muzzle end), then the next to last resort is the following:
Get it stripped down to the barreled action
Scrub it out with a chamber brush and penetrating oil, wipe "clean"
Shine a light in the chamber to see if you can see thee edges of the broken case
Using a BRASS chisel, punch, or empty fired 5.56 case, and a small hammer, get the edge of the chise/punch on the edge of the broken case and tap-tap-tap gently
The objective is to break an edge free of the chamber without driving it further up the throat, and then hitting it with penetrating oil again.

[wpage]

So trying these tips and now convinced not to cut it up. Will try the empty case idea. The freeze and heat also is in progress. I have a rotary tool with some small stones to grind some if all else fails.

[Murray]

No grinding  stones in there, please. If there's enough of the case still in the chamber, more than just the shoulder and neck, then spend a few dollars on a M-10mm flat bottom tap. You can carefully slip the tap in the chamber until you feel the case, then tap that case just one revolution or two, and then use a cleaning rod through the muzzle end to knock the case and tap right out the back. M10x1.5 got this one out easily.

[wpage]

10 mm tap idea is great...
Will attempt. All else has failed so far. Wish me luck !
Thanks to all for these excellent suggestions.