Random screen Freeze, but mouse is responsive

Question

Hi,
For the last month, I have been having random screen freezes , requiring me to hard-boot the computer. During this freeze the mouse is responsive. I wait 15 minutes to see if the freeze goes away before rebooting.
I have re-formatted the disk and installed Mountain Lion, again , but it did not help. Most of the time it seemed to happen when I am using Safari, but this is not always the case.
The hardware tests came clean.
I have isolated the console logs to the following just before the issue happens. Any suggestions on how to proceed?
9/20/13 5:24:58.068 AM
com.apple.backupd-helper[891]
Not starting scheduled Time Machine backup - time machine destination not resolvable.
9/20/13 5:25:06.836 AM
Safari[205]
CGContextClipToRect: invalid context 0x0
9/20/13 5:28:03.257 AM
fseventsd[51]
disk logger: failed to open output file /Volumes/NO NAME/.fseventsd/0000000001329d08 (No such file or directory). mount point /Volumes/NO NAME/.fseventsd
9/20/13 5:28:03.257 AM
fseventsd[51]
disk logger: failed to open output file /Volumes/NO NAME/.fseventsd/0000000001329d08 (No such file or directory). mount point /Volumes/NO NAME/.fseventsd
9/20/13 5:28:03.257 AM
fseventsd[51]
failed to unlink old log file /Volumes/NO NAME/.fseventsd/0000000001329d08 (No such file or directory)
9/20/13 5:28:03.257 AM
fseventsd[51]
failed to unlink old log file /Volumes/NO NAME/.fseventsd/0000000001329d08 (No such file or directory)
9/20/13 5:28:03.257 AM
fseventsd[51]
unmounting: failed to remove log dir /Volumes/NO NAME/.fseventsd (No such file or directory)
9/20/13 5:28:03.258 AM
fseventsd[51]
disk logger: failed to open output file /Volumes/My Passport for Mac/.fseventsd/0000000001329af1 (No such file or directory). mount point /Volumes/My Passport for Mac/.fseventsd
9/20/13 5:28:03.258 AM
fseventsd[51]
disk logger: failed to open output file /Volumes/My Passport for Mac/.fseventsd/0000000001329af1 (No such file or directory). mount point /Volumes/My Passport for Mac/.fseventsd
9/20/13 5:28:04.419 AM
DiskImages UI Agent[906]
*** -[NSMachPort handlePortMessage:]: dropping incoming DO message because the connection is invalid
9/20/13 5:28:05.565 AM
CoreServicesUIAgent[903]
Error: qtn_file_apply_to_path error: Read-only file system
9/20/13 5:28:06.687 AM
Installer[929]
CoreAnimation: warning, deleted thread with uncommitted CATransaction; set CA_DEBUG_TRANSACTIONS=1 in environment to log backtraces.
9/20/13 5:33:35.000 AM
bootlog[0]
BOOT_TIME 1379680415 0

Answer 1

Back up all data immediately as your boot drive may be failing.
If you have more than one user account, these instructions must be carried out as an administrator. I've tested them only with the Safari web browser. If you use another browser, they may not work as described.
Triple-click anywhere in the line below on this page to select it:
syslog -k Sender kernel -k Message CReq 'Channel t|GPU D|I/O|n Cause: -' | tail | open -ef
Copy the selected text to the Clipboard (command-C).
Launch the Terminal application in any of the following ways:
☞ Enter the first few letters of its name into a Spotlight search. Select it in the results (it should be at the top.)
☞ In the Finder, select Go ▹ Utilities from the menu bar, or press the key combination shift-command-U. The application is in the folder that opens.
☞ Open LaunchPad. Click Utilities, then Terminal in the icon grid.
Paste into the Terminal window (command-V).
The command may take a noticeable amount of time to run. Wait for a new line ending in a dollar sign (“$”) to appear.
A TextEdit window will open with the output of the command. Normally the command will produce no output, and the window will be empty. If the TextEdit window (not the Terminal window) has anything in it, post it — the text, please, not a screenshot. The title of the TextEdit window doesn't matter, and you don't need to post that.