Hard Disk GPT Header Corrupted

Theoretically everything is fine with your Fusion Drive. Fusion Drives look like this. Disk0 is your SSD with 121 GB and disk1 is your HDD with ~1 TB (~1.121 TB summed up).

The larger parts of your SSD (disk0s2) and your HDD (disk1s2) are pooled to a CoreStorage LVG (Fusion Drive: disk3) with a size of 967.8 GB. The rest is reserved for EFIs, a Recovery HD (alltogether ~1.3 GB) and your old Windows partition - now probably free space (~152 GB).

The logical volume 'Macintosh HD' (967.8 GB) spans disk0s2 and disk1s2. This is the first 'Macintosh HD' in picture 1. The volume 'Macintosh HD' - it's the one visible on the desktop - should ideally also have about 967.8 GB. This is the second 'Macintosh HD' in picture 1.
In fact it has only 852.67 GB (see picture 3).

In the second picture the logical volume 'Macintosh HD' is the first listed in black, the volume 'Macintosh HD' is the second listed in black, the other two 'Macintosh HD's listed in grey are the parts of your SSD and HDD dedicated to the logical volume 'Macintosh HD'.

In my opinion something went wrong after deleting various partitions with the Bootcamp Assistant/Disk Utility or in Windows.

Preparation:

• Detach any external drive (especially your external Time Machine backup drive)

• Restart to Internet Recovery Mode by pressing alt cmd R at startup.
  The prerequisites are the latest firmware update installed, either ethernet or WLAN (WPA/WPA2) and a router with DHCP activated.
  On a 50 Mbps-line it takes about 4 min (presenting a small animated globe) to boot into a recovery netboot image which usually is loaded from an apple/akamai server.

  I recommend ethernet because it's more reliable. If you are restricted to WIFI and the boot process fails, just restart your Mac until you succeed booting.

  Alternatively you may start from a bootable installer thumb drive (preferably Mavericks or Yosemite) or a thumb drive containing a full system (preferably Mavericks or Yosemite).

Now you may either repair CoreStorage or rebuild your Fusion Drive:

'Repair CoreStorage' (not recommended):

• First i would try to check the volume 'Macintosh HD' with Disk Utility. If the volume is corrupted consider a reinstall of Mac OS X.

• If the volume is ok quit Disk Utility

• Open Terminal and enter diskutil unmountDisk /dev/LVIdentifier and both diskutil unmountDisk /dev/DiskContainingApple_CoreStorageIdentifier
  In your case: first diskutil unmountDisk /dev/disk3 then diskutil unmountDisk /dev/disk0 and diskutil unmountDisk /dev/disk1
• remove the EFI NO NAME partition with gpt remove -i IndexNumberOfEFINoName DiskIdentifier:
  gpt remove -i 4 disk1
• Remount the CoreStorage disks and then the Logical Volume:
  In your case: first diskutil mountDisk /dev/disk0 and diskutil mountDisk /dev/disk1 and then diskutil mount /dev/disk3.

• enter gpt -r -vvv show /dev/diskIdentfierOfApple_CoreStorage to get infos of your HDD CoreStorage disk.
  In your case: gpt -r -vvv show /dev/disk1
  It should look like this:

  -bash-3.2# gpt -r -vvv show /dev/disk1
  gpt show: /dev/disk1: mediasize=1000204886016; sectorsize=512;         blocks=1953525168
  gpt show: /dev/disk1: PMBR at sector 0
  gpt show: /dev/disk1: Pri GPT at sector 1
  gpt show: /dev/disk1: GPT partition: type=C12A7328-F81F-11D2-BA4B-00A0C93EC93B, start=40, size=409600
  gpt show: /dev/disk1: GPT partition: type=53746F72-6167-11AA-AA11-00306543ECAC, start=409640, size=1671210848
  gpt show: /dev/disk1: GPT partition: type=426F6F74-0000-11AA-AA11-00306543ECAC, start=1671620488, size=1269760
  gpt show: /dev/disk1: Sec GPT at sector 1953525167                               
       start          size  index contents                                        
           0             1        PMBR                                            
           1             1        Pri GPT header                                  
           2            32        Pri GPT table                                   
          34             6
          40        409600      1 GPT part - C12A7328-F81F-11D2-BA4B-00A0C93EC93B
      409640    1671210848      2 GPT part - 53746F72-6167-11AA-AA11-00306543ECAC
  1671620488       1269760      3 GPT part - 426F6F74-0000-11AA-AA11-00306543ECAC
  1672890248     280634887
  1953525135            32        Sec GPT table
  1953525167             1        Sec GPT header
  

• The free space on your HDD has 280634887 blocks. Please calculate the biggest block number dividable through 8. That's 280634880 blocks (á 512 bytes) which equals 143685058560 B or ~143.7 GB. Add the size of your HDD CoreStorage Physical Volume (852666400768 B) The result is 143685058560 B + 852666400768 B = 996351459328 B

• Resize your HDD CoreStorage physical volume with diskutil cs resizeDisk HDDPVUUID newsize
  In your case: diskutil cs resizeDisk 93892BE8-2B7F-4ABD-A4C3-984495DCD98D 996351459328b
• Calculate the maximal size of your CoreStorage Logical Volume in diskutil cs list: (size disk0s2) + (size disk1s2) In your case that's 120988852224 B + 996351459328 B = 1117340311552 B. That should be the size of your refreshed Logical Volume Group.
• Resize your Logical Volume with diskutil cs resizeVolume LVUUID LVGSize-128 MB In your case that's diskutil cs resizeVolume D237FFDC-7DA4-41D7-AC13-4CC7E5E8C0A0 1117212311552b. If you get an error (There is not enough free space...) choose a smaller size like 1117148311552b.
• Quit Terminal and open Disk Utility.
• Check your expanded CoreStorage Volume for errors.
• Quit Disk Utility, choose your CS volume as startup disk and restart your Mac

'Rebuild Fusion Drive' (recommended if you have a Time Machine backup)

• Booted to Internet Recovery Mode open Utilities → Terminal in the menubar and enter:
  diskutil cs list to get the CoreStorage listing.
• Copy the Logical Volume UUID, it's the fifth listed.
• Now delete the Logical Volume with diskutil cs deleteVolume LVUUID.
  In your case: diskutil cs deleteVolume D237FFDC-7DA4-41D7-AC13-4CC7E5E8C0A0.
• Copy the Logical Volume Group UUID, it's the first listed in the listing of diskutil cs list.
• Then delete the Logical Volume Group with diskutil cs delete LVGUUID.
  In your case: diskutil cs delete 1EFE58BC-3613-44C4-86EE-D816F3B66E3E
• Enter exit and quit 'Terminal'
• Open 'Disk Utility'. Enter 'Ignore' if you are asked to fix the drives.

• Choose your SSD and partition it: 1 Partition Mac OS X Extended (Journaled), hit the Options button and choose GUID Partiton table and hit OK and Apply.
  Please check that the size is ~121 GB

  Example:

  

• Choose your HDD and partition it: 1 Partition Mac OS X Extended (Journaled), hit the Options button and choose GUID Partiton table and hit OK and Apply.
  Please check that the size is ~1 TB

  Example:

  

• Quit Disk Utility and open Terminal

• Enter diskutil list

  Example (your disk identifiers and sizes are different of course: Your volume SSD probably has the Identifier disk0s2 and the size 121 GB and your volume HDD probably has the Identifier disk1s2 and the size 1.0 TB):

  

• Enter diskutil cs create "Name" IdentifierSSD IdentifierHDD
  In your case probably diskutil cs create "Macintosh HD" disk0s2 disk1s2.

  Copy the resulting LVGUUID

  Example:

  

• Enter diskutil cs CreateVolume LVGUUID jhfs+ "Macintosh HD" 100%.

  Example:

  

• Enter diskutil cs list
  Check the size of your Logical Volume. It should have the size ~1.121 TB

  Example:

  

• Quit Terminal

• Open 'Disk Utility' and check your newly created volume for errors
• Quit 'Disk Utility'
• Attach your external Time Machine backup drive or check this answer if you use NAS or another network share.
• Open 'Restore from Time Machine Backup'
• Choose the appropriate Time Machine backup and restore your system
• Reboot to your restored system.
• Unmount and detach your Time Machine backup drive
• Open 'Terminal' and enter 'diskutil list'
• Check if your 'Recovery HD' is listed.
• If your 'Recovery HD' is missing, usually reinstalling your current system with the latest available system installer (e.g. 'Install OS X Mavericks (10.9.5)' if Mavericks is currently installed) will recreate it without loosing any data. AFAIK Recovery Partition Creator 3.8 will NOT create a Recovery HD on CoreStorage volumes.
• After reinstalling the system with the latest available system installer open App Store and install the latest security fixes.

The problem turned out to be the PMBR. My first attempt was to copy sector 0 off a GPT formatted memory stick.

Examined that with fdisk and it had partition of type EE, but start and end cylinders were the same.

I edited partition 1 and accepted defaults.

fdisk on mac will report "can't get exclusive access to write sector back, reboot required" It lies. You don't have to reboot.

After doing this diskutil was able to see both partitions.

I was unable to repair the first partition, used for time machine, and ended up reformatting it. (Even after it was a recognizable partition, diskutil would start on it, and give up in a few seconds.) (I didn't try very hard. The really important stuff is stored on drop box, on a mirrored disk on another machine.)

@Klanomath deserves credit for GPT disk. I had found this, but missed the fact that it provided Mac binaries. I'm not used to binaries on Source forge. GPT disk appears to be a useful tool for dealing with this sort of situation. It would have been a better way to go.