Windows 7 Service Pack 1 for x64-based Update fails with code 80073701

Not totally unexpected

Here's what appears to be the critical error from your attempted install...

Code:

  Stage (1): flags: 8 app: [synth3dvsc.inf-LanguagePack, Version = 6.1.7601.17514, pA = PROCESSOR_ARCHITECTURE_AMD64 (9), Culture = [l:10{5}]"en-US", VersionScope = 1 nonSxS, PublicKeyToken = {l:8 b:31bf3856ad364e35}, Type neutral, TypeName neutral, PublicKey neutral]) comp: (null) man: @0xa254458
2015-05-24 23:31:10, Info                  CBS    Exec: Resolving Package: Microsoft-Windows-InternetExplorer-Optional-Package~31bf3856ad364e35~amd64~en-US~8.0.7600.16385, Update: Internet-Explorer-Optional-amd64
2015-05-24 23:31:10, Info                  CBS    Exec: Resolving Package: Microsoft-Windows-InternetExplorer-Optional-Package~31bf3856ad364e35~amd64~en-US~8.0.7600.16385, Update: Internet-Explorer-Optional-amd64, PinDeployment: amd64_microsoft-windows-i..oyment-languagepack_31bf3856ad364e35_8.0.7600.16385_en-us_ed66154d5f1d25e1
2015-05-24 23:31:10, Error                 CSI    0000023a@2015/5/24:21:31:10.017 (F) d:\win7sp1_gdr\base\wcp\componentstore\csd_locking.cpp(324): Error STATUS_SXS_ASSEMBLY_MISSING originated in function CCSDirectTransaction::LockComponent expression: (null)
[gle=0x80004005]
2015-05-24 23:31:11, Error                 CSI    0000023b (F) STATUS_SXS_ASSEMBLY_MISSING #1229948# from CCSDirectTransaction::OperateEnding at index 0 of 1 operations, disposition 2[gle=0xd015000c]
2015-05-24 23:31:11, Error                 CSI    0000023c (F) HRESULT_FROM_WIN32(ERROR_SXS_ASSEMBLY_MISSING) #1229833# from Windows::ServicingAPI::CCSITransaction::ICSITransaction_PinDeployment(Flags = 0, a = Microsoft-Windows-InternetExplorer-Optional-Deployment-LanguagePack, Version = 8.0.7600.16385, pA = PROCESSOR_ARCHITECTURE_AMD64 (9), Culture = [l:10{5}]"en-US", VersionScope = 1 nonSxS, PublicKeyToken = {l:8 b:31bf3856ad364e35}, Type neutral, TypeName neutral, PublicKey neutral, cb = (null), s = (null), rid = [128]"Microsoft-Windows-InternetExplorer-Optional-Package~31bf3856ad364e35~amd64~en-US~8.0.7600.16385.Internet-Explorer-Optional-amd64", rah = (null), manpath = (null), catpath = (null), ed = 0, disp = 0)[gle=0x80073701]
2015-05-24 23:31:11, Info                  CBS    Failed to pin deployment while resolving Update: Internet-Explorer-Optional-amd64 from file: (null) [HRESULT = 0x80073701 - ERROR_SXS_ASSEMBLY_MISSING]
2015-05-24 23:31:11, Info                  CBS    Failed to resolve item[0] in Package: Microsoft-Windows-InternetExplorer-Optional-Package~31bf3856ad364e35~amd64~en-US~8.0.7600.16385, Update: Internet-Explorer-Optional-amd64 [HRESULT = 0x80073701 - ERROR_SXS_ASSEMBLY_MISSING]
2015-05-24 23:31:11, Info                  CBS    Failed to resolve execution update. [HRESULT = 0x80073701 - ERROR_SXS_ASSEMBLY_MISSING]
2015-05-24 23:31:11, Error                 CBS    Failed to resolve execution package: Microsoft-Windows-InternetExplorer-Optional-Package~31bf3856ad364e35~amd64~en-US~8.0.7600.16385 [HRESULT = 0x80073701 - ERROR_SXS_ASSEMBLY_MISSING]
2015-05-24 23:31:11, Info                  CSI    0000023d@2015/5/24:21:31:11.927 CSI Transaction @0x1bb6b60 destroyed
2015-05-24 23:31:11, Info                  CBS    Perf: Resolve chain complete.

I'm at a loss to understand this one at the moment, and need to do some research

You may want to consider a Repair install of Windows if you have access to the Win7 media

DISM does NOT work the same way in Windows 7 as it does in Win 8!

Please start the whole process again with a new set of diagnostics - maybe we can avoid the hassle of a repair install..

SFC is error-free, as is CheckSUR.
The error message implies registry corruption, which would normally be found by CheckSUR unless buried in the SOFTWARE hive rather than the COMPONENTS hive.

I'm neither competent nor confident in that hive

The first significant error from your latest attempt to install SP1 is this...

Code:

2015-05-25 11:20:21, Info                  CBS    SPI: Self trust verification failed for C:\Windows\System32\SPC_SP1_Wave1.xml.  0x57
2015-05-25 11:20:21, Error                 CBS    SPI: (SPCVerifyTrust:1664)Failed to CryptCATAdminCalcHashFromFileHandle GLE=0x57
2015-05-25 11:20:21, Info                  CBS    SPI: Installed SPInstall or XML not trusted.  Continuing with current version.
2015-05-25 11:20:21, Info                  CBS    SPI: Self trust verification failed for C:\Windows\System32\EventProviders\spcmsg.dll.  0x57

... which tends to suggest a corrupt download - I assume this was from the full standalone download?

It's certainly worth a try!
https://www.microsoft.com/en-us/down...s.aspx?id=5842 - you want the windows6.1-KB976932-X64.exe file

Great! - You're welcome, and good luck.