Sum by Implicit Group
The "implicit" group you mention seems to be based on the row order.  Unlike a spreadsheet or text file, a relational table is logically an unordered set of rows regardless of whether it is stored as a heap or has a clustered index. It will not be possible to write a query to provide the desired results unless you have another column to facilitate the grouping.


  I was hoping I could create a deterministic row ordering with a query similar to the following:

SELECT      * 
        , ROW_NUMBER() OVER (ORDER BY (SELECT NULL)) AS RowNum  
FROM Table WITH(TABLOCK)
OPTION (MAXDOP 1)



Sorry to be the bearer of bad news, but ordering by a literal will not guarantee deterministic ordering (and even if it did, you'd need an ORDER BY clause there). If it appears to do so, it is only by happenstance. Any incremental column will work, though.


  I am hoping this forces an allocation order scan which would then give me a deterministic row order.


An allocation-ordered scan is no more deterministic than any other implementation; you're just (unsafely) relying on a different observed behaviour.