sumproduct to count multiple criteria (once per row)

I'm nearly there but I'm stumped on the logic... (I've attached an example workbook with my problem, and my own attempt to fix it.)

At first I used COUNTIFS, but I moved the same logic over to SUMPRODUCT as I wanted an array to be tested in the 3rd criterion.

I'm looking for a formula to count the rows that:
A) are >0 in column J
B) are blank in column I
C) have the sum of columns $C:H as >0

what I've tried
=SUMPRODUCT((J9:J99>0) * (I9:I99="") * ($C9:H99>0))

which nearly gets it, BUT instead of counting the rows where the above criteria are all true (once per row), it instead counts rows multiple times if there are multiple non=blanks in the C:H range of that row...

HELP!
Thanks

Not sure what result you are expecting but try:

Formula:

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Please try

=SUMPRODUCT((J$9:J$101>0)*(I$9:I$101="")*(MMULT((N(+$C$9:H$101)),{1;1;1;1;1;1})>0))

yeah I thought wrapping it in SUM() would work too, but sadly not.
ie it gives a result, but not the result i need...
It was after trying to untangle why that didn't work that lead me to ask greater minds than mine!
Thanks though