Same as this one (also note this), only replace "king" by "pawn". (I.e. shortest 960 helpmate, the mating side only uses pawn moves.) Yes, it can be done in the near-lying 5 moves with wK remaining put, even if that is a bit tricky. But...
(Please use your brain first, then verify with Jacobi.)
I have a genuine h#3.5, which can be animated because there's no castling.
[Title "Chess960: White moving only pawns helpmates in 3.5"]
[FEN "rkrqnbbn/pppppppp/8/8/8/8/PPPPPPPP/RKRQNBBN w - - 0 0"]
1. a4 a5 2. b4 Ka7 3. f3+ Ka6 4. b5#
I have further checked with Jacobi that there are no shorter helpmates for any of the 960 game arrays. So h#3.5 is as good as one can get.
Jacobi has checked exhaustively for all h#3.5 solutions. There is only one more matrix:
[Title "Chess960: White moving only pawns helpmates in 3.5"]
[FEN "rqkrbnnb/pppppppp/8/8/8/8/PPPPPPPP/RQKRBNNB w - - 0 0"]
1. d3 b6 2. b4 Kb7 3. g3+ Ka6 4. b5#
160/960 (i.e. one in six) of the starting positions allow h#3.5 with White only moving pawns. 108 give the first matrix, 42 give the second. Each valid starting position has exactly 8 solutions.
The Jacobi instructions are:
stipulation h#3.5
condition Chess960
constraint Xa1-a1(0..0) Xb1-b1(0..0) Xc1-c1(0..0) Xd1-d1(0..0) Xe1-e1(0..0) Xf1-f1(0..0) Xg1-g1(0..0) Xh1-h1(0..0)
So White officers cannot move. The "a1-a1" substring etc excludes castling, which curiously in Jacobi is normally not prevented by "(0..0)", the command to say that each piece must make at least 0 and at most 0 moves.
It's easy to check that if White makes 5 pawn moves, with Black not moving, i.e. seriesmover, then we have 512 solutions, with 64 starting positions each having 8 solutions, e.g. (QBSSRKBR) 1.b3 2.f4 3.f5 4.f6 5.fxg7#
Can Jacobi even solve such a puzzle? I suppose one can have a program like Popeye solve 960 helpmates in 5, 4.5, 4, ... until it finds no solution to any of them, but that's not much fun to set up -- and it might not even work, because . . .
[Title "Chess 960, after 0-0-0: Helpmate in 3.5, Black moving only pawns"]
[FEN "bnrkqnrb/pppppppp/8/8/8/8/PPPPPPPP/BNKRQNRB w - - 0 0"]
1. pass a5 2. b3 a4 3. Bxg7 a3 4. Bb2 axb2#
White's first move was the Chess960 "0-0-0", which I expect the Chess Stackexchange display won't do. Black then mates on move 4, moving only the a-pawn.
This also means that 3.5 moves would suffice if each K weren't constrained to start between its R's, for example:
[Title "Helpmate in 3.5, White moving only pawns"]
[FEN "bqnrrknb/pppppppp/8/8/8/8/PPPPPPPP/BQNRRKNB w - - 0 0"]
1. h4 g6 2. h5 Bxb2 3. h6 Bg7 4. hxg7#
