How many moves until checkmate? (White's turn)

This is a 7-piece position, so the Syzygy tablebases indicate it’s a win for white if played perfectly.
However, it doesn’t tell you an exact number of moves until checkmate (like #10, #20, etc), and there’s multiple branches if you follow the DTZ sequence.
Stockfish 19 also can't figure it out. It will take too long to reach a high depth, and it will just say it's a "+3" advantage for white.

The FEN for the position is the following:
8/6k1/8/P7/5R1p/7N/r7/6K1 w - - 0 1

Since all engines agree on the move, this is the next FEN to analyse...
FEN: 8/6k1/8/P4R2/7p/7N/r7/6K1 b - - 1 1

https://lichess.org/analysis/standard/8/6k1/8/P4R2/7p/7N/r7/6K1_b_-_-_1_1

21 Engine PV1 : Kg6
01 Engine PV1 : Ra1+

Now we can start the branching, but we should focus on the most popular FEN first. Go with the flow ...
21_FEN: 8/8/6k1/P4R2/7p/7N/r7/6K1 w - - 2 2
https://lichess.org/analysis/standard/8/8/6k1/P4R2/7p/7N/r7/6K1_w_-_-_2_2

01_FEN: 8/6k1/8/P4R2/7p/7N/8/r5K1 w - - 2 2
https://lichess.org/analysis/standard/8/6k1/8/P4R2/7p/7N/8/r5K1_w_-_-_2_2

All the engine results show only their PV1 best move choices. The prompt used was : go movetime 30000

We should still assume the mainline is the majority votes.
Rd5 Majority votes 19_FEN : 8/8/6k1/P2R4/7p/7N/r7/6K1 b - - 3 2
Rb5 Minority votes 03_FEN : 8/8/6k1/PR6/7p/7N/r7/6K1 b - - 3 2

As long as it is not under 64%, I think we can still assume the mainline is in the majority votes.
I'm assuming the majority knows best, but what if they were wrong.
Ra4 Majority votes 15_FEN : 8/8/6k1/P2R4/r6p/7N/8/6K1 w - - 4 3
Kf6 Minority votes 03_FEN : 8/8/5k2/P2R4/7p/7N/r7/6K1 w - - 4 3
Ra1+ Minority vote 01_FEN : 8/8/6k1/P2R4/7p/7N/8/r5K1 w - - 4 3

I imagine the white king blocking the isolated h-pawn. The white rook creating the wall that separates the black king from the pawns. I see the knight as the helper to promote the a-pawn.

For the pawn to promote imagine a position like this. Ever square for the pawn is protected :

If we play 15_FEN than we have a solid 100% Nf2 to play next.
22_FEN : 8/8/6k1/P2R4/r6p/7N/8/6K1 w - - 4 3
https://lichess.org/analysis/standard/8/8/6k1/P2R4/r6p/7N/8/6K1_w_-_-_4_3

So this is the next FEN to analyse : 8/8/6k1/P2R4/r6p/8/5N2/6K1 b - - 5 3
https://lichess.org/analysis/standard/8/8/6k1/P2R4/r6p/8/5N2/6K1_b_-_-_5_3
Just look how hard it can be to pick the mainline with those cp values.
Now you know why I use more than one engine to discover the mainlines.
It's like having one large pruned database and I don't need Tetra drives to do it.

Next move is:
https://lichess.org/analysis/standard/8/8/6k1/P2R4/7p/8/5N2/r5K1_w_-_-_6_4

DTZ 20 + DTM 16

"We should still assume the mainline is the majority votes."

• The majority may be wrong and the minority may be right.

@RemRus said ^

DTZ 20 + DTM 16

That's what I found out at first using the Syzygy tablebases, but when you get to DTZ 1 in the following position:
8/8/4k3/P6R/7p/5N2/1K6/r7 b - - 0 1
And it's black's turn, choosing Ra4 instead of Ra2+ (which just gets your rook captured by the king), will lead you to DTM 48.
Choosing the longest route when losing would be the most reasonable thing to do.

So maybe the longest optimal route is DTZ 20 + DTM 48? That is, mate in 34 moves?

White completed the conversion, reaching mate 2 moves slower than the stated DTM-optimal 48-move result.
Even if a move was considered a blunder by the server analysis, the blunder did not avoid the checkmate.

Critical turning points to review:

Move 30...Rc8: Black missed a more resilient defensive setup, increasing the theoretical distance to conversion.
Move 33...Ra7: The rook became less effective defensively, allowing White to strengthen the passed-pawn advance.
Move 35...Ra8?: This was the decisive defensive error, permitting the forced promotion sequence and rapid transition to a winning rook ending.
Move 50.Rh8#: White completed the conversion, reaching mate 2 moves slower than the stated DTM-optimal 48-move result.

30... Black - Ply 60 : Minority votes, try out the alternatives.
FEN: r7/8/P7/2N1k3/3R4/4K3/8/8 b - - 6 30
Rc8 [35.7% (5 votes), Score 0.0629, Median Rated 3347] *
Ra7 [14.3% (2 votes), Score 0.0459, Median Rated 3514]
Kf6 [7.1% (1 votes), Score 0.0189, Median Rated 3436]
Rh8 [28.6% (4 votes), Score 0.0166, Median Rated 3405]
Rg8 [7.1% (1 votes), Score 0.0156, Median Rated 3456]
Kf5 [7.1% (1 votes), Score 0.0000, Median Rated 3119]

33... Black - Ply 66
FEN: 8/2r5/PN2k3/8/R7/4K3/8/8 b - - 12 33
Ra7 [100.0% (14 votes), Score 0.1302, Median Rated 3394] *

35... Black - Ply 70 : Minority votes, try out the alternatives.
FEN: 2N5/r7/P4k2/R7/8/4K3/8/8 b - - 16 35
Ra8 [71.4% (10 votes), Score 0.0890, Median Rated 3410] *
Rc7 [28.6% (4 votes), Score 0.0608, Median Rated 3332]

There was a total of 69 Minority votes used in this game.