Is this position a pure mirror mate?

Question

Task

A pure mirror checkmate is a checkmate position, where the mated king is surrounded by 8 empty squares which are attacked or guarded by the winning side exactly once. You will be given a valid chess position where Black is in mate. Your task is to find if that mate is a pure mirror checkmate.

Examples ^{_{(click for larger images)}}

1. This is a pure mirror mate. Squares f5, g5, are protected by the king only, h5, h6, h7 by the rook only and f6, f7, g7 by the two knights only.

   

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2. This position does not match the criteria. It is a mirror mate, but it's not pure, as d7 is guarded by the bishop and attacked by the rook.

   

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3. This position is not a pure mirror mate as well, since the king is not surrounded by 8 empty squares.

   

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4. This is a pure mate, but not a mirror mate (not all 8 squares surrounding the king are empty), therefore it doesn't meet the criteria:

   

Input

As mentioned above, your input will be a valid chess position, where black will be in mate. You may take the input as an FEN or as a grid/matrix/2D array. From the examples above:

1. first position:

   4N2N/4q3/6k1/8/5K2/8/8/7R
   
   . . . . N . . N
   . . . . q . . .
   . . . . . . k .
   . . . . . . . .
   . . . . . K . .
   . . . . . . . .
   . . . . . . . .
   . . . . . . . R
   

2. second position

   8/R7/4k3/2P3P1/3K4/7B/8/8
   
   . . . . . . . .
   R . . . . . . .
   . . . . k . . .
   . . P . . . P .
   . . . K . . . .
   . . . . . . . B
   . . . . . . . .
   . . . . . . . .
   

3. third position

   7b/8/8/5N2/8/7k/5N2/7K
   
   . . . . . . . b
   . . . . . . . .
   . . . . . . . .
   . . . . . N . .
   . . . . . . . .
   . . . . . . . k
   . . . . . N . .
   . . . . . . . K
   

4. fourth position:

   2NN4/4k3/1b2NNb1/8/6N1/8/6K1/q6n
   
   . . N N . . . .
   . . . . k . . .
   . b . . N N b .
   . . . . . . . .
   . . . . . . N .
   . . . . . . . .
   . . . . . . K .
   q . . . . . . n
   

Rules

• The chess position will always be valid and black will always be in mate.
• There will be no double checks (the king won't be attacked twice).
• You can receive input through any of the standard IO methods.
• This is code-golf, so shortest code in bytes wins!

Answer 1

Python + chess, 175 166 161 bytes

def f(e):b=chess.Board(e);s=b.attacks(k:=b.king(0));b.remove_piece_at(k);return(any(len(b.attackers(1,q))>1or b.piece_at(q)for q in s)or len(s)<8)^1
import chess

Attempt This Online!

-9 bytes thanks to rule clarification by @l4m2
-4 bytes because I realised I could just use chess.Board instead of chess.BaseBoard

Takes the fen as input and outputs 1 for a pure mirror mate and 0 otherwise. Using the chess library certainly made this easier. Explanation:

def f(e):
    b=chess.Board(e)                # Generates the board with the input fen
    k=b.king(0)                     # Finds the square of the black king
    s=b.attacks(k)                  # Get a generator of all squares surrounding the king by finding the squares he attacks
    b.remove_piece_at(k)            # Remove the king from the square to allow us to find guarded squares
    return(
        any(                        # If any of the following return True
            [len(b.attackers(1,q))>1# There is more than one attacker
                or b.piece_at(q)    # Or there is a piece present
                for q in s]         # In the squares surrounding the king
            )or                     # or
        len(s)<8                    # There are less than 8 squares surrounding the king
        )^1                         # Flip boolean and return

Answer 2

JavaScript (Node.js), 381 334 330 bytes

B=>(Q=q=>B.some((b,i)=>b.some((R,j)=>q?+q?'*kk'[(K-i)**2+(L-j)**2]==R&&!(Q[++k,[K,L]]^=1):!eval(q,x=I-i,y=J-j)&R=='.'&&Q(1,K=i,L=j):(I=i,J=j,h=(m,n,r=4,p=i+m,q=j+n,x=B[K=p])=>x&&(x[L=q]in{k,'.':0})?Q(1)|h(m,n,r,p+m,q+n):r&&h(n,-m,r-1),z={N:`x*x-5`,P:`-x`,K:`x*x>2`}[R])?Q('y*y+'+z):R in{R,Q}&&h(1,0)||R in{B,Q}&&h(1,1))))(k=0)|k<8

Try it online!

-2 bytes from Kevin Cruijssen

-4 bytes from emanresu A

foo = a=>g1(k=0)|k<8;
g1 = q=>a.some((b,i)=>b.some((c,j)=>(
    I=i,J=j,
    c=='N'?g2((x,y)=>x*x+y*y-5): // Knight
    c=='P'?g2((x,y)=>y*y-x):
    c=='K'?g2((x,y)=>x*x+y*y>2):
    c=='R'?h(1,0):
    c=='B'?h(1,1):
    c=='Q'?h(1,1)|h(1,0):
    0
)));
g2 = q=>a.some((b,i)=>b.some((c,j)=>
    !q(I-i,J-j)&  // Correct destination
    c=='.'&&      // Empty 
    g3(1,K=i,L=j) // then go
));
g3 = q=>a.some((b,i)=>b.some((c,j)=>
    [,1,1][(K-i)**2+(L-j)**2]&c=='k'? // Near king
    !(g[M=[K,L]]=!g[++k,M]) // count as an attack
    :0                      // and memory the pos
));
h = (m,n,r=4,p=i+m,q=j+n,x=a[K=p])=>
    x&&(x[L=q]=='.'|x[q]=='k')? // Not blocked
    g3(1)|h(m,n,r,p+m,q+n):      // Next step
    r&&h(n,-m,r-1) // rotate 90 degree