[right]  at (1,1.4)     $\geq2$;

[line width=1pt,auto,scale=#1]
[-](-1,0) to (1,0);
[-](-1,0) to [out=90,in=180] (1,1.5);
[-](1,1.5) to     (1,0);
(-1,0) circle (3pt);
(1,1.5) circle (3pt);
(1,0) circle (3pt);
[left]   at(-1,0)      $0$;
[below]  at (1.3,0)     $\ve[\iota]$;
[right]   at(1,1.5)   $y$;
[right]  at (1,0.7)    $=1$;
[below]  at (0,0)     $=1$;
[left]  at (0,1.4)     $\geq2$;

[-] (7,0)–(5,0);
[-] (5,0)–(5,1.5);
[-] (5,1.5)– (7,1.5);
[-]  (7,0) –(7,1.5);
(5,0) circle (3pt);
(5,1.5) circle (3pt);
(7,1.5) circle (3pt);
(7,0) circle (3pt);
[left]   at(5,0)       $0$;
[right]   at (7,0)       $\ve[\iota]$;
[right]   at (7,1.5)      $y$;
[left]   at (5,1.5)      $v$;
[below]  at (6,0)        $=1$;
[left]   at (5,.9)     $\geq 1$;
[above]   at (6,1.5)     $\geq 1$;
[right]  at (7,0.9)    $= 1$;

[line width=1pt,auto,scale=#1]
[-](4,0) to (4,2);
[-](4,2) to (6,2);
[-](6,0) to    (6,2);
[-](4,0) to    (6,0);
(4,0) circle (3pt);
(6,0) circle (3pt);
(6,2) circle (3pt);
(6,0) circle (3pt);
[left]   at (4,0)      $0$;
[left]   at (4,2)      $y$;
[right]  at (6,0)     $\ve[\iota]$;
[right]  at (6,2)     $x$;
[above]  at (5,2)  $=1$;
[left]  at (4,1)  $\geq 1$;
[below]  at (5,-0.1)  $= 1$;
[right]  at (6,1)     $\geq 1$;

[line width=1pt,auto,scale=#1]
[-](0,0) to (-1+0.25,1.5);
[-](0,0) to (2,0);
[-](2,0) to (3-0.25,1.5);
[-](3-0.25,1.5) to  (1,2.5);
[-](-1+0.25,1.5) to (1,2.5);
(0,0) circle (3pt);
(2,0) circle (3pt);
(3-0.25,1.5) circle (3pt);
(-1+0.25,1.5) circle (3pt);
(1,2.5) circle (3pt);

[left]   at(0,0)      $0$;
[right]  at (2,0)     $\ve[\iota]$;
[above]   at(1,2.5)     $y$;
[right]   at(2.75,1.5)     $x$;
[left]   at(-0.75,1.5)     $v$;

[left]  at (-.5,0.6)      $\geq 1$;
[above]  at (2.3,2.1)      $=1$;
[below]  at (1,0)     $=1$;
[right]  at (2.5,0.6)   $\geq1$;
[above]  at (0,2.1)   $\geq 0$;

[line width=1pt,auto,scale=#1]
[-](4,0) to (4,2);
[-](4,2) to (6,2);
[-](6,0) to    (6,2);
[-](4,0) to    (6,0);
(4,0) circle (3pt);
(6,0) circle (3pt);
(6,2) circle (3pt);
(4,2) circle (3pt);
[left]   at (4,0)      $0$;
[left]   at (4,2)      $v$;
[right]  at (6,0)     $\ve[\iota]$;
[right]  at (6,2)     $y$;
[above]  at (5,2)  $\geq 0$;
[left]  at (4,1)  $\geq 1$;
[below]  at (5,-0.1)  $= 1$;
[right]  at (6,1)     $\geq 2$;

[line width=1pt,auto,scale=#1]
[-](0,0) to (-1+0.25,1.5);
[-](0,0) to (2,0);
[-](2,0) to (3-0.25,1.5);
[-](3-0.25,1.5) to  (1,2.5);
[-](-1+0.25,1.5) to (1,2.5);
(0,0) circle (3pt);
(2,0) circle (3pt);
(3-0.25,1.5) circle (3pt);
(-1+0.25,1.5) circle (3pt);
(1,2.5) circle (3pt);

[left]   at(0,0)      $0$;
[right]  at (2,0)     $\ve[\iota]$;
[above]   at(1,2.5)     $y$;
[right]   at(2.75,1.5)     $x$;
[left]   at(-0.75,1.5)     $v$;

[left]  at (-.5,0.6)  $\geq 0$;
[above]  at (2.3,2.1) $\geq 2$;
[below]  at (1,0)     $=1$;
[right]  at (2.5,0.6) $\geq 1$;
[above]  at (0,2.1)   $\geq 0$;

[line width=1pt,auto,scale=#1]
[-] (0,0)–(1,1.5);
[-] (1,1.5)–(2,0);
[-] (2,0)–(0,0);
[-] (0,0)–(-1,0);
(-1,0) circle (3pt);
(0,0) circle (3pt);
(1,1.5) circle (3pt);
(2,0) circle (3pt);
[above]   at(-1,0)       $\ve[\iota]$;
[below]   at(0,0)       $0$;
[right]   at (2,0)       $x$;
[left]   at (1,1.5)      $v$;

[line width=1pt,auto,scale=#1]
[-](0,0) to [out=90,in=90]    (3,0);
[-](0,0) to [out=270,in=270]    (3,0);
(0,0) circle (3pt);
(3,0) circle (3pt);
[left]   at(0,0)      $0$;
[right]  at (3,0)     $\ve[\iota]$;
[above]  at (1,0.7)     $\geq 1$;
[below]  at (2,-0.7)     $\geq 3$;

[line width=1pt,auto,scale=#1]
[-](0,0) to [out=270,in=270]    (3,0);
[-](0,0) to (3,0);
[-](3,0) to [out=90,in=0]    (0,1);
(0,0) circle (3pt);
(3,0) circle (3pt);
(0,1) circle (3pt);
[left]   at(0,0)      $0$;
[right]  at (3,0)     $x$;
[left]  at (0,1)     $\ve[\iota]$;

[line width=1pt,auto,scale=#1]
[-] (0,0)– (2,0);
[-] (0,0)– (0,1.5);
[-](0,1.5) to [out=180,in=90]    (-2,0);
[-](0,1.5) to [out=0,in=90]    (2,0);
(0,0) circle (3pt);
(0,1.5) circle (3pt);
(2,0) circle (3pt);
(-2,0) circle (3pt);
[left]   at(0,0)      $0$;
[right]  at (2,0)     $x$;
[above]   at (0,1.5)   $v$;

[below]  at (1,0)     $=1$;
[below]  at (-0.6,1.2)     $=1$;
[above]  at (2,0.8) $\geq 2$;
[above]  at (-2,1) $\geq 2$;

[line width=1pt,auto,scale=#1]
[-] (0,0)–(1,2);
[-] (1,2)–(2,0);
[-] (2,0)–(0,0);
[-] (1,2) to [out=180,in=90] (-1,0);
(-1,0) circle (3pt);
(0,0) circle (3pt);
(1,2) circle (3pt);
(2,0) circle (3pt);
[below]   at(-1,0)       $\ve[\iota]$;
[below]   at(0,0)       $0$;
[right]   at (2,0)       $x$;
[right]   at (1,2)      $u$;
[above]  at (2.2,0.6)        $\geq 1$;
[left]   at (-1,0.9)     $\geq 2$;
[below]  at (1.2,0)    $=1$;

[line width=1pt,auto,scale=#1]
[-] (0,0)–(1,1.5);
[-] (2,0)–(1,1.5);
[-] (2,0)–(0,0);
[-] (1,1.5) to [out=180,in=90] (-1,0);
(-1,0) circle (3pt);
(0,0) circle (3pt);
(1,1.5) circle (3pt);
(2,0) circle (3pt);
[below]   at(-1,0)       $\ve[\iota]$;
[below]   at(0,0)       $0$;
[right]   at (2,0)       $x$;
[right]   at (1,1.5)      $u$;
[above]  at (2.2,0.4)        $\geq 1$;
[left]   at (-1,0.7)     $\geq 1$;
[below]  at (1.2,0)    $\geq 2$;

[line width=1pt,auto,scale=#1]
[-](4,0) to (4,2);
[-](4,2) to (6,2);
[-](6,0) to    (6,2);
[-](4,0) to    (6,0);
[-](4,2) to  [out=180,in=90] (2,0);
(2,0) circle (3pt);
(4,0) circle (3pt);
(6,0) circle (3pt);
(6,2) circle (3pt);
(6,0) circle (3pt);
[below]   at (3,0)      $\ve[\iota]$;
[below]   at (4,0)      $0$;
[above]   at (4,2)      $u$;
[right]  at (6,0)     $x$;
[right]  at (6,2)     $v$;
[above]  at (5,2)  $\geq 1$;
[left]  at (4.3,1)  $\geq 1$;
[below]  at (5,-0.1)  $\geq 1$;
[right]  at (6,1)     $\geq 1$;

[line width=1pt,auto,scale=#1]
[-](4,0) to (4,2);
[-](4,2) to (6,2);
[-](6,0) to    (6,2);
[-](4,0) to    (6,0);
[-](4,0) to  (2,0);
(2,0) circle (3pt);
(4,0) circle (3pt);
(6,0) circle (3pt);
(6,2) circle (3pt);
(4,2) circle (3pt);
[below]   at (2,0)      $\ve[\iota]$;
[below]   at (4,0)      $0$;
[above]   at (4,2)      $v$;
[right]  at (6,0)     $y$;
[right]  at (6,2)     $x$;
[right]  at (6,1)     $=1$;

[line width=1pt,auto,scale=#1]
[-](0,0) to [out=270,in=180] (2,-2);
[-](2,-2) to    (2,0);
[-](0,0) to  [out=90,in=90] (2,0);
(2,0) circle (3pt);
(0,0) circle (3pt);
(2,-2) circle (3pt);
[right]   at (2,0)      $\ve[\iota]$;
[left]   at (0,0)      $0$;
[right]   at (2,-2)      $y$;
[above]  at (1,0.5)  $\geq 1$;
[below]  at (-0.5,-1)  $\geq 2$;
[right]  at (2,-1)  $=1$;

[line width=1pt,auto,scale=#1]
[-](0,0) to (0,2);
[-](0,0) to (2,0);
[-](2,0) to (2,2);
[-](0,2) to (2,2);

(0,0) circle (3pt);
(2,0) circle (3pt);
(0,2) circle (3pt);
(2,2) circle (3pt);

[left]  at (0,0)      $0$;
[right]  at (2,0)     $y$;
[right]  at (2,2)     $x$;
[left]  at (0,2)     $\ve[\iota]=v$;

[left]  at (0,1)   $\geq 1$;
[right]  at (2,1)   $= 1$;
[below]  at (1,0)   $= 1$;
[above]  at (1,2)   $\geq 1$;

[-](6.5,0) to (6.5-1+0.25,1.5);
[-](6.5,0) to (6.5+2,0);
[-](6.5+2,0) to (6.5+3-0.25,1.5);
[-](6.5+3-0.25,1.5) to  (6.5+1,2.5);
[-](6.5+-1+0.25,1.5) to (6.5+1,2.5);

(6.5+0,0) circle (3pt);
(6.5+2,0) circle (3pt);
(6.5+3-0.25,1.5) circle (3pt);
(6.5-1+0.25,1.5) circle (3pt);
(6.5+1,2.5) circle (3pt);

[below]   at(6.5+0,0)      $0$;
[right]  at (6.5+2,0)     $y$;
[above]  at(6.5+1,2.5)     $v$;
[right]  at(6.5+2.75,1.5)     $x$;
[left]   at(6.5-0.75,1.5)     $\ve[\iota]$;

[left]  at (6.5+0,0.3)  $\geq 1$;
[above]  at (6.5+2.3,2-0.1) $\geq 0$;
[below]  at (6.5+1.4,0)     $=1$;
[right]  at (6.5+2.5,0.6) $=1$;
[above]  at (6.5-0.3,2.1)   $\geq 1$;

[line width=1pt,auto,scale=#1]
[-](0,0) to (-1+0.25,1.5);
[-](0,0) to (2,0);
[-](2,0) to (3-0.25,1.5);
[-](3-0.25,1.5) to  (1,2.5);
[-](-1+0.25,1.5) to (1,2.5);
[-](-1+0.25,1.5) to [out=220,in=90] (-2.5,0);

(0,0) circle (3pt);
(2,0) circle (3pt);
(3-0.25,1.5) circle (3pt);
(-1+0.25,1.5) circle (3pt);
(1,2.5) circle (3pt);
(-2.5,0) circle (3pt);

[below]   at(0,0)      $0$;
[right]  at (2,0)     $y$;
[above]  at(1,2.5)     $v$;
[right]  at(2.75,1.5)     $x$;
[left]   at(-0.75,1.5)     $u$;
[below]  at (-2.5,0)     $\ve[\iota]$;

[above]  at (-4,0.6) $\geq 1$;

[above]  at (2.3,2-0.1) $\geq 0$;
[below]  at (1.4,0)     $= 1$;
[right]  at (2.5,0.6) $=1$;
[above]  at (-0.3,2.1)   $\geq 0$;

[line width=1pt,auto,scale=#1]
[-](0,0) to (-1+0.25,1.5);
[-](0,0) to (2,0);
[-](2,0) to (3-0.25,1.5);
[-](3-0.25,1.5) to  (1,2.5);
[-](-1+0.25,1.5) to (1,2.5);
[-](-1+0.25,1.5) to [out=220,in=90] (-2.5,0);

(0,0) circle (3pt);
(2,0) circle (3pt);
(3-0.25,1.5) circle (3pt);
(-1+0.25,1.5) circle (3pt);
(1,2.5) circle (3pt);
(-2.5,0) circle (3pt);

[below]   at(0,0)      $0$;
[right]  at (2,0)     $y$;
[above]  at(1,2.5)     $v$;
[right]  at(2.75,1.5)     $x$;
[left]   at(-0.75,1.5)     $u$;
[below]  at (-2.5,0)     $\ve[\iota]$;

[above]  at (-3,0.6) $\geq 0$;

[above]  at (2.3,2-0.1) $\geq 0$;
[below]  at (1.4,0)     $\geq 2$;
[right]  at (2.5,0.6) $=1$;
[above]  at (-0.3,2.1)   $\geq 0$;

[line width=1pt,auto,scale=#1]
[-](4,0) to (4,2);
[-](4,2) to (6,2);
[-](6,0) to    (6,2);
[-](4,0) to    (6,0);
[-](4,0) to  (2,0);
(2,0) circle (3pt);
(4,0) circle (3pt);
(6,0) circle (3pt);
(6,2) circle (3pt);
(6,0) circle (3pt);
[below]   at (2,0)      $\ve[\iota]$;
[below]   at (4,0)      $0$;
[above]   at (4,2)      $v$;
[right]  at (6,0)     $y$;
[right]  at (6,2)     $x$;
[right]  at (6,1)     $\geq 2$;

[line width=1pt,auto,scale=#1]
[-](0,0) to (-1+0.25,1.5);
[-](0,0) to (2,0);
[-](2,0) to (3-0.25,1.5);
[-](3-0.25,1.5) to  (1,2.5);
[-](-1+0.25,1.5) to (1,2.5);
[-](-1+0.25,1.5) to [out=220,in=90] (-2.5,0);

(0,0) circle (3pt);
(2,0) circle (3pt);
(3-0.25,1.5) circle (3pt);
(-1+0.25,1.5) circle (3pt);
(1,2.5) circle (3pt);
(-2.5,0) circle (3pt);

[below]   at(0,0)      $0$;
[right]  at (2,0)      $y$;
[above]  at(1,2.5)     $v$;
[right]  at(2.75,1.5)  $x$;
[left]   at(-0.75,1.5) $u$;
[below]  at (-2.5,0)   $\ve[\iota]$;

[above]  at (-3,0.6)   $\geq 0$;
at (0.4,0.8)          $\geq 1$;
[above]  at (2.3,2-0.1) $\geq 0$;
[below]  at (1.4,0)     $\geq 1$;
[right]  at (2.5,0.6)   $\geq 2$;
[above]  at (-0.3,2.1)  $\geq 0$;

[line width=1pt,auto,scale=#1]
[-](4,0) to (4,2);
[-](4,2) to (6,2);
[-](6,0) to    (6,2);
[-](4,0) to    (6,0);
[-](4,0) to  (2,0);
(2,0) circle (3pt);
(4,0) circle (3pt);
(4,2) circle (3pt);
(6,0) circle (3pt);
(6,2) circle (3pt);
(6,0) circle (3pt);
[below]   at (2,0)      $\ve[\iota]$;
[below]   at (4,0)      $0$;
[above]   at (4,2)      $v$;
[right]  at (6,0)     $x$;
[right]  at (6,2)     $y$;
[right]  at (6,1)     $=1$;

[line width=1pt,auto,scale=#1]
[-](0,0) to  (2,2);
[-](0,0) to  (2,0);
[-](2,0) to    (2,2);
[-](2,2) to  [out=180,in=90] (-2,0);
(2,0) circle (3pt);
(0,0) circle (3pt);
(2,2) circle (3pt);
(-2,0) circle (3pt);
[left]   at (-2,0)      $\ve[\iota]$;
[left]   at (0,0)      $0$;
[right]   at (2,2)      $y$;
[right]   at (2,0)      $x$;-
at (0,0.9)   $\geq 2$;
[below]  at (1,0)  $=1$;
at (-1.5,2)  $\geq 1$;
[right]  at (2,1)  $=1$;

[line width=1pt,auto,scale=#1]
[-](0,0) to  (0,2);
[-](0,2) to  (2,2);
[-](0,0) to  (2,0);
[-](2,0) to    (2,2);
[-](0,2) to  [out=180,in=90] (-2,0);
(2,0) circle (3pt);
(0,0) circle (3pt);
(2,2) circle (3pt);
(0,2) circle (3pt);
(-2,0) circle (3pt);
[left]   at (-2,0)      $\ve[\iota]$;
[left]   at (0,0)      $0$;
[right]   at (2,2)      $y$;
[right]   at (2,0)      $x$;-
at (0.2,0.9)   $\geq 1$;
[above]  at (1,2)   $\geq 1$;
[below]  at (1,0)  $=1$;
at (-1.5,2)  $\geq 1$;
[right]  at (2,1)  $=1$;

[line width=1pt,auto,scale=#1]
[-](0,0) to (-1+0.25,1.5);
[-](0,0) to (2,0);
[-](2,0) to (3-0.25,1.5);
[-](3-0.25,1.5) to  (1,2.5);
[-](-1+0.25,1.5) to (1,2.5);
[-](-1+0.25,1.5) to [out=220,in=90] (-2.5,0);

(0,0) circle (3pt);
(2,0) circle (3pt);
(3-0.25,1.5) circle (3pt);
(-1+0.25,1.5) circle (3pt);
(1,2.5) circle (3pt);
(-2.5,0) circle (3pt);

[below]   at(0,0)      $0$;
[right]  at (2,0)      $x$;
[above]  at(1,2.5)     $v$;
[right]  at(2.75,1.5)  $y$;
[left]   at(-0.75,1.5) $u$;
[below]  at (-2.5,0)   $\ve[\iota]$;

[above]  at (-3,0.6)   $\geq 0$;
at (0.5,0.85)          $\geq 1$;
[above]  at (2.3,2-0.1) $\geq 0$;
[below]  at (1.4,0)     $=1$;
[right]  at (2.5,0.6)   $=1$;
[above]  at (-0.3,2.1)  $\geq 1$;

[line width=1pt,auto,scale=#1]
[-](0,0) to (-1+0.25,1.5);
[-](0,0) to (2,0);
[-](2,0) to (3-0.25,1.5);
[-](3-0.25,1.5) to  (1,2.5);
[-](-1+0.25,1.5) to (1,2.5);
[-](-1+0.25,1.5) to [out=220,in=90] (-2.5,0);

(0,0) circle (3pt);
(2,0) circle (3pt);
(3-0.25,1.5) circle (3pt);
(-1+0.25,1.5) circle (3pt);
(1,2.5) circle (3pt);
(-2.5,0) circle (3pt);

[below]   at(0,0)      $0$;
[right]  at (2,0)      $x$;
[above]  at(1,2.5)     $v$;
[right]  at(2.75,1.5)  $y$;
[left]   at(-0.75,1.5) $u$;
[below]  at (-2.5,0)   $\ve[\iota]$;

[above]  at (-3,0.6)   $\geq 0$;
at (0.5,0.85)          $\geq 1$;
[above]  at (2.3,2-0.1) $\geq 0$;
[below]  at (1.4,0)     $\geq 2$;
[right]  at (2.5,0.6)   $=1$;
[above]  at (-0.3,2.1)  $\geq 0$;

[line width=1pt,auto,scale=#1]
[-](4,0) to (4,2);
[-](4,2) to (6,2);
[-](6,0) to    (6,2);
[-](4,0) to    (6,0);
[-](4,0) to  (2,0);
(2,0) circle (3pt);
(4,0) circle (3pt);
(6,0) circle (3pt);
(6,2) circle (3pt);
(6,0) circle (3pt);
[below]   at (2,0)      $\ve[\iota]$;
[below]   at (4,0)      $0$;
[above]   at (4,2)      $v$;
[right]  at (6,0)     $x$;
[right]  at (6,2)     $y$;
[right]  at (6,1)     $\geq 2$;

[line width=1pt,auto,scale=#1]
[-](0,0) to (-1+0.25,1.5);
[-](0,0) to (2,0);
[-](2,0) to (3-0.25,1.5);
[-](3-0.25,1.5) to  (1,2.5);
[-](-1+0.25,1.5) to (1,2.5);
[-](-1+0.25,1.5) to [out=220,in=90] (-2.5,0);

(0,0) circle (3pt);
(2,0) circle (3pt);
(3-0.25,1.5) circle (3pt);
(-1+0.25,1.5) circle (3pt);
(1,2.5) circle (3pt);
(-2.5,0) circle (3pt);

[below]   at(0,0)      $0$;
[right]  at (2,0)     $x$;
[above]  at(1,2.5)     $v$;
[right]  at(2.75,1.5)     $y$;
[left]   at(-0.75,1.5)     $u$;
[below]  at (-2.5,0)     $\ve[\iota]$;

[above]  at (-3,0.6)   $\geq 0$;
at (0.4,0.8)          $\geq 1$;
[above]  at (2.3,2-0.1) $\geq 0$;
[below]  at (1.4,0)     $\geq 1$;
[right]  at (2.5,0.6)   $\geq 2$;
[above]  at (-0.3,2.1)  $\geq 0$;

[line width=1pt,auto,scale=#1]
(0,0)– (2,0)– (1,1.5)– cycle;
(0,0) circle (3pt);
(2,0) circle (3pt);
(1,1.5) circle (3pt);
[left]   at(0,0)       $0$;
[right]   at (2,0)       $x$;
[left]   at (1,1.5)       $v$;

[line width=1pt,auto,scale=#1]
[-](0,0) to [out=90,in=90]    (2,0);
[-](0,0) to [out=270,in=270]    (2,0);
(0,0) circle (3pt);
[left]   at(0,0)      $0$;
[right]  at (2,0)   $\geq 4$;

[line width=1pt,auto,scale=#1]
(0,1.5)– (2,0);
[line width=1pt] (2,0) –(0,0);
[line width=1pt] (0,0) –(0,1.5);
(0,0) circle (3pt);
(0,1.5) circle (3pt);
(2,0) circle (3pt);
[left]   at(0,0)      $0$;
[right]  at (2,0)     $x$;
[left]   at (0,1.5)   $v$;
[left]   at (0,.725)  $=1$;
[below]  at (1,0)     $=1$;
[above]  at (1.5,0.7)     $\geq 2$;

(4,0)–(5,1.5)– (6,0);
[line width=1pt] (4,0) –(6,0);
(4,0) circle (3pt);
(5,1.5) circle (3pt);
(6,0) circle (3pt);
[left]   at(4,0)       $0$;
[right]   at (6,0)       $x$;
[left]   at (5,1.5)      $v$;
[below]  at (5,0)        $=1$;
[left]   at (4.6,.9)     $\geq 2$;
[right]  at (5.4,0.9)    $\geq 1$;

[line width=1pt,auto,scale=#1]
[-](0,0) to [out=90,in=90]    (3,0);
[-](0,0) to [out=270,in=270]    (3,0);
(0,0) circle (3pt);
(3,0) circle (3pt);
[left]   at(0,0)      $0$;
[right]  at (3,0)     $x$;
[above]  at (1,0.7)     $\geq 3$;
[below]  at (2,-0.7)     $=1$;

[-](5,0) to [out=90,in=90]    (8,0);
[-](5,0) to [out=270,in=270]    (8,0);
(5,0) circle (3pt);
(8,0) circle (3pt);
[left]   at(5,0)      $0$;
[right]  at (8,0)     $x$;
[above]  at (6,0.7)     $\geq 2$;
[below]  at (7,-0.7)     $\geq 2$;

[line width=1pt,auto,scale=#1]
(0,0)–(1,1.5)– (2,0)–(0,0);
(0,0) circle (3pt);
(1,1.5) circle (3pt);
(2,0) circle (3pt);
[left]   at(0,0)       $0$;
[right]   at (2,0)       $x$;
[left]   at (1,1.5)      $v$;
[below]  at (1,0)        $\geq 2$;
[left]   at (0.6,.9)     $\geq 1$;
[right]  at (1.4,0.9)    $\geq 1$;

[line width=1pt,auto,scale=#1]
[-](0,2) to [out=0,in=90]    (1,1);
[-](0,0) to [out=0,in=270]    (1,1);
[dotted](0,2) to     (0,0);
(0,0) circle (3pt);
(0,2) circle (3pt);
[left]   at(0,2.1)      $u=0$;
[left]  at (0,-0.2)     $x=v$;
[right]  at (1,1)     $\geq 3$;
[left]  at (-0.3,1.3)     $= 1$;

[line width=1pt,auto,scale=#1]
[-] (0,1.5) to (2,1.5);
[line width=1pt,dotted] (0,0) –(0,1.5);
[-] (2,0) –(2,1.5);
[-] (0,0) –(2,0);

(0,0) circle (3pt);
(0,1.5) circle (3pt);
(2,0) circle (3pt);
(2,1.5) circle (3pt);
[left]   at(0,1.5)      $0$;
[right]  at (2,0)     $w$;
[left]   at (0,0)   $v$;
[right]   at (2,1.5)   $x$;
[left]   at (0,.725)  $=1$;
[below]  at (1,0)     $\geq 0$;
[above]  at (1,1.5)     $=1$;
[right]   at (2,.725)  $\geq 1$;

[line width=1pt,auto,scale=#1]
[-] (0,1.5) to (2,1.5);
[line width=1pt,dotted] (0,0) –(0,1.5);
[-] (2,0) –(2,1.5);
[-] (0,0) –(2,0);

(0,0) circle (3pt);
(0,1.5) circle (3pt);
(2,0) circle (3pt);
(2,1.5) circle (3pt);
[left]   at(0,1.5)      $0$;
[right]  at (2,0)     $w$;
[left]   at (0,0)   $v$;
[right]   at (2,1.5)   $x\neq v$;
[left]   at (0,.725)  $=1$;
[below]  at (1,0)     $\geq 0$;
[above]  at (1,1.5)     $\geq 2$;
[right]   at (2,.725)  $\geq 0$;

[line width=1pt,auto,scale=#1]
[-] (0,1.5) to (2,1.5);
[line width=1pt,dotted] (0,0) –(0,1.5);
[-] (2,0) –(2,1.5);
[-] (0,0) –(2,0);

(0,0) circle (3pt);
(0,1.5) circle (3pt);
(2,0) circle (3pt);
(2,1.5) circle (3pt);
[left]   at(0,1.5)      $0$;
[right]  at (2,0)     $w$;
[left]   at (0,0)   $v$;
[right]   at (2,1.5)   $x$;
[left]   at (0,.725)  $\geq 2$;
[below]  at (1,0)     $\geq 0$;
[above]  at (1,1.5)     $=1$;
[right]   at (2,.725)  $\geq 1$;

[line width=1pt,auto,scale=#1]
[-] (0,1.5) to (2,1.5);
[line width=1pt,dotted] (0,0) –(0,1.5);
[-] (2,0) –(2,1.5);
[-] (0,0) –(2,0);

(0,0) circle (3pt);
(0,1.5) circle (3pt);
(2,0) circle (3pt);
(2,1.5) circle (3pt);
[left]   at(0,1.5)      $0$;
[right]  at (2,0)     $w$;
[left]   at (0,0)   $v$;
[right]   at (2,1.5)   $x$;
[left]   at (0,.725)  $\geq 2$;
[below]  at (1,0)     $\geq 0$;
[above]  at (1,1.5)     $\geq 2$;
[right]   at (2,.725)  $\geq 0$;

[line width=1pt,auto,scale=#1]

[-](0,0) to [out=90,in=90]    (3,0);
[dotted](0,0) to [out=270,in=270]    (3,0);
(0,0) circle (3pt);
(3,0) circle (3pt);
[left]   at(0,0)      $0$;
[right]  at (3,0)     $y$;
[above]  at (1,0.7)     $\geq 3$;
[below]  at (2,-0.7)     $= 1$;

[line width=1pt,auto,scale=#1]
[-] (0,1.5) to (2,1.5);
[-] (0,0) –(0,1.5);
[dotted] (2,0) –(2,1.5);
[-] (0,0) –(2,0);

(0,0) circle (3pt);
(0,1.5) circle (3pt);
(2,0) circle (3pt);
(2,1.5) circle (3pt);
[left]   at(0,1.5)      $0\neq x$;
[right]  at (2,0)     $x$;
[left]   at (0,0)   $w$;
[right]   at (2,1.5)   $y$;
[left]   at (0,.725)  $\geq 0$;
[below]  at (1,0)     $\geq 0$;
[above]  at (1,1.5)     $=1$;
[right]   at (2,.725)  $=1$;

[line width=1pt,auto,scale=#1]
[-] (0,1.5) to (2,1.5);
[-] (0,0) –(0,1.5);
[dotted] (2,0) –(2,1.5);
[-] (0,0) –(2,0);

(0,0) circle (3pt);
(0,1.5) circle (3pt);
(2,0) circle (3pt);
(2,1.5) circle (3pt);
[left]   at(0,1.5)      $0\neq x$;
[right]  at (2,0)     $x$;
[left]   at (0,0)   $w$;
[right]   at (2,1.5)   $y$;
[left]   at (0,.725)  $\geq 0$;
[below]  at (1,0)     $\geq 0$;
[above]  at (1,1.5)     $\geq 2$;
[right]   at (2,.725)  $=1$;

[line width=1pt,auto,scale=0.6]
(0,0) circle (3pt);
(0,2) circle (3pt);
(3,0) circle (3pt);
(3,2) circle (3pt);
(4.5,3) circle (3pt);
(6,0) circle (3pt);
(6,2) circle (3pt);
(7.5,3) circle (3pt);
(9,0) circle (3pt);
(9,2) circle (3pt);
(12,0) circle (3pt);
(12,2) circle (3pt);

[line width=3pt,-](0,0) to  (3,0);
[line width=3pt,-](3,0) to  (6,0);
[line width=3pt,-](6,0) to  (9,0);
[line width=3pt,-](9,0) to  (12,0);
[-](0,0) to  (0,2);
[-](3,0) to  (3,2);
[-](6,0) to  (6,2);
[-](9,0) to  (9,2);
[-](12,0) to  (12,2);
[-](0,2) to  (3,2);
[-](3,2) to  (4.5,3);
[-](6,2) to  (4.5,3);
[-](6,2) to  (7.5,3);
[-](9,2) to  (7.5,3);
[-](9,2) to  (12,2);
at(-2,1)      $\sum_x\sum_{v_i}$;
[below]   at(0,0)      $0$;
[above]   at(0,2)      $v_1$;
[above]   at(3,2)      $v_2$;
[below]   at(3,0)      $v_3$;
[above]   at(4.5,3)    $v_4$;
[above]   at(6,2)      $v_5$;
[below]   at(6,0)      $v_6$;
[above]   at(7.5,3)    $v_7$;
[above]   at(9,2)      $v_8$;
[below]   at(9,0)      $v_9$;
[above]   at(12,2)      $v_{10}$;
[below]   at(12,0)      $x$;

[line width=1pt,auto,scale=0.5]
(0,0) circle (3pt);
(0,2) circle (3pt);
(3,0) circle (3pt);
(3,2) circle (3pt);

[line width=3pt,-](0,0) to  (3,0);
[-](0,0) to  (0,2);
[-](3,0) to  (3,2);
[-](0,2) to  (3,2);

at(-1.5,1)      $\sum_{v_1,v_2,v_3}$;
[below]   at(0,0)      $0$;
[above]   at(0,2)      $v_1$;
[above]   at(3,2)      $v_2$;
[below]   at(3,0)      $v_3$;

[line width=1pt,auto,scale=0.5]
(3,0) circle (3pt);
(3,2) circle (3pt);
(4.5,3) circle (3pt);
(6,0) circle (3pt);
(6,2) circle (3pt);

[line width=3pt,-](3,0) to  (6,0);
[-](6,0) to  (6,2);
[-](3,2) to  (4.5,3);
[-](6,2) to  (4.5,3);
at(1,1)      $\sup_{v_2,v_3} \sum_{v_4,v_5,v_{6}}$;
[above]   at(3,2)      $v_2$;
[below]   at(3,0)      $v_3$;
[above]   at(4.5,3)    $v_4$;
[above]   at(6,2)      $v_5$;
[below]   at(6,0)      $v_6$;

[line width=1pt,auto,scale=0.5]
(6,0) circle (3pt);
(6,2) circle (3pt);
(7.5,3) circle (3pt);
(9,0) circle (3pt);
(9,2) circle (3pt);

[line width=3pt,-](6,0) to  (9,0);

[-](6,2) to  (7.5,3);
[-](9,2) to  (7.5,3);
[-](9,0) to  (9,2);
at(4,1)      $\sup_{v_5,v_6}\sum_{v_7,v_8,v_9}$;
[above]   at(6,2)      $v_5$;
[below]   at(6,0)      $v_6$;
[above]   at(7.5,3)    $v_7$;
[above]   at(9,2)      $v_8$;
[below]   at(9,0)      $v_9$;

[line width=1pt,auto,scale=0.5]
(9,0) circle (3pt);
(9,2) circle (3pt);
(12,0) circle (3pt);
(12,2) circle (3pt);

[line width=3pt,-](9,0) to  (12,0);
[-](12,0) to  (12,2);
[-](9,2) to  (12,2);
at(7,1)      $\sup_{v_8,v_9}\sum_{v_{10},x}$;
[above]   at(9,2)      $v_8$;
[below]   at(9,0)      $v_9$;
[above]   at(12,2)      $v_{10}$;
[below]   at(12,0)      $x$;

$\leq$ ClosedSquare (OpenSquare$)^2$ OpenTriangle

[line width=1pt,auto,scale=#1]
at(-3,1)      $P^{\ssc[1],b}(x,y):=\sum_v$;
[-](0,0) to  (0,2);
[line width=2pt,-](0,0) to  (2,0);
[-](0,2) to    (2,2);
[line width=2pt,-](2,0) to    (2,2);
(0,2) circle (2pt);
(2,0) circle (2pt);
(2,2) circle (2pt);
(0,0) circle (2pt);
[left]   at(0,0)      $0$;
[left]   at(0,2)      $v$;
[right]  at (2,0)     $y$;
[right]  at (2,2)     $x$;
[right]  at (2,1)     $b$;

[line width=1pt,auto,scale=#1]
at(-3-0.2,)       $P^{\ssc[1],m}(x,y):=\sum_w$;

[-](0,0) to  (0,2);
[line width=2pt,-](0,0) to  (2,0);
[-](0,2) to    (2,2);
[line width=2pt,-](2,0) to    (2,2);
(0,2) circle (2pt);
(2,0) circle (2pt);
(2,2) circle (2pt);
(0,0) circle (2pt);
[left]   at(0,0)      $0$;
[left]   at(0,2)      $w$;
[right]  at (2,0)     $y$;
[right]  at (2,2)     $x$;
[right]  at (2,1)     $m$;

at(-3,1-.1)      $+\sum_{w,u,v}$;
[-](-0.8,1) to  (0,2);
[-](-0.8,1) to  (0,0);
[-](0,0) to  (0,2);
[line width=2pt,-](0,0) to  (2,0);
[-](0,2) to    (1,2.5);
[-](1,2.5) to    (2,2);
[line width=2pt,-](2,0) to    (2,2);
(0,2) circle (2pt);
(2,0) circle (2pt);
(2,2) circle (2pt);
(0,0) circle (2pt);
(-0.8,1) circle (2pt);
(1,2.5) circle (2pt);

[left]   at(-0.8,1)      $0$;
[below]   at (0,0)      $u$;
[above]   at(0,2)      $v$;
[below]   at(1,2.4)      $w$;
[right]  at (2,0)     $y$;
[right]  at (2,2)     $x$;
[right]  at (2,1)     $m$;

[line width=1pt,auto,scale=#1]
at(-3,1)      $P^{\ssc[1],-m}(x,y):=\sum_v$;
[-](0,0) to  (0,2);
[line width=2pt,-](0,0) to  (2,0);
[-](0,2) to    (2,2);
[-](2,0) to    (2,2);
(0,2) circle (2pt);
(2,0) circle (2pt);
(2,2) circle (2pt);
(0,0) circle (2pt);
[left]   at(0,0)      $0$;
[left]   at(0,2)      $v$;
[right]  at (2,0)     $x$;
[right]  at (2,2)     $y$;
[right]  at (2,1)     $m$;

[line width=1pt,auto,scale=#1]
at(-3-0.4,1)      $P^{\ssc[1],-m}(x,y):=\sum_w$;

[-](0,0) to  (0,2);
[line width=2pt,-](0,0) to  (2,0);
[-](0,2) to    (2,2);
[-](2,0) to    (2,2);
(0,2) circle (2pt);
(2,0) circle (2pt);
(2,2) circle (2pt);
(0,0) circle (2pt);
[left]   at(0,0)      $0$;
[left]   at(0,2)      $w$;
[right]  at (2,0)     $x$;
[right]  at (2,2)     $y$;
[right]  at (2,1)     $m$;

at(-3,1-.1)      $+\sum_{w,u,v}$;
[-](-0.8,1) to  (0,2);
[-](-0.8,1) to  (0,0);
[-](0,0) to  (0,2);
[line width=2pt,-](0,0) to  (2,0);
[-](0,2) to    (1,2.5);
[-](1,2.5) to    (2,2);
[-](2,0) to    (2,2);
(0,2) circle (2pt);
(2,0) circle (2pt);
(2,2) circle (2pt);
(0,0) circle (2pt);
(-0.8,1) circle (2pt);
(1,2.5) circle (2pt);

[left]   at(-0.8,1)      $0$;
[below]   at (0,0)      $u$;
[above]   at(0,2)      $v$;
[below]   at(1,2.4)      $w$;
[right]  at (2,0)     $x$;
[right]  at (2,2)     $y$;
[right]  at (2,1)     $m$;

[line width=1pt,auto,scale=#1]
at(-4,1)      $P^{\ssc[1],\iota,\text{step},b}(x,y):=\sum_v$;
[-](0,0) to  (0,2);
[line width=2pt,-](0,0) to  (0.7,0);
[line width=2pt,-](0.7,0) to  (2,0);
[-](0,2) to    (2,2);
[line width=2pt,-](2,0) to    (2,2);
(0,2) circle (2pt);
(2,0) circle (2pt);
(2,2) circle (2pt);
(0,0) circle (2pt);
(0.7,0) circle (4pt);

[left]   at(0,0)      $0$;
[below]   at(0.7,0)   $\ve[\iota]$;
[left]   at(0,2)      $v$;
[right]  at (2,0)     $y$;
[right]  at (2,2)     $x$;
[right]  at (2,1)     $b$;

[line width=1pt,auto,scale=#1]
at(-4,1)      $P^{\ssc[1],\iota,\text{step},-b}(x,y):=\sum_v$;
[-](0,0) to  (0,2);
[line width=2pt,-](0,0) to  (0.7,0);
[line width=2pt,-](0.7,0) to  (2,0);
[-](0,2) to    (2,2);
[-](2,0) to    (2,2);
(0,2) circle (2pt);
(0.7,0) circle (4pt);
(2,0) circle (2pt);
(2,2) circle (2pt);
(0,0) circle (2pt);
[left]   at(0,0)      $0$;
[below]   at(0.7,0)   $\ve[\iota]$;
[left]   at(0,2)      $v$;
[right]  at (2,0)     $x$;
[right]  at (2,2)     $y$;
[right]  at (2,1)     $b$;

[line width=1pt,auto,scale=#1]
at(-4,1)      $P^{\ssc[1],\iota,\text{rib},b}(x,y):=\sum_{u,v}$;
[-](0,0) to  (0.25,1.2);
[-](0.25,1.2) to  (1.4,2);
[-](1.4,2) to  (2.5,2);
[line width=2pt,-](0,0) to    (2.5,0);
[line width=2pt,-](2.5,0) to    (2.5,2);
[-](0.25,1.2) to  [out=135,in=90]   (-0.8,0);
(0,0) circle (2pt);
(0.25,1.2) circle (2pt);
(1.4,2) circle (2pt);
(2.5,2) circle (2pt);
(2.5,0) circle (2pt);
(-0.8,0) circle (2pt);
[below]   at(-0.8,0)      $\ve[\iota]$;
[below]   at(0,0)      $0$;
[above]   at(0.25,1.2)      $u$;
[left]   at(1.5,1.5)      $v$;
[right]  at (2.5,0)     $y$;
[right]  at (2.5,2)     $x$;
[right]  at (2.5,1)     $b$;

[line width=1pt,auto,scale=#1]
at(-4,1)      $P^{\ssc[1],\iota,\text{rib},-b}(x,y):=\sum_{u,v}$;
[-](0,0) to  (0.25,1.2);
[-](0.25,1.2) to  (1.4,2);
[-](1.4,2) to  (2.5,2);
[line width=2pt,-](0,0) to    (2.5,0);
[-](2.5,0) to    (2.5,2);
[-](0.25,1.2) to  [out=135,in=90]   (-0.8,0);
(0,0) circle (2pt);
(0.25,1.2) circle (2pt);
(1.4,2) circle (2pt);
(2.5,2) circle (2pt);
(2.5,0) circle (2pt);
(-0.8,0) circle (2pt);
[below]   at(-0.8,0)      $\ve[\iota]$;
[below]   at(0,0)      $0$;
[above]   at(0.25,1.2)      $u$;
[left]   at(1.5,1.5)      $v$;
[right]  at (2.5,0)     $x$;
[right]  at (2.5,2)     $y$;
[right]  at (2.5,1)     $b$;

[line width=1pt,auto,scale=#1]
at(-3.2,0.25)      $A^{l,m}(u,v,x,y):=\sum_{w}$;

[-](1,4) to  (1,-3);
[-](5,4) to  (5,-3);
[-](0,3) to  (9,3);
[-](0,0) to  (9,0);
at(0,1.5)      $l\leq 0$;
at(0,-1.5)      $l> 0$;

at(3,3.5)      $m\leq 0$;
at(7,3.5)      $m> 0$;

[line width=2pt,-](0,0) to  (2,0);
[-](2,0) to  (2,1.5);
[dotted](0,0) to    (0,1.5);
[-](0,1.5) to    (1,2);
[-](1,2) to    (2,1.5);

(0,1.5) circle (2pt);
(2,1.5) circle (2pt);
(2,0) circle (2pt);
(1,2) circle (2pt);
(0,0) circle (2pt);
[left]   at(0,0)      $u$;
[left]   at(0,0.75)      $l$;
[left]   at(0,1.5)      $v$;
[right]  at (2,0)     $x$;
[right]  at (2,0.75)     $m$;
[right]  at (2,1.5)     $y$;
[below]  at (1,1.8)   $w$;

[line width=2pt,-](0,1.5) to  (2,1.5);
[line width=2pt,-](2,0) to  (2,1.5);
[dotted](0,0) to    (0,1.5);
[-](0,0) to    (1,-.5);
[-](1,-.5) to    (2,0);

(0,1.5) circle (2pt);
(2,1.5) circle (2pt);
(2,0) circle (2pt);
(1,-.5) circle (2pt);
(0,0) circle (2pt);
[left]   at(0,0)      $v$;
[left]   at(0,0.75)      $l$;
[left]   at(0,1.5)      $u$;
[right]  at (2,0)     $x$;
[right]  at (2,0.75)     $m$;
[right]  at (2,1.5)     $y$;
[above]  at (1,-.5)   $w$;

[line width=2pt,-](0,0) to  (2,0);
[line width=2pt,-](2,0) to  (2,1.5);
[dotted](0,0) to    (0,1.5);
[-](0,1.5) to    (1,2);
[-](1,2) to    (2,1.5);

(0,1.5) circle (2pt);
(2,1.5) circle (2pt);
(2,0) circle (2pt);
(1,2) circle (2pt);
(0,0) circle (2pt);
[left]   at(0,0)      $u$;
[left]   at(0,0.75)      $l$;
[left]   at(0,1.5)      $v$;
[right]  at (2,0)     $y$;
[right]  at (2,0.75)     $b$;
[right]  at (2,1.5)     $x$;
[below]  at (1,1.8)   $w$;

[line width=2pt,-](0,1.5) to  (2,1.5);
[-](2,0) to  (2,1.5);
[dotted](0,0) to    (0,1.5);
[-](0,0) to    (1,-.5);
[-](1,-.5) to    (2,0);

(0,1.5) circle (2pt);
(2,1.5) circle (2pt);
(2,0) circle (2pt);
(1,-.5) circle (2pt);
(0,0) circle (2pt);
[left]   at(0,0)      $v$;
[left]   at(0,0.75)      $l$;
[left]   at(0,1.5)      $u$;
[right]  at (2,0)     $y$;
[right]  at (2,0.75)     $m$;
[right]  at (2,1.5)     $x$;
[above]  at (1,-.5)   $w$;

[line width=1pt,auto,scale=#1]
at(-3.5,0.75)      $\bar A^{l,m}(u,v,x,y):=\sum_{w}$;
[line width=2pt,-](0,1.5) to  (2,1.5);
[dotted](2,0) to  (2,1.5);
[dotted](0,0) to    (0,1.5);
[-](0,0) to    (1,-.5);
[-](1,-.5) to    (2,0);

(0,1.5) circle (2pt);
(2,1.5) circle (2pt);
(2,0) circle (2pt);
(1,-.5) circle (2pt);
(0,0) circle (2pt);
[left]   at(0,0)      $v$;
[left]   at(0,0.75)      $l$;
[left]   at(0,1.5)      $u$;
[right]  at (2,0)     $y$;
[right]  at (2,0.75)     $m$;
[right]  at (2,1.5)     $x$;
[above]  at (1,-.5)   $w$;

[line width=1pt,auto,scale=#1]
at(-3.2,0.25)      $C^{l,m}(u,v,x,y):=\sum_{w}$;

[-](1,4) to  (1,-3);
[-](5,4) to  (5,-3);
[-](0,3) to  (9,3);
[-](0,-1) to  (9,-1);
at(0,1.5)      $l\leq 0$;
at(0,-1.5)      $l> 0$;

at(3,3.5)      $m\leq 0$;
at(7,3.5)      $m> 0$;

[line width=2pt,-](0,0) to  (2,0);
[dotted](2,0) to  (2,1.5);
[dotted](0,0) to    (0,1.5);
[-](0,1.5) to    (1,2);
[-](1,2) to    (2,1.5);

(0,1.5) circle (2pt);
(2,1.5) circle (2pt);
(2,0) circle (2pt);
(1,2) circle (2pt);
(0,0) circle (2pt);
[left]   at(0,0)      $u$;
[left]   at(0,0.75)      $l$;
[left]   at(0,1.5)      $v$;
[right]  at (2,0)     $x$;
[right]  at (2,0.75)     $m$;
[right]  at (2,1.5)     $y$;
[below]  at (1,1.8)   $w$;
at (1,-0.4)     $|x-u|^2$;

[line width=2pt,-](0,1.5) to  (2,1.5);
[line width=2pt,dotted](2,0) to  (2,1.5);
[dotted](0,0) to    (0,1.5);
[-](0,0) to    (1,-.5);
[-](1,-.5) to    (2,0);

(0,1.5) circle (2pt);
(2,1.5) circle (2pt);
(2,0) circle (2pt);
(1,-.5) circle (2pt);
(0,0) circle (2pt);
[left]   at(0,0)      $v$;
[left]   at(0,0.75)      $l$;
[left]   at(0,1.5)      $u$;
[right]  at (2,0)     $x$;
[right]  at (2,0.75)     $m$;
[right]  at (2,1.5)     $y$;
[above]  at (1,-.5)   $w$;
[below]  at (1,-0.2)     $|x-v|^2$;

[line width=2pt,-](0,0) to  (2,0);
[line width=2pt,dotted](2,0) to  (2,1.5);
[dotted](0,0) to    (0,1.5);
[-](0,1.5) to    (1,2);
[-](1,2) to    (2,1.5);

(0,1.5) circle (2pt);
(2,1.5) circle (2pt);
(2,0) circle (2pt);
(1,2) circle (2pt);
(0,0) circle (2pt);
[left]   at(0,0)      $u$;
[left]   at(0,0.75)      $l$;
[left]   at(0,1.5)      $v$;
[right]  at (2,0)     $y$;
[right]  at (2,0.75)     $m$;
[right]  at (2,1.5)     $x$;
[below]  at (1,1.8)   $w$;
at (1,-0.4)     $|y-u|^2$;

[line width=2pt,-](0,1.5) to  (2,1.5);
[dotted](2,0) to  (2,1.5);
[dotted](0,0) to    (0,1.5);
[-](0,0) to    (1,-.5);
[-](1,-.5) to    (2,0);

(0,1.5) circle (2pt);
(2,1.5) circle (2pt);
(2,0) circle (2pt);
(1,-.5) circle (2pt);
(0,0) circle (2pt);
[left]   at(0,0)      $v$;
[left]   at(0,0.75)      $l$;
[left]   at(0,1.5)      $u$;
[right]  at (2,0)     $y$;
[right]  at (2,0.75)     $m$;
[right]  at (2,1.5)     $x$;
[above]  at (1,-.5)   $w$;
[below]  at (1,-0.2)     $|y-v|^2$;

[line width=1pt,auto,scale=#1]
at(-4,0.75)      $\Delta M^{-a,b}(u,v,x,y):=\sum_{w}$;
[line width=2pt,-](0,0) to  (2,0);
[below]  at (1,0)     $[1-\cos(k\cdot (y-u))]$;
[dotted](2,0) to  (2,1.5);
[dotted](0,0) to    (0,1.5);
[-](0,1.5) to    (1,2);
[-](1,2) to    (2,1.5);

(0,1.5) circle (2pt);
(1,2) circle (2pt);
(2,0) circle (2pt);
(2,1.5) circle (2pt);
(0,0) circle (2pt);
[left]   at(0,0)      $u$;
[left]   at(0,0.75)      $a$;
[left]   at(0,1.5)      $v$;
[right]  at (2,0)     $y$;
[right]  at (2,0.75)     $b$;
[right]  at (2,1.5)     $x$;
[above]  at (1,2)   $w$;

at(-4,0.75)      $\Delta M^{a,b}(u,v,x,y):=\sum_{w}$;
[line width=2pt,-](0,0) to  (1,-0.5);
[line width=2pt,-](1,-0.5) to  (2,0);
[below]  at (1,-.5)     $[1-\cos(k\cdot (y-v))]$;
[dotted](2,0) to  (2,1.5);
[dotted](0,0) to    (0,1.5);
[-](0,1.5) to    (2,1.5);
(0,1.5) circle (2pt);
(1,-0.5) circle (2pt);
(2,0) circle (2pt);
(2,1.5) circle (2pt);
(0,0) circle (2pt);
[left]   at(0,0)      $v$;
[left]   at(0,0.75)      $a$;
[left]   at(0,1.5)      $u$;
[right]  at (2,0)     $y$;
[right]  at (2,0.75)     $b$;
[right]  at (2,1.5)     $x$;
[right]  at (1.1,-.5)   $w$;

[line width=1pt,auto,scale=#1]
[-](5,4) to  (5,-0.5);
[-](1.5,3) to  (13.5,3);
at(3,3.5)      $a\leq 0$;
at(9,3.5)      $a> 0$;

[line width=2pt,-](0,0) to  (2,0);
[dotted](2,0) to  (2,1.5);
[dotted](0,0) to    (0,1.5);
[-](0,1.5) to    (1,2);
[-](1,2) to    (2,1.5);

(0,1.5) circle (2pt);
(2,1.5) circle (2pt);
(2,0) circle (2pt);
(1,2) circle (2pt);
(0,0) circle (2pt);
[left]   at(0,0)      $u$;
[left]   at(0,0.75)      $a$;
[left]   at(0,1.5)      $v$;
[right]  at (2,0)     $x$;
[right]  at (2,0.75)     $b$;
[right]  at (2,1.5)     $y$;
[below]  at (1,1.8)   $w$;
[below]  at (1,-0.5)   $[1-\cos(k\cdot (u-x))]$;

[-](0,1.5) to  (2,1.5);
[dotted](2,0) to  (2,1.5);
[dotted](0,0) to    (0,1.5);
[line width=2pt,-](0,0) to    (1,-.5);
[-](1,-.5) to    (2,0);

(0,1.5) circle (2pt);
(2,1.5) circle (2pt);
(2,0) circle (2pt);
(1,-.5) circle (2pt);
(0,0) circle (2pt);
[left]   at(0,0)      $v$;
[left]   at(0,0.75)      $a$;
[left]   at(0,1.5)      $u$;
[right]  at (2,0)     $x$;
[right]  at (2,0.75)     $b$;
[right]  at (2,1.5)     $y$;
[above]  at (1,-.5)   $w$;
[below]  at (0.7,-0.5)   $[1-\cos(k\cdot (v-w))]$;

[-](3,0.75+0.25) to  (3,0.75-0.25);
[-](3+0.25,0.75) to  (3-0.25,0.75);

[-](0,1.5) to  (2,1.5);
[dotted](2,0) to  (2,1.5);
[dotted](0,0) to    (0,1.5);
[-](0,0) to    (1,-.5);
[line width=2pt,-](1,-.5) to    (2,0);

(0,1.5) circle (2pt);
(2,1.5) circle (2pt);
(2,0) circle (2pt);
(1,-.5) circle (2pt);
(0,0) circle (2pt);
[left]   at(0,0)      $v$;
[left]   at(0,0.75)      $a$;
[left]   at(0,1.5)      $u$;
[right]  at (2,0)     $x$;
[right]  at (2,0.75)     $b$;
[right]  at (2,1.5)     $y$;
[above]  at (1,-.5)   $w$;
[below]  at (1.4,-0.5)   $[1-\cos(k\cdot (w-y))]$;

[red,line width=1 pt] (3,-1) to (3,3);
[red,line width=1 pt] (3,-1) to (4,-1);
[red,line width=1 pt] (3,0) to (4,0);
[red,line width=1 pt] (3,1) to (4,1);
[red,line width=1 pt] (4,2) to (4,1);
[red,line width=1 pt] (2,2) to (4,2);
[red,line width=1 pt] (2,2) to (2,3)   ;
[red] at(4,3)   $B_{t_N}^\omega$;
[red] (3,0) circle (3pt);
[red] at(2.5,-0.2)   $\overline b^\omega_{t_N}$;
[red] (2,2) circle (2pt);
[red] (3,1) circle (2pt);

(0,0) circle (3pt);
[line width=1 pt] (0,0) to (0,2);
[line width=1 pt] (-1,1) to (3,1);
[line width=1 pt] (-1,2) to (0,2);
[line width=1 pt] (1,1) to (1,2);
[line width=1 pt] (1,2) to (4,2);
[line width=1 pt] (2,2) to (2,3);
[line width=1 pt] (1,3) to (2,3);
at(0,3)   $B_{s_N}^\omega$;
at(-0.3,-0.2)   $\overline b^\omega_{s_N}$;
(2,2) circle (2pt);
(3,1) circle (2pt);

at(2,2)   $w_a$;
at(3,1)   $w_b$;

[red,line width=1 pt] (4,1) to (1,1);
[red,line width=1 pt] (3,0) to (3,2);
[red,line width=1 pt] (2,0) to (2,3);
[red,line width=1 pt] (1,1) to (1,2);
[red,line width=1 pt] (1,2) to (-1,2);
[red,line width=1 pt] (0,2) to (0,3);
[red] at(3,3)   $B_{t_N}^\omega$;
[red] (4,1) circle (3pt);
[red] at(4.5,1.2)   $\overline b^\omega_{t_N}$;
[red] (2,0) circle (2pt);
[red] (1,1) circle (2pt);
[red] (0,2) circle (2pt);

(0,0) circle (3pt);
[line width=1 pt] (0,0) to (0,1);
[line width=1 pt] (0,1) to (1,1);
[line width=1 pt] (0,1) to (-1,1);
[line width=1 pt] (-1,1) to (-1,3);
[line width=1 pt] (0,1) to (0,2);
[line width=1 pt] (0,2) to (1,2);
[line width=1 pt] (0,2) to (0,3);
at(-2,1)   $B_{s_N}^\omega$;
at(-0.3,-0.2)   $\overline b^\omega_{s_N}$;
(1,1) circle (2pt);
(0,2) circle (2pt);
(-1,3) circle (2pt);

at(1.6,1)   $w_a$;
at(-0.2,2)   $w_b$;
at(-1.2,3)   $w_c$;

at(-.65,.5)   $0$;
[gray] (0,0) – (-0.5,0.5)–(0,1);
[line width=1 pt] (-0.5,0.5) to (0,0);
[line width=1 pt] (-0.5,0.5) to (0,1);
[line width=1 pt] (0,0) to (4,0);
[line width=1 pt] (0,1) to (4,1);
[line width=1 pt] (0,0) to (0,1);
[line width=1 pt] (4,0) to (4,1);
[line width=1 pt] (1,0) to (1,1);
[line width=1 pt] (2,0) to (2,1);
[line width=1 pt] (3,0) to (3,1);
[line width=2.5 pt] (0,0) to (4,0);
[line width=1 pt] (3,1) to (4,1);
(1.5,1) circle (3pt);
(2.5,1) circle (3pt);
at(4.3,0)   $x$;

[red,line width=1 pt] (-0.46 ,0.5-0.15) to (0,-0.1);
[red,line width=1 pt] (0,-0.1) to (1-,-0.1);
[red,line width=1 pt] (1+,-0.1) to (2-,-0.1);
[red,line width=1 pt] (2+,-0.1) to (3-,-0.1);
[red,line width=1 pt] (3+,-0.1) to (4-,-0.1);

[red] at(0.25,-0.5)   $x_1$;
[red] at(1.5,-0.5)   $x_2$;
[red] at(2.5,-0.5)   $x_3$;
[red] at(3.5,-0.5)   $x_4$;

at(-.65,.5)   $0$;
[gray] (0,0) – (-0.5,0.5)–(0,1);
[line width=1 pt] (-0.5,0.5) to (0,0);
[line width=1 pt] (-0.5,0.5) to (0,1);
[line width=1 pt] (0,0) to (4,0);
[line width=1 pt] (0,1) to (4,1);
[line width=1 pt] (0,0) to (0,1);
[line width=1 pt] (4,0) to (4,1);
[line width=1 pt] (1,0) to (1,1);
[line width=1 pt] (2,0) to (2,1);
[line width=2.5 pt] (3,0) to (3,1);
[line width=2.5 pt] (0,0) to (3,0);
[line width=2.5 pt] (3,1) to (4,1);
(1.5,1) circle (3pt);
(2.5,1) circle (3pt);
at(4.1,1.2)   $x$;

[red,line width=1 pt] (-0.46 ,0.5-0.15) to (0,-0.1);
[red,line width=1 pt] (0,-0.1) to (1-,-0.1);
[red,line width=1 pt] (1+,-0.1) to (2-,-0.1);
[red,line width=1 pt] (2+,-0.1) to (3-,-0.1);
[red,line width=1 pt] (3+,-0.1) to (4+0.1,-0.1);
[red,line width=1 pt] (4+0.1,-0.1) to (4+0.1,1);

[red] at(0.25,-0.5)   $x_1$;
[red] at(1.5,-0.5)   $x_2$;
[red] at(2.5,-0.5)   $x_3$;
[red] at(4,-0.5)      $x_4$;

at(-.65,.5)   $0$;
[gray] (0,0) – (-0.5,0.5)–(0,1);
[line width=1 pt] (-0.5,0.5) to (0,0);
[line width=1 pt] (-0.5,0.5) to (0,1);
[line width=1 pt] (0,0) to (4,0);
[line width=1 pt] (0,1) to (4,1);
[line width=1 pt] (0,0) to (0,1);
[line width=1 pt] (4,0) to (4,1);
[line width=2.5 pt] (1,0) to (1,1);
[line width=2.5 pt] (2,0) to (2,1);
[line width=2.5 pt] (3,0) to (3,1);
[line width=2.5 pt] (0,0) to (1,0);
[line width=2.5 pt] (1,1) to (2,1);
[line width=2.5 pt] (2,0) to (3,0);
[line width=2.5 pt] (3,1) to (4,1);
(1.5,0) circle (3pt);
(2.5,1) circle (3pt);
at(4.1,1.2)   $x$;

[red,line width=1 pt] (-0.46 ,0.5-0.15) to (0,-0.1);
[red,line width=1 pt] (0,-0.1) to (1-,-0.1);
[red,line width=1 pt] (1+,-0.1) to (2-,-0.1);
[red,line width=1 pt] (2+,-0.1) to (3-,-0.1);
[red,line width=1 pt] (3+,-0.1) to (4+0.1,-0.1);
[red,line width=1 pt] (4+0.1,-0.1) to (4+0.1,1);

[red] at(0.25,-0.5)   $x_1$;
[red] at(1.5,-0.5)   $x_2$;
[red] at(2.5,-0.5)   $x_3$;
[red] at(4,-0.5)   $x_4$;

at(-.65,.5)   $0$;
[gray] (0,0) – (-0.5,0.5)–(0,1);
[line width=1 pt] (-0.5,0.5) to (0,0);
[line width=1 pt] (-0.5,0.5) to (0,1);
[line width=1 pt] (0,0) to (4,0);
[line width=1 pt] (0,1) to (4,1);
[line width=1 pt] (0,0) to (0,1);
[line width=1 pt] (4,0) to (4,1);
[line width=2.5 pt] (1,0) to (1,1);
[line width=1 pt] (2,0) to (2,1);
[line width=2.5 pt] (3,0) to (3,1);
[line width=2.5 pt] (0,0) to (1,0);
[line width=2.5 pt] (1,1) to (3,1);
[line width=2.5 pt] (3,0) to (4,0);
(1.5,0) circle (3pt);
(2.5,0) circle (3pt);
at(4.3,0)   $x$;

[red,line width=1 pt] (-0.46 ,0.5-0.15) to (0,-0.1);
[red,line width=1 pt] (0,-0.1) to (1-,-0.1);
[red,line width=1 pt] (1+,-0.1) to (2-,-0.1);
[red,line width=1 pt] (2+,-0.1) to (3-,-0.1);
[red,line width=1 pt] (3+,-0.1) to (4-,-0.1);

[red] at(0.25,-0.5)   $x_1$;
[red] at(1.5,-0.5)   $x_2$;
[red] at(2.5,-0.5)   $x_3$;
[red] at(3.5,-0.5)   $x_4$;

[line width=1pt,auto,scale=#1]

[-](0,0) to  (0,2);
[line width=2pt,-](0,0) to  (2,0);
[-](0,2) to    (2,2);
[dotted](2,0) to    (2,2);
(0,2) circle (3pt);
(2,0) circle (3pt);
(2,2) circle (3pt);
(0,0) circle (3pt);
(0.7,0) circle (4pt);

[left]   at(0,0)      $0$;
[below]   at(0.7,0)   $\ve[\iota]$;
[left]   at(0,2)      $v$;
[right]  at (2,0)     $x$;
[right]  at (2,2)     $y$;
[right]  at (2,1)     $a$;

[-](0,0) to  (0.25,1.2);
[-](0.25,1.2) to  (1.4,2);
[-](1.4,2) to  (2.5,2);
[line width=2pt,-](0,0) to    (2.5,0);
[dotted](2.5,0) to    (2.5,2);
[-](0.25,1.2) to  [out=135,in=90]   (-0.8,0);
(0,0) circle (3pt);
(0.25,1.2) circle (3pt);
(1.4,2) circle (3pt);
(2.5,2) circle (3pt);
(2.5,0) circle (3pt);
(-0.8,0) circle (3pt);
[below]   at(-0.8,0)      $\ve[\iota]$;
[below]   at(0,0)      $0$;
[above]   at(0.25,1.2)      $u$;
at(1.5,1.5)      $v$;
[right]  at (2.5,0)     $x$;
[right]  at (2.5,2)     $y$;
[right]  at (2.5,1)     $a$;

[line width=1pt,auto,scale=#1]

[-](0,0) to  (0,2);
[line width=2pt,-](0,0) to  (2,0);
[-](0,2) to    (2,2);
[dotted](2,0) to    (2,2);
(0,2) circle (3pt);
(2,0) circle (3pt);
(2,2) circle (3pt);
(0,0) circle (3pt);
(0.7,0) circle (4pt);

[left]   at(-0.5,1)      $\sum_\iota$;

[left]   at(0,0)      $0$;
[below]   at(0.7,0)   $\ve[\iota]$;
[left]   at(0,2)      $v$;
[right]  at (2,0)     $x$;
[right]  at (2,2)     $y$;
[right]  at (2,1)     $a$;
at (1,-1)     $[1-\cos(k\cdot x)]$;
[right]  at (3,1)     $=$;

[-](0,0.7) to  (0,2);
[line width=2pt,-](0,0) to  (0,0.7);
[line width=2pt,-](0,0) to  (2,0);
[-](0,2) to    (2,2);
[dotted](2,0) to    (2,2);
(0,2) circle (3pt);
(2,0) circle (3pt);
(2,2) circle (3pt);
(0,0) circle (3pt);
(0,0.7) circle (4pt);

[left]   at(-0.5,1)      $\sum_\iota$;

[left]   at(0,0)      $0$;
[right]   at(0,0.7)   $\ve[\iota]$;
[left]   at(0,2)      $v$;
[right]  at (2,0)     $x$;
[right]  at (2,2)     $y$;
[right]  at (2,1)     $a$;
at (1,-1)     $[1-\cos(k\cdot (x-\ve[\iota]))]$;
[right]  at (3.5,1)     $=$;

[-](0,0) to  (0,2);
[line width=2pt,-](0,0) to  (2,0);
[-](0,2) to    (2,2);
[dotted](2,0) to    (2,2);
(0,2) circle (3pt);
(2,0) circle (3pt);
(2,2) circle (3pt);
(0,0) circle (3pt);

[left]   at(0,0)      $0$;
[left]   at(0,2)      $v$;
[right]  at (2,0)     $x$;
[right]  at (2,2)     $y$;
[right]  at (2,1)     $a$;
at (1,-1)     $[1-\cos(k\cdot x)]$;

[-](0,0) to  (0,2);
[line width=2pt,-](0.7,0) to  (2,0);
[-](0,0) to  (0.7,0);
[-](0,2) to    (2,2);
[dotted](2,0) to    (2,2);
(0,2) circle (3pt);
(2,0) circle (3pt);
(2,2) circle (3pt);
(0,0) circle (3pt);
(0.7,0) circle (4pt);

[left]   at(-1,1)      $\sum_\iota$;

[left]   at(0,0)      $0$;
[below]   at(0.7,0)   $\ve[\iota]$;
[left]   at(0,2)      $v$;
[right]  at (2,0)     $x$;
[right]  at (2,2)     $y$;
[right]  at (2,1)     $a$;
[right]  at (-0.1,1)     ${\tiny \geq n}$;
at (1,-1)     $[1-\cos(k\cdot (x-\ve[\iota]))]$;
[right]  at (3,1)     $=$;

[-](0,0.7) to  (0,2);
[-](0,0) to  (0,0.7);
[line width=2pt,-](0,0) to  (2,0);
[-](0,2) to    (2,2);
[dotted](2,0) to    (2,2);
(0,2) circle (3pt);
(2,0) circle (3pt);
(2,2) circle (3pt);
(0,0) circle (3pt);
(0,0.7) circle (4pt);

[left]   at(-0.5,1)      $\sum_\iota$;
[right]  at (-0.1,1.4)     ${\tiny \geq n}$;
[left]   at(0,0)      $0$;
[right]   at(0,0.7)   $\ve[\iota]$;
[left]   at(0,2)      $v$;
[right]  at (2,0)     $x$;
[right]  at (2,2)     $y$;
[right]  at (2,1)     $a$;
at (1,-1)     $[1-\cos(k\cdot x)]$;
[right]  at (3.5,1)     $=$;

[-](0,0) to  (0,2);
[line width=2pt,-](0,0) to  (2,0);
[-](0,2) to    (2,2);
[dotted](2,0) to    (2,2);
(0,2) circle (3pt);
(2,0) circle (3pt);
(2,2) circle (3pt);
(0,0) circle (3pt);

[left]   at(0,0)      $0$;
[left]   at(0,2)      $v$;
[right]  at (2,0)     $x$;
[right]  at (2,2)     $y$;
[right]  at (-0.2,1)     ${\Small \geq n+1}$;
[right]  at (2,1)     $a$;
at (1,-1)     $[1-\cos(k\cdot x)]$;

[line width=1 pt] (0,0) to (-1,0);
[line width=1 pt] (-1,0) to (-2,0);
[line width=1 pt] (-2,0) to (-2,1);
[line width=1 pt] (0,0) to (0,-1);
[line width=1 pt] (-1,0) to (-1,2);
[line width=1 pt] (0,-1) to (0,-2);
[line width=1 pt] (0,-1) to (-1,-1);
[line width=1 pt] (-1,-1) to (-1,-2);
[line width=1 pt] (0,0) to (1,0);
[line width=1 pt] (1,0) to (2,0);
[line width=1 pt] (1,0) to (1,-1);
[line width=1 pt] (0,1) to (0,2);
[line width=1 pt] (-1,2) to (0,2);
[line width=1 pt] (0,1) to (-1,1);
[red,line width=2 pt] (0,0) to (0,1);
[red,line width=2 pt] (0,1) to (1,1);
[red,line width=2 pt] (1,1) to (2,1);
[red,line width=2 pt] (1,1) to (1,2);
[red,line width=2 pt] (2,1) to (2,2);
[red,line width=2 pt] (1,2) to (2,2);
[red,line width=2 pt] (2,2) to (2,3);
[red,line width=2 pt] (2,2) to (3,2);
[red,line width=2 pt] (2,3) to (3,3);
(0,0) circle (3pt);
(2,2) circle (3pt);
at(-0.2,-0.2)   $x$;
at(2.2,2.2)   $y$;
[red] at(0.7,3)   $B^A(x,y)$;

[line width=1 pt] (0,0) to (-1,0);
[line width=1 pt] (-1,0) to (-2,0);
[line width=1 pt] (-2,0) to (-2,1);
[line width=1 pt] (0,0) to (0,-1);
[line width=1 pt] (0,-1) to (2,-1);
[line width=1 pt] (2,0) to (2,-1);
[line width=1 pt] (0,-1) to (0,-2);
[line width=1 pt] (0,-1) to (-1,-1);
[line width=1 pt] (-1,-1) to (-1,-2);
[line width=1 pt] (1,0) to (1,-1);
[line width=1 pt] (1,0) to (2,0);
[line width=1 pt] (0,0) to (1,0);
[red,line width=2 pt] (0,0) to (1,0);
[line width=1 pt] (0,1) to (1,1);
[line width=1 pt] (0,1) to (0,2);
[line width=1 pt] (0,0) to (0,1);
[line width=1 pt] (-1,2) to (0,2);
[line width=1 pt] (0,1) to (-1,1);
[red,line width=2 pt] (1,0) to (1,1);
[red,line width=2 pt] (2,1) to (2,2);

[red,line width=2 pt] (1,1) to (1,2);
[red,line width=2 pt] (1,2) to (2,2);
[red,line width=2 pt] (2,2) to (2,3);
[red,line width=2 pt] (2,2) to (3,2);
[red,line width=2 pt] (2,3) to (3,3);
(0,0) circle (3pt);
(2,2) circle (3pt);
at(-0.2,-0.2)   $x$;
at(2.2,2.2)   $y$;
[red] at(0.7,3)   $B^A(x,y)$;

[line width=1pt,auto,scale=#1]

[gray] (4,0) – (5,1)–(5,-1);
[- latex](4,0) to  (5,1);
[latex-](4,0) to  (5,-1);
[-latex](5,1) to  (5,-1);
(4,0) circle (3pt);
[left]   at(4,0)      $0$;
[-](5+0.03,-1-0.2) to (5+0.03,-1+0.2);
[-](5+0.3-0.03,-1-0.2) to   (5+0.3-0.03,-1+0.2);
[below]   at(5.2,-1)      $b_1$;
[above]   at(5,1)      $u$;
(5,1) circle (3pt);

[-latex](5.5,1) to  (6.5,1.2);
[latex-](6.5,1.2) to  (7.5,1);
[-latex](5.8,-1) to  (7.5,-1);
[latex-](7.5,1) to  (7.5,-1);
[-](7.5+0.03,1-0.2) to (7.5+0.03,1+0.2);
[-](7.5+0.3-0.03,1-0.2) to   (7.5+0.3-0.03,1+0.2);
[-latex](7.5,-1) to  (9, -1);
[-latex](7.8,1) to  (9,  1);
[-latex](9,-1) to  (9,  1);
(9,-1) circle (3pt);
(9,1) circle (3pt);
(7.5,-1) circle (3pt);
(6.5,1.2) circle (3pt);
[right]   at(9,1)      $x$;
[right]   at(9,-1)      $w_2$;

[above]   at(6.5,1.2)      $w_1$;
[above]   at(7.75,1)      $b_{t_1+1}$;
[below]   at(7.5,-1)      $y$;

every node=[font=]
[above]  at(4.9,0.5)    $I_{S2}$;
[below]  at(5,-0.5)   $I_{S3}$;
[right]  at(5.5,0)    $I_{S4}$;
[above]  at(6.5,-1)   $II_B$;
[below]  at(6.2,1.2)      $I_{S1}$;
[below]  at(7,1.1)      $III_S$;
[right]  at(7.4,0)        $III_B$;
[above]  at(8.5,-1)   $II_S$;
[below]  at(8.5,1)   $IV_B$;
[right]  at(9,0)    $IV_S$;

[gray] (4,0) – (5,1)–(5,-1);
[-latex](4,0) to  (5,1);
[latex-](4,0) to  (5,-1);
[-latex](5,1) to  (5,-1);
(4,0) circle (3pt);
[left]   at(4,0)      $0$;
[-](5+0.03,-1-0.2) to (5+0.03,-1+0.2);
[-](5+0.3-0.03,-1-0.2) to   (5+0.3-0.03,-1+0.2);
[below]   at(5.2,-1)      $b_1$;
[above]   at(5,1)      $u$;
(5,1) circle (3pt);

[-latex](5.5,1) to  (6.5,1.2);
[latex-](6.5,1.2) to  (7.5,1);
[-latex](5.8,-1) to  (7.5,-1);
[latex-](7.5,1) to  (7.5,-1);
[-](7.5+0.03,-1-0.2) to (7.5+0.03,-1+0.2);
[-](7.5+0.3-0.03,-1-0.2) to   (7.5+0.3-0.03,-1+0.2);
[-latex](7.8,-1) to  (9, -1);
[-latex](7.5,1) to  (9,  1);
[-latex](9,-1) to  (9,  1);
(9,-1) circle (3pt);
(9,1) circle (3pt);
(7.5,1) circle (3pt);
(6.5,1.2) circle (3pt);
[right]   at(9,-1)      $x$;
[right]   at(9,1)      $w_2$;

[above]   at(6.5,1.2)      $w_1$;
[below]   at(7.75,-1)      $b_{t_1+1}$;
[above]   at(7.5,1)      $y$;

every node=[font=]
[above]  at(4.9,0.5)    $I_{S2}$;
[below]  at(5,-0.5)   $I_{S3}$;
[right]  at(5.5,0)    $I_{S4}$;
[above]  at(6.5,-1)   $II_B$;
[below]  at(6.2,1.2)      $I_{S2}$;
[below]  at(7,1.1)      $II_{S1}$;
[right]     at(7.4,0)        $II_{S3}$;
[above]  at(8.5,-1)   $III_B$;
[below]  at(8.5,1)   $II_{S1}$;
[right]  at(9,0)    $III_S$;

[line width=1pt,auto,scale=#1]
every node=[font=]

[-](5,1) to  (5,-1);
[left]   at(5,-1)      $0$;
[-](5+0.03,-1-0.2) to (5+0.03,-1+0.2);
[-](5+0.5-0.03,-1-0.2) to   (5+0.5-0.03,-1+0.2);
[below]   at(5.5,-1)      $e_1$;
[above]   at(5,1)      $w_1$;
(5,1) circle (3pt);
(7,1) circle (3pt);
(7,-1) circle (3pt);
[-](5,1) to  (7,1);
[-](5.5,-1) to  (7,-1);
[right]   at(7,1)      $y$;
[right]   at(7,-1)      $x$;

[-](4,0) to  (5,1);
[-](4,0) to  (5,-1);
[-](5,1) to  (5,-1);
(4,0) circle (3pt);
[left]   at(4,0)      $0$;
[-](5+0.03,-1-0.2) to (5+0.03,-1+0.2);
[-](5+0.5-0.03,-1-0.2) to   (5+0.5-0.03,-1+0.2);
[below]   at(5.5,-1)      $e_1$;
[left]   at(5,-1)      $\bb_0\neq 0$;

[above]   at(5,1)      $v$;
(5,1) circle (3pt);
(6,1.2) circle (3pt);
(7,1) circle (3pt);
(7,-1) circle (3pt);

[-](5,1) to  (6,1.2);
[-](6,1.2) to  (7,1);
[-](5.5,-1) to  (7,-1);
[right]   at(7,1)      $y$;
[right]   at(7,-1)      $x$;
[above]   at(6,1.2)      $w_1$;

[-](4,-1) to  (5,0.5);
[-](5,0.5) to  (5,-1);
(4,-1) circle (3pt);
(5,-1) circle (3pt);
[below]   at(5,-1)      $0$;
[below]   at(4,-1)      $e_1$;
[-](5+0.03,-1-0.2) to (5+0.03,-1+0.2);
[-](5+0.5-0.03,-1-0.2) to   (5+0.5-0.03,-1+0.2);
[above]   at(5-0.3,0.5)      $u$;
(5,0.5) circle (3pt);
(6,1.2) circle (3pt);
(7,1) circle (3pt);
(7,-1) circle (3pt);

[-](5,0.5) to  (6,1.2);
[-](6,1.2) to  (7,1);
[-](5.5,-1) to  (7,-1);
[right]   at(7,1)      $y$;
[right]   at(7,-1)      $x$;
[above]   at(6,1.2)      $w_1$;

[-](4,0) to  (5,1);
[-](4,0) to  (5,-1);
[-](3,0) to  (4,0);
[-](5,1) to  (5,-1);
(4,0) circle (3pt);
[above]   at(4,0)      $0$;
[below]   at(3,0)      $e_1$;
[-](5+0.03,-1-0.2) to (5+0.03,-1+0.2);
[-](5+0.5-0.03,-1-0.2) to   (5+0.5-0.03,-1+0.2);
[above]   at(5,1)      $v$;
(5,1) circle (3pt);
(6,1.2) circle (3pt);
(7,1) circle (3pt);
(7,-1) circle (3pt);

[-](5,1) to  (6,1.2);
[-](6,1.2) to  (7,1);
[-](5.5,-1) to  (7,-1);
[right]   at(7,1)      $y$;
[right]   at(7,-1)      $x$;
[above]   at(6,1.2)      $w_1$;

[-](4,0) to  (5,1);
[-](4,0) to  (5,-1);
[-](4.5,0.5) to  [out=135,in=90]   (3,0);
[-](5,1) to  (5,-1);
(4,0) circle (3pt);
(4.5,0.5) circle (3pt);
[below]   at(4,0)      $0$;
[above]   at(4.5,0.5)      $u$;
[below]   at(3,0)      $e_1$;
[-](5+0.03,-1-0.2) to (5+0.03,-1+0.2);
[-](5+0.5-0.03,-1-0.2) to   (5+0.5-0.03,-1+0.2);
[above]   at(5,1)      $v$;
(5,1) circle (3pt);
(6,1.2) circle (3pt);
(7,1) circle (3pt);
(7,-1) circle (3pt);

[-](5,1) to  (6,1.2);
[-](6,1.2) to  (7,1);
[-](5.5,-1) to  (7,-1);
[right]   at(7,1)      $y$;
[right]   at(7,-1)      $x$;
[above]   at(6,1.2)      $w_1$;

[-](4,0) to  (5,1);
[-](4,0) to  (5,-1);
[-](4.5,-0.5) to  [out=225,in=270]   (3,0);
[-](5,1) to  (5,-1);
(4,0) circle (3pt);
(4.5,-0.5) circle (3pt);
[above]   at(4,0)      $0$;
[below]   at(4.5,-0.5)      $u$;
[above]   at(3,0)      $e_1$;
[-](5+0.03,-1-0.2) to (5+0.03,-1+0.2);
[-](5+0.5-0.03,-1-0.2) to   (5+0.5-0.03,-1+0.2);
[above]   at(5,1)      $v$;
(5,1) circle (3pt);
(6,1.2) circle (3pt);
(7,1) circle (3pt);
(7,-1) circle (3pt);

[-](5,1) to  (6,1.2);
[-](6,1.2) to  (7,1);
[-](5.5,-1) to  (7,-1);
[right]   at(7,1)      $y$;
[right]   at(7,-1)      $x$;
[above]   at(6,1.2)      $w_1$;

[line width=1pt,auto,scale=#1]
every node=[font=]
at(-3-0.4,1)      $P^{\ssc[1],\iota,-m}(x,y):=\sum_w$;

at(1,0)      $\sum_{v,w,u,\bb_1}$;
[gray] (4,0) – (5,1)–(5,-1);
[-](4,0) to  (5,1);
[line width=2pt](4,0) to  (5,-1);
[dotted](4.5,0.5) to  [out=135,in=45]   (3,0);
[dotted](4.5,-0.5) to  [out=225,in=270]   (3,0);
[dotted](3,0) to  [out=345,in=210] (5,0);
[dotted](3,0) to  [out=90,in=150] (5.5,1.1);
[-](5,1) to  (5,-1);
(4,0) circle (3pt);
(5,-1) circle (3pt);
[gray,draw=black] (4.5,0.5) circle (3pt);
[gray,draw=black] (4.5,-0.5) circle (3pt);
[gray,draw=black] (5,0) circle (3pt);
[gray,draw=black] (5.5,1.1) circle (3pt);

[below]   at(4,0)      $0$;
at(4.6,0)      $u$;
[below]   at(3,0)      $e_\iota$;
[above]   at(5,1)      $v$;
[below]   at(5,-1)      $\bb_1$;
(5,1) circle (3pt);
(6,1.2) circle (3pt);
(7,1) circle (3pt);
(7,-1) circle (3pt);
(3,0) circle (3pt);

[-](5,1) to  (6,1.2);
[-](6,1.2) to  (7,1);
[line width=2pt](5,-1) to  (7,-1);
[-](7,-1) to  (7,1);
[right]   at(7,0)      $m$;
[right]   at(7,1)      $y$;
[right]   at(7,-1)      $x$;
[above]   at(6,1.2)      $w$;

at(2,0)      $+\sum_{v,w,\bb_1}$;
[gray] (4,0) – (5,1)–(5,-1);
[-](4,0) to  (5,1);
[line width=2pt](4,0) to  (5,-1);
[-](5,1) to  (5,-1);

(4,0) circle (3pt);
[left]   at(4,0)      $0$;
[-](5+0.03,-1-0.2) to (5+0.03,-1+0.2);
[-](5+0.5-0.03,-1-0.2) to   (5+0.5-0.03,-1+0.2);
[below]   at(5.5,-1)      $e_\iota$;
[left]   at(5.1,-1.2)      $\bb_1$;

[above]   at(5,1)      $v$;
(5,1) circle (3pt);
(6,1.2) circle (3pt);
(7,1) circle (3pt);
(7,-1) circle (3pt);

[-](5,1) to  (6,1.2);
[-](6,1.2) to  (7,1);
[line width=2pt](5.5,-1) to  (7,-1);
[-](7,1) to  (7,-1);

[right]   at(7,1)      $y$;
[right]   at(7,0)      $m$;
[right]   at(7,-1)      $x$;
[above]   at(6,1.2)      $w$;

§ AN OVERVIEW TO THE NOBLE DIAGRAMMATIC BOUNDS

In this section we discuss how we bound the NoBLE coefficients. We explain the concepts that we use to obtain sharp bound on these diagrams, each of which forms a numerical improvement on the bounds on classical lace-expansion coefficients. Then, we follow that up with the technical definitions of these concepts. We end this section with a discussion of how we bound double connections for LAs, as these are quite central in our bounds for LAs.

§.§ Basic bounds and their improvements

We explain here how we obtain sharp bounds on

\begin{align}
\bar \Xi^{\ssc[1]}_z(x) =& \sum_{\omega\in\Wcal(x)} z^{|\omega|} Z[0,|\omega|]  J^\ssc[1][0,|\omega|]\nnb
=& \sum_{\omega\in\Wcal(x)}  z^{|\omega|}  \prod_{i=0}^{|\omega|} z^{|S^{\omega}_i|}
\indic{S^{\omega}_s\text{ and }S^{\omega}_t\text{ are only intersecting for $s=0$ and $t=|\omega|$}   },
\lbeq{bound-Xi1-1}
\end{align}

by using four ideas, one at a time. We restrict here to LTs, as these give rise to the simplest diagrams, which already display all the main ingredients to our bounds.

We compare the improvement that we obtain in our bounds by providing numerical bounds for the initial point $z=z_I$ resulting from the numerical analysis performed in our Mathematica notebooks. Our proof is based upon the fact that the values at $z_I$ and at $z_c$  are actually quite close. Thus, even though $z_c>z_I$,
such bounds should give us a clear idea of how large such bounds are, and what the effect of the improvements is.

We recall LTInitalBound-goal and combine it with the bound $g^\iota_{z_I}\leq \e$, see gjgzrelation-Initial, to conclude, for $z_I=(2d-1)^{-1}\e^{-1}$,

\begin{align}
\lbeq{bound-Gzi}
\bar G_{z_I}(x)\leq g_{z_I}^\iota B_{1/(2d-1)}(x)=\e B(x),
\end{align}

where $\e$ is the Euler number and $B(x)$ critical two-point function of the NBW, see NBWGen. This bound is independent of the lace expansion and we can compute its value numerically. While this bound does not hold at the critical point $z_c$, its value is good enough to compare the bounds discussed in this section. All stated numerical results are computed for dimension $d=18$, and are aimed to convey the numerical improvements in the bounds.

Plain-vanilla bound.
We focus on $\bar \Xi^{\ssc[1]}_z$ and start with the simplest possible approach, which relies on the bounds as they have been stated up to now in the literature.
The coefficient $\bar \Xi^{\ssc[1]}_z$ involves three points: $0$, $x$ and a point where $S^{\omega}_0$ and $S^{\omega}_{|\omega|}$ intersect, which we denote by $w$.
Bounding the connections between each of the points by individual independent LTs we obtain

\begin{align}
\lbeq{bound-Xi1-2pre}
\bar \Xi^{\ssc[1]}_z(x) \leq & \bar G_z(x)\sum_{w} \bar G_z(w-x) \bar G_z(w),
\end{align}

which, combined with bound-Gzi, implies, for $z\leq z_I$,

\begin{align}
\lbeq{bound-Xi1-2}
\sum_x\bar \Xi^{\ssc[1]}_{z_I}(x) \leq & (\bar G_{z_I})^{\star 3}(0)\leq \e^3  B^{\star 3}(0)=22.322\dots
\end{align}

This is of course a very bad bound that does not allow us to successfully apply the lace-expansion method, so we improve it using the four ideas explained below.

First improvement: extracting trivial lines.
A simple way to improve this bound is to consider four cases depending on which lines are trivial: $x=0$, then $w=x\neq 0$ and $w=0\neq x$ and the remaining cases.
In the case $x=0$, we extract the contribution of the first rib $g_z^\iota$ and the first bond of the backbone
with all its possible directions $2dz$, and bound the remaining rib-walk, that goes back to the origin, by $\bar G_z(\ve[1])$.
Thus, the case $x=0$ is bounded by $2d zg_z^\iota(D\star \bar G_z)(0)$.

For $w=x\neq 0$ and $w=0\neq x$,  $2$ non-trivial two-point functions are required. For $w\nin\{0,x\}$, instead, all three connections are non-trivial.
We bound the non-trivial connections by $\bar G_{z_I}(x) \leq \e B(x)\leq \tfrac {2d\e}{2d-1}(D\star B)(x)$ for $x\neq 0$ and conclude that

\begin{align}
\sum_x\bar \Xi^{\ssc[1]}_{z_I}(x) \leq &
\e\Big(\frac {2d}{2d-1}\Big)^2 (D^{\star 2}\star B)(0)+ 2\e^2 \Big(\frac {2d}{2d-1}\Big)^2 (D^{\star 2}\star B^{\star 2})(0)\nnb
&+ \e^3\Big(\frac {2d}{2d-1}\Big)^3 (D^{\star 3}\star B^{\star 3})(0)=0.85541\ldots
\lbeq{bound-Xi1-3}
\end{align}

This is an enormous improvement compared to  bound-Xi1-2. The simple idea of extracting trivial lines, by splitting between different cases, is a basic technique that is used for all coefficients in Section <ref>.

Second improvement: improved counting of one-point functions by rib allocation.
We observe that we extract a single one-point function at every line in an intersection point. This gives rise to the three factors of $\e$ in bound-Xi1-3.
However, when the different lines come from the same rib, there in fact is only one one-point function, so that we are overcounting one one-point function (leading to a factor $\e$)
every time we split a rib. Thus, for the diagram in bound-Xi1-2, we would need only one factor $\e$ instead of three.
Let us explain one way to obtain a bound without overcounting.

In bound-Xi1-2pre, the terms $\bar G_z(w)$ and $\bar G_z(w-x)$ bound the ribs $S^{\omega}_0$ and $S^{\omega}_{|\omega|}$, which, being ribs, are just LTs.
The indicator $J^\ssc[1][0,|\omega|]$ in bound-Xi1-1 guarantees that only those $\omega$ contribute for which all in-between ribs do not intersect, so that the rib-walk between
$\tb_1^\omega=u$ and $\tb^{\omega}_{|\omega|}=v$ also describes a LT and can be bounded by $\bar G_z(v-u)$.
Extracting the case that $|\omega|=1$, for which $\tb_1^\omega=\tb^{\omega}_{|\omega|}$,
so that there are no in-between ribs, we obtain

\begin{align}
\lbeq{bound-Xi1-4pre}
\bar \Xi^{\ssc[1]}_z(x) \leq
&\indic{x=0}2d zg_z^\iota(D\star \bar G_z)(0) \\
&+\indic{x\neq 0} \sum_w \bar G_z(w) \bar G_z(w-x)
\Big( \indic{|x|=1} z+ z^2  \sum_{\stackrel{u,v\in\Zd}{|u|=|x-v|=1}}
\bar G_z(v-u)\Big),\nn
\end{align}

where $z$ and $z^2$ correspond to the weight of the first and last bond of the original backbone.
Considering the four cases for $0,x,w$, used in the first improvement, we obtain

\begin{align}
\sum_x\bar \Xi^{\ssc[1]}_{z_I}(x) \leq &
3\e\Big(\frac {2d}{2d-1}\Big)^2 (D^{\star 2}\star B)(0) + \e \Big(\frac {2d}{2d-1}\Big)^3 (D^{\star 3}\star B^{\star 2})(0)\nnb
&+ 2\e\Big(\frac {2d}{2d-1}\Big)^3 (D^{\star 3}\star B^{\star 2})(0) +\e\Big(\frac {2d}{2d-1}\Big)^4 (D^{\star 3}\star B^{\star 3})(0)=0.5725\dots
\lbeq{bound-Xi1-4}
\end{align}

The improvement of bound-Xi1-3 to bound-Xi1-4 is mostly realized by the proper handling
of the weights of the one-point function $\gj\approx \e$ at the origin and at $x$, which are bounded twice in bound-Xi1-3.

In general, the diagram of $\Xi^{\ssc[N]}_z$ has $4N-1$ connections for LTs. Allocating the one-point functions properly will save us approximately a factor $\e^{4N-2}$ for all $N$. This allocation of rib weights is a central technical problem, especially as we require various different allocations for the weights
when we bound the coefficient with spatial weight $|x|^2$. Using splitting arguments of such spatial terms, that are standard in lace-expansion analyses, we split the weight along a path connecting $0$ and $x$,
e.g. using $|x|^2\leq N\sum_{i=1}^N |x_i|^2$, where the $(x_i)_{i=1}^N$ are the displacements along the bottom lines of the diagram,
so that $x=\sum_{i=1}^N x_i$. For each of these partial weights $|x_i|^2$, we want to bound the corresponding connection
by $\bar G_z$, as $f_3$ only provides us with a bound on $|x_i|^2 \bar G_z(x_i)$.
This imposes severe restrictions on our rib-weight allocation, and we need to be really careful in such arguments.
We discuss this in more detail in Section <ref>, see also Remark <ref>.

Third improvement: using the non-backtracking nature of the diagram.
Now, we use the non-backtracking property of the rib-walk for the first time. This property implies that any loop needs at least $4$ bonds,
as direct reversals are prohibited and loops on the lattice require an even number of bonds.
For example the self-loop of the rib-walk back to $x=0$ will take at least $4$ steps and can be bounded by

\begin{align}
(2d {z_I}g_{z_I}^\iota)^3 (D^{\star 3}\star \bar G_{z_I})(0)
\leq \left(\frac {2d }{2d-1}\right)^3 \frac {2d\e} {2d-1} (D^{\star 4}\star B)(0).
\end{align}

Further, we can omit the case $w\in\{0,x\}$ for the case that $b_1^\omega=(0,x)$, see property (v) of Definition <ref>.
Going through $8$ different combinations of how the $4$ steps can be distributed, we obtain

\begin{align}
\sum_x\bar \Xi^{\ssc[1]}_{z_I}(x) \leq &
4\e\big(\frac {2d}{2d-1}\big)^4 (D^{\star 4}\star B)(0) + 3\e \big(\frac {2d}{2d-1}\big)^4 (D^{\star 4}\star B^{\star 2})(0)\nnb
&+\e\big(\frac {2d}{2d-1}\big)^4 (D^{\star 3}\star B^{\star 3})(0)=0.0755\dots
\lbeq{bound-Xi1-5}
\end{align}

The improvement from bound-Xi1-4 to bound-Xi1-5 demonstrate the power of the NoBLE and its non-backtracking property. To use this property for the other diagrams, we will split depending on the lengths of lines shared by two loops. This splitting into a total of five cases of lengths of shared lines will create matrix-based bounds. This will become apparent in Section <ref>.

Fourth improvement: using repulsiveness in diagrams. We can actually improve this bound even further. For this we have to understand that the paths connecting $0$, $x$ and $w$ do not intersect, so that the loop is actually a self-avoiding polygon. We call such a polygon a repulsive diagram. For this to work, we have to choose the point $w_1$ as a first intersection point and bound it by a repulsive diagram. Both terms are defined within the next two sections. Using these repulsive diagrams, we improve the bound bound-Xi1-5 to

\begin{align}
\lbeq{bound-Xi1-6}
\sum_x\bar \Xi^{\ssc[1]}_{z_I}(x) \leq & 0.02562\dots
\end{align}

This is explained in more detail in Section <ref>.

Summary. By the above four improvements combined, we have now improved our estimate on this simple diagram $\sum_x\bar \Xi^{\ssc[1]}_{z_I}(x)$ by a factor that is close to 1000. Since we need all the numerical precision that we can get our hands upon, so as to be able to apply our methods in the lowest possible dimensions, such estimates are crucial to the success of our method.

Extension to lattice animals.
Lattice animals (LAs) allow for double connections. Despite this additional difficulty, for $\bb_0=0$ and $\tb_{|\omega|}=x$, we can use the same bounds bound-Xi1-2pre-bound-Xi1-6 based on exactly the same arguments. For $\bb_0\neq 0$ we need to bound the sausage, which connects the points $0,\bb_0$, as well as the first intersection point $w_1$. We explain this in Section <ref>, where we also define the required notions. In Section <ref>, we discuss how we use a symmetry argument to improve the bound for the important case $w_1\in\{0,\bb_0\}$ of a double connection.

§.§ Allocation of one-point functions

In the preceding section, we disassembled the rib-walk into up to three pieces: the initial rib, the final rib and the walk in between, and have argued that rib-weight allocation is a crucial ingredient to the success of our method. In the following, we also need to split individual ribs/sausages. Below we give a rigorous description of the split of a rib/LT and explain how we ensure that we do not overcount one-point functions by an optimal rib-weight allocation.

After this, we explain the allocation of one-point functions for the more involved diagrams. In the process we define the concept of first intersection point of two sausages, that will allow use to choose bond-disjoint paths connecting the corner points of the loops. In the next section we use this choice of bond-disjoint paths to define repulsive diagrams.

Splitting the weight of a rib: planted animals/trees.
Our aim is to bound the contribution $z^{|T|}$ of LTs that contain the vertices $0,x,v\in\Zd$ efficiently. We identify the last vertex $u$ that the path $0\leftrightarrow x$ and the path $0\leftrightarrow v$ have in common, split the tree into three trees and ignore the avoidance constraints between them to obtain

\begin{align}
\sum_{T} \indic{0,x,v\in T} z^{|T|}&\leq \sum_{T_1,T_2,T_3}\sum_{u\in\Zd}
\indic{0,u\in T_1}\indic{x,u\in T_2}\indic{u,v\in T_3} z^{|T_1|+|T_2|+|T_3|}\nnb
\lbeq{ribweightLT-1}
&=\sum_{u} \bar G_{z}(u) \bar G_{z}(x-u)\bar G_{z}(v-u),
\end{align}

which corresponds to the plain-vanilla bound of the last section. Thus, it also counts three ribs at $u$, whereas there really only is one.

To avoid this, we define an adapted version of the two-point function that only counts the trivial first rib/sausage $A^\omega_0=\varnothing$, i.e.,

\begin{eqnarray}
\lbeq{defTwoPointFunctionTildeTotal}
\tilde G_{z}(x)&=&\sum_{\omega\in\Wcal(x)}z^{|\omega|} Z[0,|\omega|]K[0,|\omega|]\indic{A^\omega_0=\varnothing}.
\end{eqnarray}

Using this two-point function, we obtain the bound

\begin{align}
\lbeq{ribweightLT-2}
\sum_{T} \indic{0,x,v\in T} z^{|T|}\leq \sum_{u} \bar G_{z}(u)\tilde G_{z}(x-u)\tilde G_{z}(v-u),
\end{align}

which is approximately $\e^2$ smaller than ribweightLT-1.

Planted animals.
For LTs, the point $u$ at which we split the tree is unique, which is not the case for LA. We define the concepts of backbone, connecting planted animal and
first point of intersection, to provide a rigorous way to split the sausage for LAs. We only give the definition for LAs and sausage-walks, as each LT is also a LA and each rib-walk is also a sausage-walk.

Before defining these notions, we discuss the lexicographic order on paths, which is a way to identify paths uniquely. We say that $x\in \Zd$ has a lower lexicographic order than $y\in \Zd$, if there exists an $i\in \{1,2,\dots,d \}$ with $x_i<y_i$ and $x_j=y_j$ for $j=1,2,\dots, i-1$. A bond $b$ is defined as a tuple of two vertices $b=(\bb,\tb)$, so that we can view $b$ also to be a vector in $\Zbold^{2d}$ and use the same order relations as for vertices. A path $(b_1,b_2,\dots,b_n)$ has a lower lexicographic order than the path $(t_1,t_2,\dots,t_m)$, if either there exists a $i\in \{1,2,\dots, \min(n,m)\}$, such that $b_i$ has a lower order than $t_i$ and $b_j=t_j$ for $j=1,2,\dots, i-1$ or if $n<m$ and $b_i=t_i$ for $i=1,2,\dots, n$.

Let $A$ be a lattice animal containing $x,y\in\Zd$, with $x\neq y$.
A path from $x$ to $y$ in $A$ is a sequence of bonds $(b_i)_{i=1,\dots,N}$ such that $b_i\in A,
\tb_i\neq \tb_j$ for all $i\neq j$, and $\bb_1=x,\tb_i=\bb_{i+1},\tb_N=y$ for $i=1,\dots,N-1$.
We define $b^A(x,y)$ to be the path from to $x$ to $y$ in $A$ with the lowest lexicographic order.

For $x=y\in A$, we define $b^A(x,y)=\varnothing$.

Let $A$ be a lattice animal containing $x,y\in\Zd$, with $x\neq y$, and let $p$ be a path from $x$ to $y$ in $A$.
Let $S$ be the subset of $A\setminus\{p\}$ of all bonds for which at least one vertex of the bond is connected to $x$ via bonds in $A\setminus \{ p\}$.
We call $B^A(x,y;p)=A\setminus S$ the connecting planted animal from $x$ to $y$ along $p$.

Unless stated otherwise, we consider the path $p$ to be the backbone $p=b^A(x,y)$ and omit $p$ from the notation.

For a sausage-/rib-walk $\omega$, we define $B^\omega_i(x,y)=B^{A^\omega_i}(x,y)$.

Two examples of a connecting planted animal.

We next define the notion of a first intersection point. We do this in a more general setting, encoded by a lace $L$, where some of the LAs are forced to intersect each other, while others are not. This will make sure that we can use this definition for all NoBLE lace-expansion coefficients:

For a sausage-walk $\omega$, a lace $L=\{s_1t_1,s_2t_2,$ $\dots,$ $s_Mt_M\}$ and $i\leq M$, we define $W_i(\omega)$  to be the set of vertices $w$ that are contained in both $A^\omega_{s_i}$ and $A^\omega_{t_i}$, such that $b^{A^\omega_{s_i}}(\tb^\omega_{s_i},w)$ and $A^\omega_{t_i}(\tb^\omega_{t_i},w)$ only intersect at $w$. If $W_i(\omega)$ is non-empty, then we define $w_i(\omega)$ to be the smallest in the set $W_i(\omega)$.
We call $w_i(\omega)$ the first intersection point of $A^\omega_{s_i}$ and $A^\omega_{t_i}$.

Note that if the end of the $s_i$th pivotal bond $\tb^\omega_{s_i}$ is in $S^\omega_{t_i}$, then $W_i(\omega)=\{\tb^\omega_{s_i}\}$.
We illustrate the concept of a first intersection point in Figure <ref>.

Two examples of the set of first intersection points $W_N(\omega)=\{w_a,w_b,w_c\}$.
In the picture the $s_N$th and $t_N$th sausage of the sausage-walk $\omega$ intersect.
We choose the point $w_N(\omega)$ to be the unique smallest representative of $W_N(\omega)$ in the lexicographic order.

Non-trivial sausages.
If we wish to extend the analysis to LAs, then we need to deal with non-trivial sausages. A lattice animal $A$ is called a non-trivial sausage if it contains three vertices $0,x,w$ (with $x\neq 0$), such that $0$ and $x$ are doubly connected in $A$, as shown in the first image of Figure <ref>. We now show how we bound these sausages.
In Section <ref>, we explain how we obtain improved bounds for the special case $w\in\{0,x\}$, which is a major contribution for LAs.

We define the point $u$ to be a point that every path from $0$ to $w$ and $x$ to $w$ shares. By construction, the points $0,u,x$ are pairwise doubly connected, so that there exist four bond-disjoint paths: $p_1(u,w), p_2(u,x), p_3(0,u)$ and $p_4(0,x)$ connecting the indicated points in $A$. We bound the contribution of such LAs by

\begin{align}
\sum_{A\colon 0,x,w\in A} \indic{0\dbct{A} x}z^{|A|}\leq& 2d z \sum_{u}\tilde G_{z}(u-w)(D\star \tilde G_{z})(u-x)\tilde G_{z}(u)\bar G_{z}(x).
\lbeq{split-LA-sausage-Gbar}
\end{align}

We obtain this bound by decomposing each LA $A$ into four parts as shown in Figure <ref>. Using the pictured decomposition, we know that $A_4$ is an animal containing $x$, which we bound by $\bar G_z(x)$. Further, $A_1$ and $A_3$, respectively, only include one bond that contains $u$ and $0$, respectively.
Thus, we bound them using the modified two-point function $\tilde G_z$. The animal $A_2$ is the interesting case as $u$ and $x$ are only connected by one bond each. If $u\neq x$, then we bound this contribution by the step involved by $2d z D(\cdot)$ and the remainder by $\tilde G_z$, creating the bound $2dz(D\star \tilde G_z)(x-u)$.
If $u=x$ then $A_2=\varnothing$ and $A_3$ only has one bond at the origin and one at $x$, so we bound $A_3$ by $2dz(D\star \tilde G_z)(x)$.

[anchor=south west,inner sep=0] at (0,0) ;
at(0.1,2.4)   $0$;
at(5.5,1.7)   $x$;
at(2.2,3.6)   $u$;
at(4.8,5.1)   $w$;
at(1,4.5)   $A$;

*$A$ containing $0,x,w$ and the last common point $u$ on the paths from $0$ to $w$ and $x$ to $w$.

[anchor=south west,inner sep=0] at (0,0) ;
at(0.1,2.4)   $0$;
at(5.5,1.7)   $x$;
at(2.2,3.6)   $u$;
at(4.8,5.1)   $w$;
[blue] at(2.2,5.0)   $A_1$;
at(0,3.1)   $R_1$;

*Let $A_1=B^{A}(u,w; p_1(u,w))$ and $R_1=A\setminus A_1$.

[anchor=south west,inner sep=0] at (0,0) ;
at(0.1,2.4)   $0$;
at(5.5,1.7)   $x$;
at(2.2,3.6)   $u$;
at(4.8,5.1)   $w$;
[blue] at(2.2,5.0)   $A_1$;
[green] at(5,3)   $A_2$;
at(0,3.1)   $R_2$;

*Let $A_2=B^{R_1}(u,x; p_2(u,x))$ and $R_2=R_1\setminus A_2$.

[anchor=south west,inner sep=0] at (0,0) ;
at(0.1,2.4)   $0$;
at(5.5,1.7)   $x$;
at(2.2,3.6)   $u$;
at(4.8,5.1)   $w$;
[blue] at(2.2,5.0)   $A_1$;
[green] at(5,3)   $A_2$;
[red] at(0,3.1)   $A_3$;
at(2.7,1)   $A_4$;

*Let $A_3=B^{R_2}(0,u; p_3(0,u))$ and $A_4=A\setminus(A_1\cup A_2 \cup A_3)$.

The way that we split a LA to create the bound split-LA-sausage-Gbar.

There are multiple ways to split such animals, and depending on the situation we might want to use a different combination of where to attribute the rib weights. To give an example, we can create animals as discussed above in which each line is bounded by $\tilde G_z$. For this, we modify the split shown in Figure <ref> as follows:

* Let $R_3=B^{R_2}(x,0; p_4(x,0))$, and $R_4=R_2\setminus R_3$.

* Let $\bar A_3=B^{R_2}(0,u; p_3(0,u))$ and $\bar A_4=R_3\setminus \bar A_3$.

* Let $\bar A_2=R_4\cup A_2$.

The split into $A_1,\bar A_2,\bar A_3$ and $\bar A_4$ is very much alike the one shown in Figure <ref>, the only difference
being that every bond incident to $x$ that is not part of the path $p_4(x,0)$ is now part of $\bar A_2$ instead of $\bar A_4$.
This creates the bound

\begin{align}
\lbeq{split-LA-sausage-second}
\sum_{A\colon 0,x,w\in A} \indic{0\dbct{A} x}z^{|A|}\leq& \sum_{u}\tilde G_{z}(u-w)\tilde G_{z}(u-x)\tilde G_{z}(u)\tilde G_{z}(x).
\end{align}

Which of the different bounds is chosen depends on the precise nature of the bound that we are deriving. In fact, optimizing over such decisions is numerically an important ingredient of our method.

Bound on the sausage-walk in $\Xi^{\ssc[1]}_z(x)$.
Here we continue our bound on $\Xi^{\ssc[1]}_z(x)$ started in Section <ref>, now specializing to LAs with a non-trivial first sausage.
We have bounded the case $\bb_1=0$ by splitting the sausage-walk into three parts:

* The first sausage $S^\omega_0$ that connects $0$ and $w_1(\omega)$, bounded by $\bar G_z(w)$;

* The last sausage $S^\omega_{|\omega|}$ that connects $x$ and $w_1(\omega)$, bounded by $\bar G_z(w-x)$;

* The remaining sausage-walk without the first and last sausage, bounded by $2dz(D\star\tilde G_z)(x)$.

For $\bb_1\neq 0$, which is not possible for LTs, we need to split the first sausage $S^\omega_0$.
The first sausage contains $0,\bb_1$ and $w_1(\omega)$ and doubly connects $0$ and $\bb_1$.
Thus, $S^\omega_0$ is a non-trivial sausage as discussed in the previous paragraph.
Combining these bounds we arrive at

\begin{align}
\lbeq{bound-xi1-structure}
\Xi^{\ssc[1]}_z(x)\leq& 2dz \sum_{w} \bar G_{z}(w) \bar G_{z}(w-x)(D\star \tilde G_{z})(x)\\
&+    2dz\sum_{w,\bb,u}\bar G_{z}(w-x)(D\star \tilde G_{z})(x-\bb)\tilde G_{z}(u-w)\tilde G_{z}(u-x)\tilde G_{z}(u)\tilde G_{z}(\bb),\nn
\end{align}

where the second term is absent for LTs.

The allocation of one-point functions.
As can be seen above, we have some choice of how we split the sausages. This becomes quite complex when more lines are involved.
Let us discuss the example $\Xi^{\ssc[2]}_z$, which is the simplest diagram of the more involved cases.

[1]

The diagram of a contribution to $\Xi^{\ssc[2]}_z(x)$. We indicate the lace by $L=\{0t_1, s_2|\omega|\}$.
The left diagram correspond to $s_2<t_1$ and the right diagram to $s_2=t_1$.
For reference we label the individual lines by roman capital numbers.

In the following, we define how we split the diagram of $\Xi^{\ssc[2]}_z(x)$ for $s_2<t_1$, as shown in Figure <ref>.
Consider $\omega$ and a lace $L$ that contributes to $\Xi^{\ssc[2]}_z(x)$.
We know that $S^\omega_{0},S^\omega_{t_1}$ and $S^\omega_{s_2},S^\omega_{|\omega|}$ intersect,
at their first intersection points $w_1(\omega)$ and $w_2(\omega)$ (recall Definition <ref>).

Let $u$ be the last common point of the paths $0\conn w_1(\omega)$ and $\bb_1\conn w_1(\omega)$ in $S_0^\omega$.
Let $y$ be the last point that the paths $\tb_1\conn w_2(\omega)$ and $\bb_{t_1+1}\conn w_1(\omega)$ in $\omega$ share.

For $\bb_1=0$, we have that $u=0$ and define $I_{S1}:=B^{S_0^{\omega}}_0(0,w_1)$ and $I_{S2}=I_{S3}=I_{S4}=\varnothing$.

For $\bb_1\neq 0$, the first sausage is non-trivial. We use the split that created split-LA-sausage-second, which formally corresponds to

\begin{align*}
I_{S1}=&B^{S_0^{\omega}}_0(u,w_1), \quad R_1=S_0^{\omega}\setminus I_{S1},\quad  A_2=B^{R_1}_0(u,x),\\
R_2=&R_1\setminus A_2,\quad R_3=B^{R_2}_0(x,0), \quad R_4=R_2\setminus R_3,\\
I_{S2}=&B^{R_2}_0(0,u), \quad I_{S3}=R_3\setminus I_{S2}, \quad \text{and}\quad  I_{S4}=R_4\cup A_2.
\end{align*}

Then, we split the three branches from the backbone, that create two intersections at $w_1$ and $w_2$, namely
$II_S:=B^ \omega_{t_1}(y,w_2)$, $III_S:=B^\omega_{s_2}(\bb_{s_2+1},w_1)$ and $IV_S:=B^\omega_{|\omega|}(x,w_2)$.
The trimmed sausage-walk then create $II_B$, $III_B$, $IV_B$.
By this construction, all ten paths connecting the labeled points are bond-disjoint and each piece is a LA.

All pieces with sub-index $S$ are planted animals and can be bounded by $\tilde G_z$.
For the backbone lines $II_B,III_B,IV_B$, we have to decide where to bound the sausages at the labeled points.
We can bound two of the pieces using $\tilde G_z$, taking into account only the weight of either the first or final sausage.
The third piece has to be bounded using $\bar G_z$, which bounds the sausages on both ends.
Thus, we bound only one connection by $\bar G_z$, rather than all four, which is a major improvement.
Using a slightly different split of the sausage-walk, we are actually free to choose any line to be bounded by $\bar G_z$.
This will be useful when dealing with weighted diagrams, we discuss this in detail in Section <ref>.

For the case $t_1=s_2$, as shown in the right diagram of Figure <ref>, we need to
split the sausage $S^\omega_{t_1}$. For this we define $y$ to the be point that all connections in  $S^\omega_{t_1}$
from $\bb_{t_1+1}$ to $w_1$  and from $\bb_{t_1+1}$ to $w_2$ share and define

\begin{align*}
II_{S1}=&B^{S_{t_1}^{\omega}}_0(y,w_1),\qquad II_{S2}=&B^{S_{t_1}^{\omega}}_0(y,w_2), \quad
II_{S3}=&B^{S_{t_1}^{\omega}\setminus(II_{S1}\cup II_{S2}) }_0(\bb_{t_1+1},y).
\end{align*}

The rest is spit in the same way as described for the case $s_2<t_1$.

[1]

Picture of a possible $\Xi^{\ssc[4]}_z$ diagram. The backbone is marked by a thicker line and the ribs of $x_i$ have different colors. Arrow tips indicate that the one-point function is allocated at this endpoint. Thus, all lines except for one are $\tilde G$'s. Only one line has two arrow tips, which corresponds to a factor $\bar{G}$. This line can be located anywhere in the diagram. The numbers are labels for the corresponding two-point functions.

In Figure <ref>, we have ten connections, nine of which can be bounded by $\tilde G_z$, and one by $\bar G_z(k)$.
In general, the diagram of $\Xi^{\ssc[N]}_z$ has $4N+2$ connections, and only one of them needs to be bounded by $\bar G_z$.
This means that by bounding each line by $\bar G_z$ instead of $\tilde G_z$,  as done previously in literature,
any bound on $\hat \Xi^{\ssc[N]}_z$ is unnecessarily a factor $\e^{4N+1}$ too large, at least for $\bb_1\neq 0$.
For $\bb_1=0$ three lines are trivial ($I_{S2}=I_{S3}=I_{S4}=\varnothing$), so that this unnecessary factor is still $\e^{4N-2}$.
We can and will choose which of the lines is bounded by $\bar G_z(k)$ in a way that sensitively depends on the precise structure of a diagram.
In Section <ref>, we explain in detail how we do this for the lace-expansion coefficients with the spatial weight $|x|^2$,
for which it is the most relevant. For bounds without the spatial weights, we usually use the split shown in Figure <ref>.
There, we use the bound $\bar G_z(x)\leq \gj \tilde G_z(x)$ for $x\neq 0$ for the single $\bar G_z$ factor, which simply extracts one
one-point function $\gj$, after which all connections of the diagram are given by $\tilde G_z$.

§.§ Repulsive diagrams

In this section, we define the basic diagrams used to bound the NoBLE coefficients.
The ribs/sausages of the NoBLE coefficients have numerous avoidance constraints.
We incorporate some of these in our diagrams to improve our numerical bounds.
Here we explain how this can be done using so-called repulsive diagrams.

To motivate the precise definition of these diagrams, we review the avoidance structure of the first square $(1,2,3,4)$ in Figure <ref>.
In the diagrams the vertices $x_i$ denote points on the backbone, the vertices $w_i$ denote the $i$th first intersection point (recall Definition <ref>), and $u_i$ the vertices where we have to split a sausage/rib.
Alike the split of $\Xi^{\ssc[N]}_z$ given above, the split given in Figure <ref> has the following interpretation:

$\rhd$ Line $2$ is the connected planted animal $B^{\omega}_0(0,w_1)$;

$\rhd$ Line $3$ is the connected planted animal $B^{\omega}_{t_1}(x_1,w_1)$;

$\rhd$ Line $4$ is the connected planted animal $B^{\omega}_{t_1}(w_1,u_1)$;

$\rhd$ Line $1$ is the backbone $b^\omega(0,x_1)$, together with the sausages $(A^\omega_{i})_{i=1,\dots,t_1-1}$ and the (trimmed) animal
$A^{\omega}_{t_1}\setminus (B^{\omega}_{t_1}(x_1,w_1)\cup B^{\omega}_{t_1}(w_1,u_1))$.

These lines obey the following avoidance constraints:

$\rhd$ Relation between $1$ and $2$: By the definition of $J[0,|\omega|]$, the sausage $A_{t_1}^\omega$ is the first sausage to intersect with the sausage $A_0^\omega$.
Thus, lines $2$ and $1$ only have the origin as starting point in common and are otherwise vertex-disjoint;

$\rhd$ Relation between $1$ and $3,4$: The connections $3$ and $4$ are all in $A_{t_1}^\omega$.
The indicator $J[0,|\omega|]$ ensures that $A_{t_1}^\omega$ does not intersect any of the intermediate sausages $(A_i^\omega)_{i=1,2,\dots,t_1-1}$.
Further, $1$ bounds the part of $A_{t_1}^\omega$ that has not been split off as planted animal into 3, indicated by the tip of the arrow in  Figure <ref>.
Thus, $1$ is bond-disjoint from $3$ and $4$;

$\rhd$ Relation between $3$ and $4$: Both are part of the sausage $A_{t_1}^\omega$ and are thus bond-disjoint;

$\rhd$ Relation between $2$ and $3,4$: We know that $A_{0}^\omega$ and $A_{t_1}^\omega$ intersect in at least one point.
We choose $w_1=w_1(\omega)$ to be the first intersection point, as defined in Definition <ref>.
By this choice the backbones $b^{A^\omega_0}(0,w_1)$ and $b^{A^\omega_{t_1}}(x_1,w_1)$ only intersect at $w_1$.

We define repulsive diagrams to bound such diagrams. As seen for $2$ and $3,4$, the constraint that the pieces are bond-disjoint is too strong.
For this reason, we denote by repulsiveness of a diagram that we can find bond-disjoint paths connecting the corner points of the diagram.

In the following, we define the skeleton of a diagram that encodes the avoidance constraints of the backbone lines, the lengths of the backbone lines
and the information whether the start or the end points of a connection is the root of a planted animal. This is formalized as follows:

Let $x_0=0\in\Zd$ and $n\in\{1,2,3,4\}$. For each $i\in\{1,\dots,n\}$, let $x_i\in\Zd, l_i\in\Nbold$, the index $j_i\in\{l_i,\underline l_i\}$, $s_i\in\{+,-\}$, and let $\omega^i$ be a sausage-walk from $x_{i-1}$ to $x_i$. We define $S^{s_1,\dots,s_n}_{j_1,\dots,j_n}(\omega^1,\dots,\omega^n)$ to be the indicator that the following holds:

* There exists a sequence of paths $(p^i)_{i=1}^n$ such that $p^i$ is a path from $x_{i-1}$ to $x_i$ and $p^i$ uses only edges in $\omega^i$, each path $p^i$ describes a self-avoiding walk and all paths are pairwise bond-disjoint;

* For each $i\in\{1,\dots,n\}$, we can choose $p^i$ in item (1) such that, $|p^i|\geq l_i$ when $j_i=l_i$, while $|p^i|=l_i$ when $j_i=\underline l_i$;

* For each $i\in\{1,\dots,n\}$, the sausages of the walk $(A^{\omega^i}_j)_{j=0,\dots,|\omega^i|}$ do not intersect, so that $\omega^i$ describes a LA.

* For each $i\in\{1,\dots,n\}$,
if $s_i=+$ then $A^{\omega^i}_{0}=\varnothing$, while $A^{\omega^i}_{|\omega^i|}=\varnothing$ when $s_i=-$.

We next use this skeleton to define the repulsive bubble, triangle and square diagrams as follows:

For $i=1,2,3,4$ let $x_i\in \Zd,\ l_i\in\Nbold,\ j_i\in\{l_i,\underline l_i\}$ and $x_0=0$. We define

\begin{align}
\diagRepulsiveLetter{B}_{j_1,j_2}&(x_1,x_2)
\lbeq{defLTBubble}
=\max_{s_1,s_2\in\{+,-\}}\sum_{\omega^1\in\Wcal(x_0,x_1)} \sum_{\omega^2\in\Wcal(x_1,x_2)}
\left(\prod_{i=1}^2 z^{|\omega^i|} Z[0,|\omega^i|]\right) S^{s_1,s_2}_{j_1,j_2}(\omega^1,\omega^2),\\
\diagRepulsiveLetter{T}_{j_1,j_2,j_3}&(x_1,x_2,x_3)\nnb
\sum_{\footnotesize\begin{array}{c}
\omega^1\in\Wcal(x_0,x_1)\\
\omega^2\in\Wcal(x_1,x_2)\\
\omega^3\in\Wcal(x_2,x_3)
\end{array}}
\left(\prod_{i=1}^3 z^{|\omega^i|} Z[0,|\omega^i|]\right) S^{s_1,s_2,s_3}_{j_1,j_2,j_3}(\omega^1,\omega^2,\omega^3),
\lbeq{defLTTriangle} \\
\diagRepulsiveLetter{S}_{j_1,j_2,j_3,j_4}&(x_1,x_2,x_3,x_4)\nnb
&\qquad\qquad\qquad\qquad\times\left(\prod_{i=1}^4 z^{|\omega^i|} Z[0,|\omega^i|]\right) S^{s_1,s_2,s_3,s_4}_{j_1,j_2,j_3,j_4}(\omega^1,\omega^2,\omega^3,\omega^4).
\lbeq{defLTSquare}
\end{align}

Further, for $i\in\{1,\dots,n\}$ and $x\in\Zd$, we define

\begin{align}
\lbeq{defLTDouble}
\diagRepulsiveLetter{D}_i(x)=\sum_{A\colon 0,x\in A} \indic{0\dbct{A} x}\indic{d_A(0,x)\geq i}z^{|A|}.
\end{align}

Each line of these repulsive diagrams in defLTBubble, defLTTriangle and defLTSquare are planted animals and can therefore be bounded using the modified two-point functions $\tilde G_{z}$. The double connection in defLTDouble is special, and we will treat this separately in Section <ref> below.

To use the constraints on the lengths of connections arising from loops consisting of at least 4 bonds in the NoBLE, we further define, for $l\in\mathbb{N} $ and $j\in {l,\underline l}$,

\begin{eqnarray}
\tilde G_{j,z}(x)=\sum_{\omega\in\Wcal(x_0,x)}  z^{|\omega|} Z[0,|\omega|] S^{+}_{j}(\omega),
\end{eqnarray}

and note that $\tilde G_{0,z}(x)=\tilde G_{z}(x)$.
Using $\tilde G_{j,z}$, we can obviously bound the repulsive diagrams by ignoring the self-avoidance constraints as

\begin{eqnarray}
\lbeq{badBoundSquareLTLA}
\diagRepulsiveLetter{T}_{j_1,j_2,j_3}(x_1,x_2,x_3)&\leq &\tilde G_{j_1,z}(x_1)\tilde G_{j_2,z}(x_2-x_1)\tilde G_{j_3,z}(x_2-x_3).
\end{eqnarray}

However, as demonstrated by the fourth improvement in Section <ref>, the use of
repulsiveness improves our numerical bounds drastically, so we do not use the worse bounds as in badBoundSquareLTLA.

§.§ Bounds on double connections

The bound on contributions in which two-points are doubly connected, that is, connected by two-bond disjoint paths,
is of central interest for LAs. This is especially relevant since a large contribution to the NoBLE lace-expansion coefficients for LAs is given by

\begin{align}
\lbeq{split-LA-doublePrimitiveRecall}
\bar \Xi^{\ssc[0]}_z(x)= \sum_{A\colon 0,x\in A} \indic{0\dbct{A} x}z^{|A|}.%\leq  \min\big\{ \tilde G_{z}(x)^2,  2d z (D\star \tilde G_{z})(x)\bar G_{z}(x)\big\}.
\end{align}

Using our notation we bound this $\diagRepulsiveLetter{D}_0(x)$, defined in defLTDouble.

Let us briefly discuss a numerical aspect of our bounds. Using the techniques described in the previous section, we obtain

\begin{align}
\lbeq{split-LA-doublePrimitive}
\Xi^{\ssc[0]}_z(x)\leq  \min\big\{ 2d z (D\star \tilde G_{z})(x)\bar G_{z}(x), \tilde G_{z}(x)^2\big\}.
\end{align}

We now explain how we can improve this bound by a factor close to $2$. This is possible as the two paths that created the double connection are interchangeable, so that
split-LA-doublePrimitive bounds each possible animal twice.

For the technical derivation of this improvement, we consider yet another split of the LA $A$ given as follows:

* Let $p_1=(b^1_1,b^1_2,\dots,b^1_{n})$ and $p_2(0,x)=(b^2_1,b^2_2,\dots,b^2_{m})$ be two bond-disjoint paths between $0$ and $x$ in $A$,
that we choose in some unique way for each $A$, e.g. $p_1$ is the smallest path in the lexicographic order such that there exists such a bond-disjoint $p_2$, and then $p_2$ is the smallest of these bond-disjoint paths, so that $p_1$ is smaller in lexicographic order than $p_2$ as we will assume from now on.

* Let $A_1=B^{A}(0,b^1_{n};p_1)$.

\begin{align}
\sum_{A\colon 0,x\in A} \indic{0\dbct{A} x}z^{|A|}=&\sum_{A} \indic{0\dbct{A} x}z^{|A_1|}z^{|A\setminus A_1|}%\nnb
\lbeq{split-LA-double-tmp}
%=&\sum_{A} z^{|A_2|} \indic{0\dbct{A} x} z^{|A\setminus A_2|}
\leq 2d z (D\star \tilde G_{z})(x)\bar G_{z}(x).
\end{align}

By symmetry, we could also have performed the split of the animal $A$ according to the path $p_2$, instead.

We next improve upon this bound. Since the paths $p_1$ and $p_2$ are bond-disjoint, we conclude that $\tb^1_1=\ve[\iota]\neq \tb^2_1=\ve[\kappa]$, with $\iota<\kappa$ since $p_1$ is smaller than $p_2$ in the lexicographic order. Remove the bonds in the paths $p_1$ and $p_2$ from $A$. That divides the LA into at most $n+m+2$ disjoint LAs containing the vertices of $p_1$ and $p_2$. Thus, now defining $A_0$ the LA containing $0$, $A_1=B^{A'}(b^1_1,x;p_1)\setminus A_0$, where $A'$ is the LA obtained from $A_2$ by removing $p_2$ and keeping the parts that do not contain vertices in $p_2$. Then, by definition of $B^{A'}(b^1_1,x;p_1)$, $A_1$ is disjoint from $p_2$, and the sausage at $\tb_1^1$ is trivial in $A_1$. Finally, define $A_2=A\setminus (A_0\cup A_1)$ (which contains the bonds in $p_2$). Note that, by construction,  the sausage of $x$ in $A_2$ is trivial, so that $A_2$ is a planted LA, while also the sausage at $0$ is trivial. Then,

\begin{align}
\sum_{A\colon 0,x\in A} \indic{0\dbct{A} x}z^{|A|}=
&\sum_{\iota}\sum_{\kappa>\iota}  \sum_{A} \indic{0\dbct{A} x}\indic {\tb^1_1=\ve[\iota]}
\indic {\tb^2_1=\ve[\kappa]}z^{|A_0|} z^{|A_1|} z^{|A\setminus (A_0\cup A_1)|}\nnb
\leq & \sum_{\iota}\sum_{\kappa>\iota} z^2  \gj \tilde G_{z}(x-\ve[\iota]) G_{z}(x-\ve[\kappa])\nnb
\leq & \frac{1}{2} (2d z  \gj)^2 (D\star \tilde G_{z})(x)(D\star \bar G_{z})(x),
\lbeq{split-LA-double-tmp2}
\end{align}

as the bound split-LA-double-tmp does not assume anything on the order of the paths. Here, in the first inequality, $z^2$ corresponds to the weight of the bonds $b_1^1, b_1^2$, the factor $\gj$ comes from the LA $A_0$, the factor $\tilde G_{z}$ from the LA $A_1$, and the factor $\bar G_{z}$ from the LA $A_2$.
Using the same arguments we obtain the factor $1/2$ for all other terms in split-LA-doublePrimitive. In our final numerical bounds, we will drastically improve upon this bound by using the repulsive nature of the two paths $p_1, p_2$.

§ THE DIAGRAMMATIC BOUNDS

In the preceding section we have explained the technical details of our method of bounding LT/LA diagrams.
In this section we first combine the repulsive diagrams defLTBubble-defLTSquare, to larger diagrams that we call building blocks.
Then, we combine these building blocks to the create bounds on the NoBLE coefficients.
We close this section with an overview of where to find all bounds required for the analysis, as stated in <cit.>.

In Section <ref> we prove the bounds for $N=0$ and sketch how the remaining bounds are proven in the example of $N=1,2$. The rigorous proof of these bounds is tedious and is based on techniques that are relatively standard in lace-expansion analyses. For this reason we omit many details.

§.§ Building blocks

In this section we combine the simple diagram, defined in defLTBubble-defLTSquare, to larger diagrams, that we call building blocks. These building blocks are designed
to encode the following information:

$\rhd$ the repulsiveness of the paths involved;

$\rhd$ the lengths of lines shared by two loops;

$\rhd$ the role of the lines shared by two loops, meaning whether it is part of the backbone or not, and, if so, how the backbone is connected to the line in question.

We encode the repulsiveness by defining the blocks using the repulsive diagram defLTBubble-defLTSquare. In the following we discuss the other two features.

Aim: encoding backbone-paths.
In Figure <ref>, we have shown an example of the diagram contributing to $\Xi^\ssc[4]_z$.
However, the form shown in Figure <ref> is only one of eight possible patterns that the backbone can form. These patterns arise by the different cases for the underlying lace, i.e. whether $s_{i+1}=t_i$ or $s_{i+1}<t_i$ for $i=1,2,3$. In Figure <ref> we show all the possible backbone patterns for the $\Xi^{\ssc[4]}$-diagrams. One function of the building block will be to encode all possible patterns at once, so as to encode the roles that the shared lines play compared to the backbone.

Picture of all possible $\Xi^{\ssc[4]}_z$ diagrams.
The backbone is marked by a thick line and the four first intersection points are marked as dots.
The leftmost diagram on the second row corresponds to the diagram shown in Figure <ref>.

Aim: encoding of the line lengths.
When creating a bound on the NoBLE coefficients we want to make full use of all the avoidance constraints of a sausage-walk, as well as the fact that any closed loop consists of at least four bonds. To be able to do this, we consider different cases for the lengths of paths that are shared by two squares. Here, the length of a path is given by the number of bonds it uses.

Our solution.
We use the indices $l,m\in\{-2,-1,0,1,2\}$ to encode the information of the length and role of a shared line.
We explain their meaning at the example of the diagram given in Figure <ref>.

Example of a description of the backbone-path and the length of the shared line using the indices $(a_i)_i$.

As indicated in Figures <ref>, <ref> and <ref>, we always draw the diagrams in such a way that the origin is on the left side and the first piece of the sausage-walk from $\bb_0$ to $\bb_{s_2}$ is the lower line of the first square. A positive index $m_i$ indicates that the backbone ends on the upper part of a piece of the diagram.
Interactions between neighboring squares will be particularly important, in which case there will be two indices, for example in $A^{m_{i-1},m_i}$. In this case, $m_{i-1}>0$ indicates that the backbone in the $(i-1)$st square starts on the upper part of the diagram, while $m_i>0$ indicates that the backbone ends on the upper part of a piece of the diagram.

By $|m_i|=1$, we denote that the vertical line consists of exactly one bond and $|m_i|=2$ denotes that
the vertical line consists of at least two bonds. If $m_i=0$, then the vertical line is trivial, i.e., it consists of no bonds.
In the case that $m_i=0$, we draw the following piece of the backbone on the lower part of the following square.

Informal definition.
In this section, we informally define the diagrams, also called building blocks, that we use to bound the NoBLE coefficients. The precise properties of the building blocks originate from the properties of the NoBLE coefficients, as we will see when we bound the coefficients in Section <ref>. The list below consists of 9 basic building blocks:

* Open square $A^{l,m}$.

We use this building block to bound the intermediate pieces of the diagrams appearing in the NoBLE coefficients. It consists of an open square, with 4 of the 5 vertices fixed and one being summed over.

We label the vertices in the diagram $A^{l,m}$ such that $u$ is the starting vertex and $x$ is the ending vertex of the backbone.
The indices $l,m$ identify the lengths of the connections between $u,v$ and $x,y$, respectively, i.e., the number of bonds in these connections.
For example: If $m=0$ then $x=y$. If $|m|=1$ then $x$ and $y$ are direct neighbors and are directly connected. The case where $|m|=2$ summarizes all the other cases.

A positive index $l,m$ indicates that the backbone starts/ends at the top. This means that if $lm\geq 0$, then the connection $x$ to $y$ is not part of the backbone.
The horizontal parts of the backbones (i.e., the paths from $u$ to $x$ or from $u$ to $y$) are always non-trivial, in that they use at least one bond.
If the backbone $b^A(u,x)$ consists of only one bond, then we know that $w\nin \{u,x\}$.
Closing the diagram, by including $u\conn v$, always creates a diagram with at least four bonds.

* Double open triangle $\bar A^{l,m}$.

This diagram follows the same rules as the open square $A^{l,m}$, the difference being that the connection $x$ to $y$ does not contribute (i.e., that line is absent and used in another building block). We only draw the example with $l,m>0$. The role of $l$ and $m$ in $\bar A^{l,m}$ is the same as for $A^{l,m}$.

* Weighted open triangle $C^{l,m}$.

This diagram follows the same rules as the double open triangle $\bar A^{l,m}$.
Additionally, the lower connections are not planted animals. This means that the connection with spatial weight
might have a non-trivial one-point function at both end-points, thus should be bounded by $\bar G_z$.

* Closed fundamental block $P^{\ssc[1],m}.$

This diagram is used to bound the first square of $\Xi^{\ssc[N]}_z$ and has the following shape for $m>0$:

For $m=0$, we have that $x=y$, apart from that, the diagram remains the same.

Each triangle, square and pentagon consists of at least four bonds. The first diagram characterizes the case where the first step of the backbone starts at the origin, i.e., $\bb_1^\omega=0$.
In the following, we use $u=\bar{b}_1^\omega$ to denote the first step of the backbone. Note that only $\bb_1^\omega= 0$ contributes for LT.

If the backbone between $u$ and $x$ consists of only one step, then $w\nin\{u,x\}$.
The pieces of the backbone $b^A(u,x)$, $b^A(0,y)$ and $b^A(x,y)$, respectively, are non-trivial.
Note that the left diagram is the same diagram as $A^{0,m}(0,0,x,y)$.

* Open fundamental block $\bar P^{\ssc[1],m}.$

This diagram corresponds to the closed fundamental block $P^{\ssc[1],m}$, in which the connection $x$ to $y$ does not contribute (but we keep information about its length using the index $m$).

* Weighted fundamental block $\Delta^{\text {start},m}$.

This diagram corresponds to the open fundamental block $\bar P^{\ssc[1],m}$, where the lower connection is weighted by its displacement, i.e., $|x|^2$ for $m\leq 0$ and $|y|^2$ for $m\geq 0$.

* Weighted terminal block $\Delta^{\text {end},m}$.

This diagram is defined by $\Delta^{\text {end},m}(u,v,x)=C^{m,0}(u,v,x,x)$ and is used to bound the last square of $\Xi^{\ssc[N]}_z$ and $\Xi^{\ssc[N],\iota}_z$.

* Iota-fundamental block $P^{\ssc[1],\iota,m}.$

This diagram is used to bound the first square of $\Xi^{\ssc[N],\iota}_z$ and has the following structure for $m<0$:

This diagram is similar to $P^{\ssc[1],m}$. The sausage-walks that we bound with this diagram have the property that either $\ve[\iota]$ is part of the first sausage or that $\bb_1^\omega$ ends in $\ve[\iota]$. If $\ve[\iota]$ is in the first sausage, then we extend the idea of split-LA-sausage-second.
We identify the last point $v$ that the double connection $0$ to $\bb_1$ and a connection $0$ to $w$ share.
Then, we identify the last common point $u$ of the connection $0$ to $\ve[\iota]$ to the rest.
This point $u$ could be on all possible connections $0$ to $\bb_1$, $0$ to $v$, $\bb_1$ to $v$ as well as $v$ to $w$, as indicated in the figure above.

* Weighted iota-fundamental block $\Delta^{\text {iota},{\sss I},m},\Delta^{\text {iota},{\sss II},m}$.

These diagrams have the same shape as $P^{\ssc[1],\iota,m}$, but the connection $x$ to $y$ does not contribute. Further, $\Delta^{\text {iota},{\sss I},m}$ is weighted by a factor $|x|^2$ and $\Delta^{\text {iota},{\sss II},m}$ by a factor $|x-\ve[\iota]|^2$ for $m\leq 0$.

The formal definition of these diagrams is given in Appendix <ref>.
In this version we omit the detailed definition of these diagrams, as they do not create additional  insight and are rather lengthly. There definition is given in the appendix of the extended version <cit.>.

Elements of the bounds.
Here we define the quantities that we use to state the bounds on the NoBLE coefficients.
In the following, we define the vectors $\vec P,\vec P^{\iota},\vec E,\vec \Delta^{\rm{start}},\vec \Delta^{\rm{end}},\vec \Delta^{\iota}\in\Rbold^{5}$
and the matrices ${ \bf {\bar A}},{\bf A},{\bf C}\in\Rbold^{5\times 5}$ by their entries. For convenience, we use $\{-2,-1,0,1,2\}$ as indices for the entries of the vectors and matrices.

\begin{align}
\lbeq{defOfBoundingElement1}
(\vec P)_m&= \sum_{x,y} P^{\ssc[1],m}(x,y),\qquad\qquad
(\vec P^\iota)_m= \sum_{x,y} P^{\ssc[1],\iota,m}(x,y),\\
(\vec \Delta^{\rm{start}})_m&= \sup_{y}\sum_{x} \Delta^{\rm{start},m}(x,x+y), \quad
(\vec \Delta^{\rm{end}})_m= \sup_{y} \sum_{x} \Delta^{\rm{start},m}(x,x+y),\\
(\vec \Delta^{\iota,{\sss I}})_m &=\sup_{y} \sum_{x} \Delta^{\iota,{\sss I},m}(x,x+y),\qquad
(\vec \Delta^{\iota,{\sss II}})_m =\sup_{y} \sum_{x} \Delta^{\iota,{\sss II},m}(x,x+y),
\end{align}

\begin{align}
\lbeq{defOfBoundingElement2a}
({\bf A})_{l,m}&= \sup_{v} \sum_{x,y} A^{l,m}(0,v,x,y),\qquad\qquad
({\bf C})_{l,m}= \sup_{v,y} \sum_{x} C^{l,m}(0,v,x,x+y),\\
(\vec E^{\rm{open}})_m&= \sum_{v,x} A^{m,0}(x,x,v,0), \qquad\qquad (\vec E^{\rm{closed}})_m= \sum_{v,x} A^{0,m}(x,x,v,0).
\lbeq{defOfBoundingElement2}
\end{align}

Using the bootstrap assumption $f_i(z)<\Gamma_i$ we can compute explicit numerical bound for each of these quantities. We use these quantities to formulate the bounds on the NoBLE coefficients.

§.§ The bounds

Now we state the required bound on the coefficients. For this we use the notation $\indAnimal$, as used in conj-zc, where $\indAnimal$ is $1$ in bounds for LAs and $0$ in bounds for LTs. We have to be especially careful in our bounds for $N=0$ and $N=1$, which turn out to be the major contributions to the NoBLE coefficients. Therefore, we distinguish between the bounds for $N=0$, $N=1$ and $N\geq 2$. We start by formulating the bounds on $\Xi^{\ssc[0]}$ and $\Psi^{\ssc[0],\kappa}$:

Let $0\leq z<z_c$. Then,

\begin{align}
\lbeq{Bound-Xi0}
\sum_{x\in\Zd}\Xi^{\ssc[0]}_z(x)&\leq \frac {\indAnimal } {g_z} \sum_x\diagRepulsiveLetter{D}_{1}(x),\qquad\quad
\sum_{x\in\Zd}\Xi^{\ssc[0]}_{{\sss R},z}(x) \leq \frac {\indAnimal} {g_z} \diagRepulsiveLetter{D}_{2}(x),\\
\lbeq{Bound-Psi0RemII}
\sum_{x\in\Zd} \Psi^{\ssc[0],\kappa}_{{\sss R, II},z}(x)&\leq \frac {2d-2}{2d} \frac {\indAnimal} {g^\iota_z} \diagRepulsiveLetter{D}_{2}(x,0),
\end{align}

\begin{align}
\lbeq{Bound-Xi0W}
\sum_{x\in\Zd}|x|^2\Xi^{\ssc[0]}_z(x)&\leq \frac {\indAnimal} {2}
\sum_x|x|^2  G_z(x) 2dz (D \star \tilde G_z)(x),\\
\lbeq{Bound-Xi0RemW}
\sum_{x\in\Zd}|x|^2\Xi^{\ssc[0]}_{{\sss R},z}(x)&\leq \frac {\indAnimal} {2}
\sum_x|x|^2  G_z(x) (2dz)^2g_z (D^{\star 2} \star \tilde G_z)(x),\\
\lbeq{Bound-Psi0RemIIW}
\sum_{x\in\Zd} |x|^2 \Psi^{\ssc[0],\kappa}_{{\sss R, II},z}(x)&\leq \frac {2d-2}{2d} \frac {\indAnimal} {2}
\sum_x|x|^2  G_z(x)  (2dz)^2g_z (D^{\star 2} \star \tilde G_z)(x).
\end{align}

\begin{align}
\lbeq{Bound-Psi0RemI}
\sum_{x\in\Zd} \Psi^{\ssc[0],\kappa}_{{\sss R, I},z}(x)\leq& \frac {\indAnimal} {g_z^\iota}\big(
(2d-1) \diagRepulsiveLetter{B}_{\underline 1,3}(\ve[1],0)+\frac {2d-2}{2d} \sum_x\diagRepulsiveLetter{B}_{2,2}(x,0)\big),\\
% we can improve the first term to
%\frac {2d-1}{2d} \sum_x\diagRepulsiveLetter{B}_{\underline 1,5}(x,0)+\frac {2d-1}{2d}\frac {2d-2}{2d} \sum_x\diagRepulsiveLetter{B}_{\underline 1,\underline 3}(x,0)
\lbeq{Bound-Psi0RemIW}
\sum_{x\in\Zd} |x-\ve[\kappa]|^2 \Psi^{\ssc[0],\kappa}_{{\sss R, I},z}(x)\leq &
4d\indAnimal \diagRepulsiveLetter{B}_{\underline 1,3}(\ve[1],0)\\
&+ \frac {\indAnimal} {2}\sum_x (1+|x|^2)(2dz)^2g_z (D^{\star 2} \star \tilde G_z)(x) G_z(x).\nn
\end{align}

We continue by formulating the bounds on $\Xi^{\ssc[0],\iota}$ and $\Pi^{\ssc[0],\iota,\kappa}$. To state the bound on $\Xi^{\ssc[0],\iota}$ and the remainder of the split $\Xi^{\ssc[0],\iota}_{{\sss R, I},z}$ and $\Xi^{\ssc[0],\iota}_{{\sss R, II},z}$, we use the abbreviations $\alpha_{n,m}$, $\beta_{n}$ and $\gamma_{n}$, that will only be used in this lemma, and which, for $n,m\in\{0,1,2\}$, are given by

\begin{align}
\lbeq{BoundNZero-XiIota-alpha}
\alpha_{n,m}=&
\frac {\indAnimal}{g_z}\tilde G_z(\ve[\iota]) \sum_{x} \diagRepulsiveLetter{D}_{n}(x)
+\frac {\indAnimal}{g_z}  \indic{m=0} \diagRepulsiveLetter{D}_{1}(\ve[\iota])
+\frac {\indAnimal}{g_z}  \sum_{x} \diagRepulsiveLetter{T}_{1,\max\{1,m\},n} (\ve[\iota],x,0)\\
&+\frac {\indAnimal}{g_z}
\big( \indic{n\leq 1}  \sup_{y} \tilde G_{2,z}(y) (2d-1)zg_z \tilde G_{3,z}(\ve[\iota])
+\sup_{y} \tilde G_{m,z}(y) \sum_{x\neq 0} \diagRepulsiveLetter{D}_{2}(x)\big)\nnb
&+\frac {\indAnimal}{g_z}   \sum_{u}\diagRepulsiveLetter{B}_{1,1}(u,\ve[\iota])  \sup_{u} \sum_{x\neq 0 }\diagRepulsiveLetter{B}_{n,1}(x,u)\big),\nnb
\lbeq{BoundNZero-XiIota-beta}
\beta_{n}=&2dz\frac {\indAnimal}{g_z}\tilde G_z(\ve[\iota]) \sum_{x} |x|^2\bar G_z(x)
\big(\frac 1 2  (D\star \tilde G_{n,z})(x)+(D\star \tilde G_z)(x-\ve[\iota])\big)\\
&+2dz\frac {\indAnimal}{g_z}  \sup_{y} \tilde G_{1,z}(y)\sum_{x}|x|^2\bar G_z(x)
\big(\frac 1 2 (D\star \tilde G_{1,z})(x)+(D\star \tilde G_{z}\star \tilde G_{1,z})(x)\big),\nnb
\lbeq{BoundNZero-XiIota-gamma}
\gamma_n=& \frac {\indAnimal}{g_z}\tilde G_z(\ve[\iota]) \sum_{x}
\big(\diagRepulsiveLetter{D}_{1}(x)+|x|^2\bar G_z(x) dz (D\star \tilde G_z)(x)\big)\\
&+\frac {\indAnimal}{g_z} \big(\tilde G_z(\ve[\iota])+ \sup_{y} G_{n,z}(y)\big) \sum_{x} |x-\ve[\iota]|^2\bar G(x-\ve[\iota]) 2dz (D\star \tilde G_z)(x)\nnb
&+2\frac {\indAnimal}{g_z}  \sum_{u}\diagRepulsiveLetter{B}_{1,1}(u,\ve[\iota])\sup_{u} \sum_{x}\diagRepulsiveLetter{B}_{1,1}(x,u)\nnb
&+4dz \frac {\indAnimal}{g_z}  \sup_{y} \tilde G_{1,z}(y)\sum_{x}|x|^2\bar G_z(x)(D\star \tilde G_z\star \tilde G_{1,z})(x).\nn
\end{align}

Then the bounds on $\Xi^{\ssc[0],\iota}$ and $\Pi^{\ssc[0],\iota,\kappa}$ read as follows:

Let $0\leq z< z_c$. Then,

\begin{align}
\lbeq{Bound-XiIota0}
\sum_{x\in\Zd}\Xi^{\ssc[0],\iota}_z(x)\leq& G_z(\ve[\iota]) +\alpha_{1,0},\\
\lbeq{Bound-XiIota0Wa}
\sum_{x\in\Zd}|x|^2\Xi^{\ssc[0],\iota}_z(x)\leq&\beta_{0},\\
\lbeq{Bound-XiIota0Wb}
\sum_{x\in\Zd}|x-\ve[\iota]|^2\Xi^{\ssc[0],\iota}_z(x)\leq& G_z(\ve[\iota]) +\gamma_1.
\end{align}

The remainder terms of the split are bounded in a similar way as

\begin{align}
\lbeq{Bound-XiIota0RemI}
\sum_{x\in\Zd}\Xi^{\ssc[0],\iota}_{{\sss R, I},z}(x)\leq& \alpha_{1,2},\qquad \qquad
\sum_{x\in\Zd}|x-\ve[\iota]|^2\Xi^{\ssc[0],\iota}_{{\sss R, I},z}(x)\leq \gamma_2,\\
\lbeq{Bound-XiIota0RemII}
\sum_{x\in\Zd}\Xi^{\ssc[0],\iota}_{{\sss R, II},z}(x)\leq& \alpha_{2,1},\qquad\qquad
\sum_{x\in\Zd}|x|^2\Xi^{\ssc[0],\iota}_{{\sss R, II},z}(x)\leq\beta_{1}.
\end{align}

The explicit terms of the splits are bounded by

\begin{align}
\lbeq{Bound-XiIotaAlphaI}
\Xi^{\ssc[0],\iota}_{\alpha,{\sss I},z}(\ve[\iota])\leq &\frac {\indAnimal}{g_z}\diagRepulsiveLetter{D}_{1}(\ve[\iota]),\quad\qquad\qquad
\Xi^{\ssc[0],\iota}_{\alpha,{\sss II},z}(0)\leq G_z(\ve[\iota]),\\
\lbeq{Bound-XiIotaAlphaIWa}
\sum_\kappa \Xi^{\ssc[0],\iota}_{\alpha,{\sss I},z}(\ve[\kappa]+\ve[\iota])\leq&
G_z(\ve[\iota])+ \frac {\indAnimal}{g_z}\sum_{\kappa\colon \kappa\neq -\iota}\diagRepulsiveLetter{T}_{1,\underline 1,2}(\ve[\iota],\ve[\kappa]+\ve[\iota],0)
+\frac {zg_z^\iota}{g_z}\indAnimal\sum_{\kappa\colon \kappa\neq -\iota}\diagRepulsiveLetter{B}_{2,2}(\ve[\iota]+\ve[\kappa],0),\\
\lbeq{Bound-XiIotaAlphaIIWb}
\sum_\kappa \Xi^{\ssc[0],\iota}_{\alpha,{\sss II},z}(\ve[\kappa])\leq&
\frac {\indAnimal}{g_z}\diagRepulsiveLetter{D}_{1}(\ve[\iota])+\frac {\indAnimal z g_z^\iota}{g_z}\sum_{\kappa\colon \kappa\neq -\iota}\sup_{x\neq 0}\tilde G_{1,z}(x)\sum_{u}\diagRepulsiveLetter{B}_{1,1}(u,\ve[\kappa]),\\
&+ \indAnimal\sum_{\kappa\colon \kappa\neq -\iota} \Big(\frac{1}{g_z}\diagRepulsiveLetter{T}_{1,2,\underline 1} (\ve[\iota],\ve[\kappa],0)
+ \indAnimal  z G_{3,z}(\ve[\kappa])\big(\tilde G_{1,z}(\ve[\iota])+\tilde G_{2,z}(\ve[\kappa]-\ve[\iota])\big)\Big).\nn
\end{align}

We bound the terms in the split of $\Pi^{\ssc[0],\iota,\kappa}_z$ as

\begin{align}
\lbeq{Bound-Pi0AlphaI}
\sum_{\kappa}\Pi^{\ssc[0],\iota,\kappa}_{\alpha,z}(\ve[\iota])\leq&  \indAnimal(2d-1)z\diagRepulsiveLetter{D}_{1}(\ve[\iota]),\\
\lbeq{Bound-Pi0RemI}
\sum_{x\in\Zd}\sum_{\iota}\Pi^{\ssc[0],\iota,\kappa}_{{\sss R},z}(x)\leq& (2d-1)zg_z G_z(\ve[\iota]) +zg_z\alpha_{1,1},\\
\lbeq{Bound-Pi0RemIW}
\sum_{x,\iota,\kappa}|x-\ve[\iota]|^2\Pi^{\ssc[0],\iota,\kappa}_{{\sss R},z}(x+\ve[\kappa])\leq&
2(2d)^2zg_z G_z(\ve[\iota]) +(2d)^2zg_z\gamma_1+4d(d+1)zg_z\alpha_{1,1}.
\end{align}

Let $0\leq z< z_c$. Then,

\begin{align}
\lbeq{Bound-Xi1}
\sum_{x\in\Zd}\Xi^{\ssc[1]}_z(x)&\leq \frac {g_z^\iota}{g_z} (\vec P)_0,\\
\lbeq{Bound-Xi1Rem}
\sum_{x\in\Zd}\Xi^{\ssc[1]}_{{\sss R},z}(x) &\leq \frac {g_z^\iota}{g_z}\Big( (\vec P)_0
- \sum_{x\in\Zd} A^{0,0}(0,0,x,x)+2\diagRepulsiveLetter{B}_{2,2}(x,0)+\sum_w \diagRepulsiveLetter{T}_{2,1,1}(x,w,0)
\Big),\\
\lbeq{Bound-Psi1RemI}
\sum_{x\in\Zd} \Psi^{\ssc[1],\kappa}_{{\sss R, I},z}(x)&\leq
\frac {g_z^\iota}{g_z} \Big( (\vec P)_0 - \sum_{x\in\Zd} A^{0,0}(0,0,x,x)+2\indic{\ve[\kappa]\neq x}\diagRepulsiveLetter{B}_{\underline 1,3}(x,0)\Big)\\
&+\frac {g_z^\iota}{g_z} \sum_{x\in\Zd} \Big(2\diagRepulsiveLetter{B}_{2,2}(x,0)+\sum_w \diagRepulsiveLetter{T}_{1,1,1}(x,w,0)\Big),\nnb
\lbeq{Bound-Psi1RemII}
\sum_{x\in\Zd} \Psi^{\ssc[1],\kappa}_{{\sss R, II},z}(x)&\leq \frac {g_z^\iota}{g_z} \Big( (\vec P)_0
- \sum_{x\in\Zd} A^{0,0}(0,0,x,x)+2\diagRepulsiveLetter{B}_{2,2}(x,0)+\sum_w \diagRepulsiveLetter{T}_{2,1,1}(x,w,0)
\Big),
\end{align}

\begin{align}
\lbeq{Bound-Xi1W}
\sum_{x\in\Zd}|x|^2\Xi^{\ssc[1]}_z(x)&\leq (\vec \Delta^{\rm{start}})_0, \qquad
\sum_{x\in\Zd}|x|^2\Xi^{\ssc[1]}_{{\sss R},z}(x)\leq(\vec \Delta^{\rm{start}})_0,\\
\sum_{x\in\Zd} |x|^2 \Psi^{\ssc[1],\kappa}_{{\sss R, II},z}(x)&\leq(\vec \Delta^{\rm{start}})_0,\qquad
\lbeq{Bound-Psi1RemIW}
\sum_{x\in\Zd} |x-\ve[\kappa]|^2 \Psi^{\ssc[1],\kappa}_{{\sss R, I},z}(x)\leq(\vec \Delta^{\rm{start}})_0+\sum_{x\in\Zd} \Psi^{\ssc[1],\kappa}_{{\sss R, I},z}(x).
\end{align}

Further, $\Xi^{\ssc[1],\iota}_z$ is bounded by

\begin{align}
\lbeq{Bound-XiIota1}
\sum_{x\in\Zd}\Xi^{\ssc[1],\iota}_z(x)\leq& \frac {g_z}{\gj}(\vec P^\iota)_0,\\
\lbeq{Bound-XiIota1Wa}
\sum_{x\in\Zd}|x|^2\Xi^{\ssc[1],\iota}_z(x)\leq&(\vec \Delta^{\iota,{\sss I}})_0,\\
\lbeq{Bound-XiIota1Wb}
\sum_{x\in\Zd}|x-\ve[\iota]|^2\Xi^{\ssc[1],\iota}_z(x)\leq &(\vec \Delta^{\iota,{\sss II}})_0.
\end{align}

Let $0\leq z<z_c$ and $N\geq 2$. Then,

\begin{align}
\lbeq{Bound-Xi2}
\sum_{x\in\Zd}\Xi^{\ssc[N]}_z(x)&\leq \frac {g_z}{\gj} \vec P^T{\bf A}^{N-2}{\vec E}^{\rm{open}}, \quad \sum_{x\in\Zd}\Xi^{\ssc[N],\iota}_z(x)\leq \frac {g_z}{\gj} (\vec P^{\iota})^T{\bf A}^{N-2}{\vec E}^{\rm{open}},\\
\sum_{x\in\Zd}|x|^2\Xi^{\ssc[N]}_z(x)\leq& \ N  \left( (\vec \Delta^{\rm{start}})^T{\bf A}^{N-2}{\vec E^{\rm{closed}}}+\vec P^T{\bf A}^{N-2}\vec \Delta^{\rm{end}}\right)\nn \\[-5mm]
&\qquad\qquad\qquad+N\indic{N> 2}\sum_{M=0}^{N-3} \vec P {\bf A}^{M} {\bf C} {\bf A}^{N-3-M}\vec E^{\rm{closed}},
\lbeq{Bound-Xi2W}
\\
\sum_{x\in\Zd}|x|^2\Xi^{\ssc[N],\iota}_z(x)\leq& \ N \left((\vec \Delta^{\iota,{\sss I}})^T{\bf A}^{N-2}{\vec E^{\rm{closed}}}+(\vec P^{\iota})^T{\bf A}^{N-2}\vec \Delta^{\rm{end}}\right)\nn \\[-5mm]
&\qquad\qquad\qquad+N\indic{N> 2}\sum_{M=0}^{N-3} \vec P^\iota {\bf A}^{M} {\bf C} {\bf A}^{N-3-M}\vec E^{\rm{closed}},
\lbeq{Bound-XiIota2Wa}
\\
\sum_{x\in\Zd}|x-\ve[\iota]|^2\Xi^{\ssc[N],\iota}_z(x)\leq& \ N  \left((\vec \Delta^{\iota,{\sss II}})^T{\bf A}^{N-2}{\vec E^{\rm{closed}}}+(\vec P^{\iota})^T{\bf A}^{N-2}\vec \Delta^{\rm{end}}\right)\nn \\[-5mm]
&\qquad\qquad\qquad+N\indic{N> 2}\sum_{M=0}^{N-3} \vec P^\iota {\bf A}^{M} {\bf C} {\bf A}^{N-3-M}\vec E^{\rm{closed}}.
\lbeq{Bound-XiIota2Wb}
\end{align}

Schematic representation of the bound on $\hat \Xi^{\ssc[N]}_z(0)$, given in Bound-Xi2. The backbone is marked by a thicker line.
See also Figures <ref> and <ref>.

Having stated our main bounds on the NoBLE coefficients, we next state some additional bounds that improve the numerical accuracy of our method:

Let $0\leq z<z_c$. Then,

\begin{align}
\lbeq{Differencebound-1}
\Xi^\ssc[0]_{\alpha,z}(0)-\Xi^\ssc[1]_{\alpha,z}(0)\leq&0,\qquad \qquad \Xi^\ssc[0]_{\alpha,z}(\ve[1])-\Xi^\ssc[1]_{\alpha,z}(\ve[1])\leq 0,\\
\lbeq{Differencebound-2}
\Xi^\ssc[1]_{\alpha,z}(0)-\Xi^\ssc[0]_{\alpha,z}(0)\leq & 2dz\gj G_{3,z}(\ve[\iota]),\\
\Xi^\ssc[1]_{\alpha,z}(\ve[1])-\Xi^\ssc[0]_{\alpha,z}(\ve[1])\leq&
\frac {g_z^\iota} {g_z} \Big(2\diagRepulsiveLetter{B}_{3,\underline 1}(\ve[1],0)+\sum_v \diagRepulsiveLetter{T}_{\underline 1,1,1}(\ve[1],v,0)\Big),
\lbeq{Differencebound-3}
\end{align}

\begin{align}
\lbeq{Differencebound-4}
&\sum_\iota \big(\Psi^{\ssc[1],1}_{\alpha,{\sss I},z}(\ve[1]+\ve[\iota])-\Psi^{\ssc[0],1}_{\alpha,{\sss I},z}(\ve[1]+\ve[\iota])\big) \leq\sum_\iota\sum_w\diagRepulsiveLetter{T}_{0,0,\underline 2}(w,\ve[1]+\ve[\iota],0),
\\
\lbeq{Differencebound-5}
&\sum_\iota \big(\Psi^{\ssc[0],1}_{\alpha,{\sss I},z}(\ve[1]+\ve[\iota])  -\Psi^{\ssc[1],1}_{\alpha,{\sss I},z}(\ve[1]+\ve[\iota]) \big)\leq {\indAnimal}{g^\iota_z}\sum_\iota\diagRepulsiveLetter{B}_{2,\underline 2}(\ve[1]+\ve[\iota],0).
\end{align}

\begin{align}
\sum_\iota \big(\Psi^{\ssc[1],1}_{\alpha,{\sss II},z}&(\ve[\iota])  -\Psi^{\ssc[0],1}_{\alpha,{\sss II},z}(\ve[\iota])\big)\leq
(2d-1) \Big(2\diagRepulsiveLetter{B}_{3,\underline 1}(\ve[1],0)+\sum_v \diagRepulsiveLetter{T}_{\underline 1,1,1}(\ve[1],v,0)\Big),
\lbeq{Differencebound-6}\\
\lbeq{Differencebound-7}
\sum_\iota \big(\Psi^{\ssc[0],1}_{\alpha,{\sss II},z}&(\ve[\iota])  -\Psi^{\ssc[1],1}_{\alpha,{\sss II},z}(\ve[\iota])\big)
\leq 0.
\end{align}

§.§ Summary of the bounds

In this section, we give an overview of where to find the bounds stated in <cit.>. We emphasize that, next to the diagrammatic bounds proven in this document, <cit.> also requires a method to conclude numerical bounds. In <cit.>, we describe this method. In [9], we provide the source code that computes the numerical bounds. This source code allows us to apply the analysis of [12] to obtain our mean-field results. We give a summary of where to find most bounds in Table <ref>.

Bound                 defined in     Bound      defined in     Bound      defined in

$\beta_{\sss \Xi}^\ssc[0]$     Bound-Xi0
$\beta_{\sss \Xi}^\ssc[1]$      Bound-Xi1
$\beta_{\sss \Xi}^\ssc[N],N\geq 2$     Bound-Xi2

$\beta_{\sss \Xi^\iota}^\ssc[0]$     Bound-XiIota0
$\beta_{\sss \Xi^\iota}^\ssc[1]$     Bound-XiIota1
$\beta_{\sss \Xi^\iota}^\ssc[N],N\geq 2$     Bound-Xi2

$\beta_{{\sss \Delta \Xi}}^\ssc[0]$     Bound-Xi0W
$\beta_{{\sss \Delta \Xi}}^\ssc[1]$     Bound-Xi1W
$\beta_{{\sss \Delta \Xi}}^\ssc[N],N\geq 2$      Bound-Xi2W

$\beta_{{\sss \Delta \Xi^{\iota}},0}^\ssc[0]$     Bound-XiIota0Wa
$\beta_{{\sss \Delta \Xi^{\iota}},0}^\ssc[1]$     Bound-XiIota1Wa
$\beta_{{\sss \Delta \Xi^{\iota}},0}^\ssc[N],N\geq 2$     Bound-XiIota2Wa

$\beta_{{\sss \Delta \Xi^{\iota}},\iota}^\ssc[0]$     Bound-XiIota0Wb
$\beta_{{\sss \Delta \Xi^{\iota}},\iota}^\ssc[1]$     Bound-XiIota1Wb
$\beta_{{\sss \Delta \Xi^{\iota}},\iota}^\ssc[N],N\geq 2$       Bound-XiIota2Wb

$ \beta_{{\sss\Xi_\alpha(0)}}^\ssc[0-1]$    Differencebound-1
$ \beta_{{\sss\Xi_\alpha(\ve[1])}}^\ssc[0-1]$    Differencebound-1
$ \beta_{{\sss\Xi_\alpha(0)}}^\ssc[1-0]$    Differencebound-2

$\beta_{{\sss\Xi_\alpha(\ve[1])}}^\ssc[1-0]$   Differencebound-3
$\beta_{{\sss\Xi^{\iota}_\alpha,I}}^\ssc[0]$     	Bound-XiIotaAlphaI
$\beta_{{\sss \sum}{\sss\Xi^{\iota}_\alpha,I}}^\ssc[0]$    Bound-XiIotaAlphaIWa

$\beta_{{\sss\Xi^{\iota}_\alpha,II}}^\ssc[0]$     	Bound-XiIotaAlphaI
$\beta_{ {\sss\sum \Xi^{\iota}_\alpha,II}}^\ssc[0]$     	Bound-XiIotaAlphaIIWb
$\beta_{{\sss\sum \Psi^{\iota}_\alpha,I}}^\ssc[0-1]$      Differencebound-5

$\beta_{{\sss\sum \Psi^{\iota}_\alpha,II}}^\ssc[0-1]$     Differencebound-7
$\beta_{{\sss\sum \Psi^{\iota}_\alpha,I}}^\ssc[1-0]$      Differencebound-4
$\beta_{{\sss\sum \Psi^{\iota}_\alpha,II}}^\ssc[1-0]$      Differencebound-6

$\beta_{\sss\sum \Pi_\alpha}^\ssc[0]$    Bound-Pi0AlphaI

$\beta_{\sss\Xi,R}^\ssc[0]$     Bound-Xi0
$\beta_{\Delta \sss\Xi,R}^\ssc[0]$      Bound-Xi0RemW
$\beta_{\sss\Psi,R,I}^\ssc[0]$     Bound-Psi0RemI

$\beta_{\Delta \sss\Psi,R,I}^\ssc[0]$     Bound-Psi0RemIW
$\beta_{\sss\Psi,R,II}^\ssc[0]$      Bound-Psi0RemII
$\beta_{\Delta \sss\Psi,R,II}^\ssc[0]$     Bound-Psi0RemIIW

$\beta_{\sss\Xi,R}^\ssc[1]$         Bound-Xi1Rem
$\beta_{\Delta \sss\Xi,R}^\ssc[1]$      Bound-Xi1W
$\beta_{\sss\Psi,R,I}^\ssc[1]$     Bound-Psi1RemI

$\beta_{\Delta \sss\Psi,R,I}^\ssc[1]$     Bound-Psi1RemIW
$\beta_{\sss\Psi,R,II}^\ssc[1]$     Bound-Psi1RemII
$\beta_{\Delta \sss\Psi,R,II}^\ssc[1]$     Bound-Psi1RemIW

$\beta_{\sss\Xi^\iota,R,I}^\ssc[0]$     Bound-XiIota0RemI
$\beta_{\Delta \sss\Xi^\iota,R,I}^\ssc[0]$     Bound-XiIota0RemI
$\beta_{\sss\Xi^\iota,R,II}^\ssc[0]$     Bound-XiIota0RemII

$\beta_{\Delta \sss\Xi^\iota,R,II}^\ssc[0]$     Bound-XiIota0RemII
$\beta_{\sss\Pi,R}^\ssc[0]$    Bound-Pi0RemI
$\beta_{\Delta \sss\Pi,R}^\ssc[0]$     Bound-Pi0RemIW

An overview where to find the bounds stated in  <cit.>.

In the following, we discuss some bounds that are not stated in Table <ref>.  These are also required for the analysis, but are in general very easy to obtain.
We start by bounding $\aabz/\aaz$, as formulated in <cit.>,

\begin{align*}
\frac {\aabz}{ \aaz}=\frac {zg_z}{zg_z^\iota}=1+\frac {g_z-g_z^\iota}{g_z^\iota}=1+\frac {\bar G_z(\ve[\iota])}{g_z^\iota} \leq 1 +2dz (D\star \bar G_z)(\ve[\iota])
\end{align*}

Regarding $\betaaa$, we remark that by the definition of $z_I$ in DefinitionOfzI, for each $z>z_I$,

\begin{align*}
zg_z \geq {z_I}g_{z_I}=\frac 1 {(2d-1)\e}:=\underline{\beta}_{\mu},
\end{align*}

as formulated in <cit.>.
<cit.> also states lower bonds on $\Psi^\ssc[0]$ and $\Pi^\ssc[0]$, which we could obtain using simple combinatorics. We numerically found that the benefit of these extra bounds is minimal, so that we simply use the trivial lower $\underline{\beta}_{\sss \Psi}^\ssc[0]  = \underline{\beta}_{\sss \sum \Pi}^\ssc[1]=\underline{\beta}_{ \sss \sum \Pi_\alpha}^\ssc[0]=0$ for these coefficients, which are by definition non-negative.

Further, we assume that the geometric series bounding our NoBLE coefficients converge, which follows if their base is smaller than one as formulated in <cit.>, i.e.,

\begin{align}
\lbeq{analys-assumed-invertablecondition}
\frac {(2d-1)\aabz}{1-\aaz}\sum_{N=0}^\infty\beta_{{\sss \Xi}^{\iota}}^\ssc[N]<1.
\end{align}

For example, this shows that the matrix inverse in lace-exp-eq is well defined. Finally, the inequalities $\lowaf-\betadeltaRfzlow>0$ and $\lowcp-\betaap-\beta_{{\sss R,\Phi}}>0$,  which guarantees that the numerator and denominator of $\hat{G}_z(k)$ in lace-exp-eq are non-negative, see <cit.>.
These are numerical conditions that are verified explicitly in the Mathematica notebooks. These conditions are relatively weak, in the sense that empirically, we observe that the bootstrap analysis, which in particular includes an improvement of bounds, is much more likely to fail.

§ PROOF OF THE BOUNDS ON THE NOBLE COEFFICIENTS

In this section, we explain how the bounds stated in the preceding section are proven. In Section <ref>, we have already explained the basic idea of the bounds. We start by proving the bounds on the relatively simple coefficients arising for $N=0$ and $N=1$. Then, we explain how the coefficients for general $N\geq 2$ are bounded. After this, we discuss bounds on the difference of coefficients.

§.§ Proof of bounds for $N=0$

Here we prove Lemma <ref> and <ref>.
For $N=0$, the coefficients are created by LAs in which $0$ and $x$ are doubly connected, see defJNanimal. In Section
<ref> we have explained how to bound the contribution of such LAs, so we add here only the missing details.

In Section <ref> we have already proven the bound on $\Xi^{\ssc[0]}_{z}$.  From $\Xi^{\ssc[0]}_{{\sss R},z}$, we have extracted the contributions in which $0$ and $x$ are connected by a direct bond, see LA-Split-Def-1. Considering this, the bounds Bound-Xi0 and Bound-Xi0RemW follow directly.
The only difference between $\Xi^{\ssc[0]}_{{\sss R},z}$ and $\Psi^{\ssc[0],\kappa}_{{\sss R, II},z}$ is that $x-\ve[\kappa]\nin A$ for $\Psi^{\ssc[0],\kappa}_{{\sss R, II},z}$.
Since each double connection requires at least two neighbors of $x$ to be part of $A$, the spatial symmetry of the LAs implies that

\begin{align}
\sum_{x\in \Zd} \Psi^{\ssc[0],\kappa}_{{\sss R, II},z}(x)=\frac 1 {2d} \sum_{x\in \Zd, \kappa} \Psi^{\ssc[0],\kappa}_{{\sss R, II},z}(x)\leq \frac {2d-2} {2d} \sum_{x\in \Zd}\Xi^{\ssc[0]}_{{\sss R, II},z}(x),
\end{align}

which implies Bound-Psi0RemII and Bound-Psi0RemIIW.

The coefficient $\Psi^{\ssc[0],\kappa}_{{\sss R, I},z}(x)$ was created by extracting the shortest contributions with $x=\ve[\kappa]$ and $x$'s that are neighbors of $\ve[\kappa]$. The contribution of the remaining $(2d-1)$ neighbors of the origin is bounded using $\diagRepulsiveLetter{B}_{\underline 1,3}(\ve[1],0)$. For all other $x$, i.e., those with $|x|>1$, we use $\Psi^{\ssc[0],\kappa}_{{\sss R, II},z}(x)\leq \Psi^{\ssc[0],\kappa}_{{\sss R, I},z}(x)$ to conclude the bounds stated in Bound-Psi0RemI.

In Bound-Psi0RemIW, we bound the contribution due to all neighbors of origin by the first term.
The contribution due to all other $x$ are bounded by the second term. Regarding the neighbors $x=\ve[\iota]$ of the origin, with $\iota\in\{\pm1,\dots,\pm d\}$, we note that

\begin{align}
\begin{cases}
2 & \text{ for~} |\iota|\neq |\kappa|,\\
4 & \text{ for } \iota=-\kappa,\\
0 & \text{ for } \iota= \kappa.
\end{cases}
\end{align}

Summing over $\iota$ creates the stated factor $4d$.
For the other $x$, we know that any path to $x$ contains at least two bonds.
To be able to use a symmetry argument, we ignore that we extracted terms and as well as the condition $x-\ve[\kappa]\nin A$, to obtain the simple bound

\begin{align}
(2dz)^2g_z (D^{\star 2} \star \tilde G_z)(x)  \bar G_z(x)
\end{align}

Then, we use $|x-\ve[\kappa]|^2=(|x|^2 -2x_\kappa+1)$ and that the term involving $-2x_\kappa$ sums to zero by symmetry, to obtain

\begin{align}
\lbeq{Bound-Psi0RemIIW-proofLine}
\sum_x |x-\ve[\kappa]|^2 (2dz)^2g_z (D^{\star 2} \star \tilde G_z)(x)  \bar G_z(x)=\sum_x (|x|^2+1 ) (2dz)^2g_z (D^{\star 2} \star \tilde G_z)(x)  \bar G_z(x),
\end{align}

which is the second term in Bound-Psi0RemIW.
This proves Bound-Psi0RemIW, and completes the proof of Lemma <ref>.

By definition,

\begin{align}
\Xi^{\ssc[0],\iota}_z(x)= \frac {1}{g_z}    \sum_{A\colon 0,x,\ve[\iota]\in A} z^{|A|}\indic{0\dbct{A} x}.
\end{align}

For LTs, this simplifies to

\begin{align}
\Xi^{\ssc[0],\iota}_z(x)= \frac {\delta_{0,x}}{g_z}  \bar G_z(\ve[\iota])=\delta_{0,x}G_z(\ve[\iota]).
\end{align}

For LAs we recall that we have bounded such animals already between split-LA-sausage-Gbar and split-LA-sausage-second.

We improve these bounds by using repulsive diagrams (recall Section <ref>) and by considering four cases for the point $u$, where $u$ is the summand appearing in split-LA-sausage-Gbar, thus yielding the bound $\alpha_{1,1}$ (see BoundNZero-XiIota-alpha). Let us explain the four cases for $u$:

(a) For $u=0$, the diagram is given by a double connection from $0$ to $x$ and a simple path from $0$ to $\ve[\iota]$ that we bound by

\begin{align}
G_z(\ve[\iota])+ \frac {1}{g_z} \tilde G_z(\ve[\iota]) \sum_{x\neq 0} \diagRepulsiveLetter{D}_{1}(x,0).
\end{align}

The first term is given explicitly in Bound-XiIota0 and the second term corresponds to the first term of $\alpha_{1,1}$.

(b) For $u=\ve[\iota]$, the point $\ve[\iota]$ is on the double connection between $0$ and $x$.
For $x=\ve[\iota]$, we have a simple double connection to $\ve[\iota]$, otherwise a repulsive triangle, as stated in the second and third term of $\alpha_{1,1}$.

(c) In the contributions in which $u=x\nin \{0,\ve[\iota]\}$ we find a double connection between $0$ and $x$, bounded by $\diagRepulsiveLetter{D}_{1}(x,0)$  and a connection from $x$ to $\ve[\iota]$, bounded by $\tilde G_{m,z}(x-\ve[\iota])$. We improve this bound slightly using the parity of the lattice:
If $0$ and $x$ are directly connected, the length of the connection between $x$ to $\ve[\iota]$ is at least $2$.

(d) In the remaining contributions, we have that $u\neq \{0,x,\ve[\iota]\}$, so that these LAs can be bounded by

\begin{align}
\sum_{u}\diagRepulsiveLetter{B}_{1,1}(u,\ve[\iota])  \sup_{u} \sum_{x}\diagRepulsiveLetter{B}_{1,1}(x,u),
\end{align}

which corresponds to the last term of $\alpha_{1,1}$. This completes the proof of Bound-XiIota0.

The diagrams with the additional weights $|x|^2$ and $|x-\ve[\iota]|^2$ are bounded in a similar manner. As the numerical bounds, using the bootstrap function $f_3$, require a $G_z$ to carry the spatial weight, we express the bounds using two-point functions instead of repulsive diagrams. For the weight $|x-\ve[\iota]|^2$, two additional steps are made: Firstly, for $u=0\neq x$ we first split the weights using symmetry, as shown in Bound-Psi0RemIIW-proofLine, and then bound the two arising terms. This leads to the first line of $\gamma_1$ in BoundNZero-XiIota-gamma. Secondly, for $u\neq \{0,\ve[\iota],x\}$, we use that $|x-\ve[\iota]|^2\leq 2(|x|^2+1)$, see <cit.>, and bound the arising terms individually. This completes the proofs of Bound-XiIota0Wa and Bound-XiIota0Wb.

For the bounds on $\Xi^{\ssc[0],\iota}_{{\sss R, I},z}$ and $\Xi^{\ssc[0],\iota}_{{\sss R, II},z}$ stated in Bound-XiIota0RemI-Bound-XiIota0RemII, we review which terms have been extracted from $\Xi^{\ssc[0],\iota}_z(x)$ to yield $\Xi^{\ssc[0],\iota}_{{\sss R, I},z}$ and $\Xi^{\ssc[0],\iota}_{{\sss R, II},z}$. For $\Xi^{\ssc[0],\iota}_{{\sss R, II},z}(x)$, the LAs in which $x$ is directly connected to the origin have been removed. For this reason, in $\Xi^{\ssc[0],\iota}_{{\sss R, II},z}$ all paths from $0$ to $x$ have a length of at least $2$ instead of $1$. This is  expressed by the subscripts $n$ of $\alpha_{n,m}$ and $\beta_{n}$. To obtain $\Xi^{\ssc[0],\iota}_{{\sss R, I},z}(x)$ from  $\Xi^{\ssc[0],\iota}_z(x)$, we have extracted all LAs in which $x$ is directly connected to $\ve[\iota]$. This means that in $\Xi^{\ssc[0],\iota}_{{\sss R, I},z}(x)$, the path from $\ve[\iota]$ to $x$ in contributing LAs have length at least $2$.
The bounds Bound-XiIota0RemI and Bound-XiIota0RemII are created by repeating the consideration of cases for $u$, as given above for the bound on bound $\Xi^{\ssc[0],\iota}_z(x)$, while keeping these additional restrictions.

We continue with the bounds on $\Xi^{\ssc[0],\iota}_{\alpha,{\sss I},z}$ and $\Xi^{\ssc[0],\iota}_{\alpha,{\sss II},z}$ in Bound-XiIotaAlphaI–Bound-XiIotaAlphaIIWb. Note that for LTs, most of these terms are zero due to the lack of double connections, so we restrict to LAs. Recall split-LTs-N0, as well as split-LTs-N0a–split-LTs-N0b.

By their definitions in LA-Split-Def-XiIotaAlphaa and LA-Split-Def-XiIotaAlphab,

\begin{align}
\Xi^{\ssc[0],\iota}_{\alpha,{\sss I},z}(\ve[\iota])= &  \frac {\indAnimal}{g_z} \sum_{A\colon 0,\ve[\iota]\in A} z^{|A|}\indic{0\dbct{A} \ve[\iota]}
\leq \frac {\indAnimal}{g_z}\diagRepulsiveLetter{D}_{1}(\ve[\iota]),\\
\Xi^{\ssc[0],\iota}_{\alpha,{\sss II},z}(0)=&\Xi^{\ssc[0],\iota}_{\alpha,{\sss I},z}(0)=\frac 1 {g_z} \bar G_z(\ve[\iota])= G_z(\ve[\iota]),
\end{align}

which proves Bound-XiIotaAlphaI. To obtain the bound in Bound-XiIotaAlphaIWa, we again use LA-Split-Def-XiIotaAlphaa, and remark that, for $x\neq 0$ with $|x-\ve[\iota]|^2=1$,
we can write $x=\ve[\kappa]+\ve[\iota]$ with $\kappa\neq -\iota$ to get

\begin{align}
\Xi^{\ssc[0],\iota}_{\alpha,{\sss I},z}(\ve[\kappa]+\ve[\iota])=&\frac {1}{g_z} \sum_{A\colon 0,\ve[\kappa]+\ve[\iota],\ve[\iota]\in A} z^{|A|}\indic{0\dbct{A} \ve[\kappa]+\ve[\iota]} \indic{d_A(\ve[\iota],\ve[\kappa]+\ve[\iota])=1}.
\end{align}

Thus, $\ve[\kappa]+\ve[\iota]$ and $\ve[\iota]$ are directly connected in the LA. We distinguish between the case where $\ve[\iota]$ is on the double connection or not to obtain