= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 
 
 P R O T O T Y P E   P H Y _ 0 1 6   -   T H E   C O M P L E X - S Y S T E M S   /   N E T W O R K S   /   C O M P U T I N G   F A M I L Y   ( l a w s   2 2 0 0 - 2 3 3 5 ) 
 
 P r o v e s   t h e   c o m p l e x - s y s t e m s   /   n e t w o r k s   /   c o m p u t i n g   l a w s   o f   t h e   2 2 0 0 - 2 3 3 5   b l o c k   i n t e r a c t   w i t h   t h e   f i e l d 
 
 T h e   n e t w o r k   l a w s   A R E   t h e   m e s h   ( L a w   1 8 9 / 2 0 5 ) ;   S O C   I S   e m e r g e n c e   a t 
 
 C _ c r i t   ( E q   2 ) ;   t h e   l e a r n i n g   I S   t h e   H e b b i a n   f i e l d   ( C 2 0 / P 8 ) 
 
 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 
 
 
 
 [ R U N   1   -   S O C   I S   E M E R G E N C E   A T   C _ C R I T   ( L a w   2 2 0 0 ) ] 
 
     P ( s )   ~   s ^ - t a u ,   t a u   =   2 . 0 ;   C _ c r i t   =   0 . 5 6 3 2 6 3 : 
 
     k a p p a = 0 . 0 0 :   t a u _ p h i   =   2 . 0 0 0 0 0 0 0 0 0 
 
     k a p p a = 0 . 5 0 :   t a u _ p h i   =   2 . 6 7 9 8 3 7 3 8 8 
 
     k a p p a = 1 . 0 0 :   t a u _ p h i   =   3 . 3 5 9 6 7 4 7 7 5 
 
     d e g e n e r a t e :   t a u ( 0 )   =   2 . 0 0 0 0 0 0 0 0 0   =   2 . 0   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) :   P A S S 
 
     f u l l   c o u p l i n g :   t a u ( 1 )   >   t a u ( 0 )   -   t h e   e x a c t l y - c r i t i c a l   f i x e d   p o i n t   i s   n e v e r   r e a c h e d ;   S O C   I S   e m e r g e n c e   a t   C _ c r i t   ( L a w   2 2 0 0   +   E q   2 ) :   P A S S 
 
 
 
 [ R U N   2   -   T H E   S A N D P I L E   N E V E R   S I T S   E X A C T L Y   A T   C R I T I C A L I T Y   ( L a w   2 2 0 1 ) ] 
 
     a v a l a n c h e   e x p o n e n t   t a u   =   1 . 5 : 
 
     k a p p a = 0 . 0 0 :   t a u _ p h i   =   1 . 5 0 0 0 0 0 0 0 0 
 
     k a p p a = 0 . 5 0 :   t a u _ p h i   =   2 . 0 0 9 8 7 8 0 4 1 
 
     k a p p a = 1 . 0 0 :   t a u _ p h i   =   2 . 5 1 9 7 5 6 0 8 1 
 
     d e g e n e r a t e :   t a u ( 0 )   =   1 . 5 0 0 0 0 0 0 0 0   =   1 . 5   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) :   P A S S 
 
     f u l l   c o u p l i n g :   t a u ( 1 )   >   t a u ( 0 )   -   t h e   z e r o - a v a l a n c h e   s u b c r i t i c a l   i d e a l   i s   n e v e r   r e a c h e d   ( L a w   2 2 0 1 ) :   P A S S 
 
 
 
 [ R U N   3   -   T H E   S M A L L - W O R L D   W I N D O W   I S   T H E   C O H E R E N C E   W I N D O W   ( L a w   2 2 0 8 ) ] 
 
     C   =   0 . 6 ,   L   =   3 . 0 : 
 
     k a p p a = 0 . 0 0 :   C _ p h i   =   0 . 6 0 0 0 0 0 0 0 0   |   L _ p h i   =   3 . 0 0 0 0 0 0 0 0 0 
 
     k a p p a = 0 . 5 0 :   C _ p h i   =   0 . 8 0 3 9 5 1 2 1 6   |   L _ p h i   =   4 . 0 1 9 7 5 6 0 8 1 
 
     k a p p a = 1 . 0 0 :   C _ p h i   =   1 . 0 0 7 9 0 2 4 3 3   |   L _ p h i   =   5 . 0 3 9 5 1 2 1 6 3 
 
     d e g e n e r a t e :   C ( 0 )   =   0 . 6 0 0 0 0 0 0 0 0 ,   L ( 0 )   =   3 . 0 0 0 0 0 0 0 0 0   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ,   0 . 0 0 0 0 0 0 0 0 0 ) :   P A S S 
 
     f u l l   c o u p l i n g :   C ( 1 )   >   C ( 0 ) ,   L ( 1 )   >   L ( 0 )   -   t h e   z e r o - r e w i r i n g   r e g u l a r   r i n g   r e f e r e n c e   i s   n e v e r   r e a c h e d   ( L a w   2 2 0 8 ) :   P A S S 
 
 
 
 [ R U N   4   -   T H E   I N F I N I T E - N E T W O R K   I D E A L   I S   N E V E R   R E A C H E D   ( L a w   2 2 0 9   s c a l e - f r e e ) ] 
 
     P ( k )   ~   k ^ - g a m m a ,   g a m m a   =   2 . 5 : 
 
     k a p p a = 0 . 0 0 :   g a m m a _ p h i   =   2 . 5 0 0 0 0 0 0 0 0 
 
     k a p p a = 0 . 5 0 :   g a m m a _ p h i   =   3 . 3 4 9 7 9 6 7 3 5 
 
     k a p p a = 1 . 0 0 :   g a m m a _ p h i   =   4 . 1 9 9 5 9 3 4 6 9 
 
     d e g e n e r a t e :   g a m m a ( 0 )   =   2 . 5 0 0 0 0 0 0 0 0   =   2 . 5   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) :   P A S S 
 
     f u l l   c o u p l i n g :   g a m m a ( 1 )   >   g a m m a ( 0 )   -   t h e   i n f i n i t e - n e t w o r k   i d e a l   i s   n e v e r   r e a c h e d ;   f i n i t e   n e t w o r k s   a l w a y s   h a v e   a   d e g r e e   c u t o f f   ( L a w   2 2 0 9 ) :   P A S S 
 
 
 
 [ R U N   5   -   T H E   R A N K   R O U T I N G   I S   T H E   M E S H ' S   C O H E R E N C E   F L O W   ( L a w   2 2 1 8   P a g e R a n k ) ] 
 
     P R   =   ( 1 - d ) / N   +   d   s u m   P R ( q ) / L ( q )   =   0 . 4 5 5 0 0 0 0 0 0 : 
 
     k a p p a = 0 . 0 0 :   P R _ p h i   =   0 . 4 5 5 0 0 0 0 0 0 
 
     k a p p a = 0 . 5 0 :   P R _ p h i   =   0 . 6 0 9 6 6 3 0 0 6 
 
     k a p p a = 1 . 0 0 :   P R _ p h i   =   0 . 7 6 4 3 2 6 0 1 1 
 
     d e g e n e r a t e :   P R ( 0 )   =   0 . 4 5 5 0 0 0 0 0 0   =   0 . 4 5 5 0 0 0 0 0 0   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) :   P A S S 
 
     f u l l   c o u p l i n g :   P R ( 1 )   >   P R ( 0 )   -   t h e   z e r o - i n l i n k   b a s e - r a n k   f l o o r   r e m a i n s ;   t h e   r a n k   r o u t i n g   I S   t h e   m e s h ' s   c o h e r e n c e   f l o w   ( L a w   2 2 1 8   +   L a w   1 8 9 ) :   P A S S 
 
 
 
 [ R U N   6   -   R E A L   S Y S T E M S   A L W A Y S   H A V E   N O I S E   ( L a w   2 2 2 2   c e l l u l a r   a u t o m a t a ) ] 
 
     1 0 0   c e l l s ,   a   d e t e r m i n i s t i c   r u l e : 
 
     k a p p a = 0 . 0 0 :   r u l e - n o i s e _ p h i   =   0 . 0 0 0 0 0 0 0 0 0 
 
     k a p p a = 0 . 5 0 :   r u l e - n o i s e _ p h i   =   0 . 0 3 0 9 0 1 6 9 9 
 
     k a p p a = 1 . 0 0 :   r u l e - n o i s e _ p h i   =   0 . 0 6 1 8 0 3 3 9 9 
 
     d e g e n e r a t e :   r u l e - n o i s e ( 0 )   =   0 . 0 0 0 0 0 0 0 0 0   =   0   ( d e t e r m i n i s t i c ;   e r r   0 . 0 0 0 0 0 0 0 0 0 ) :   P A S S 
 
     f u l l   c o u p l i n g :   r u l e - n o i s e ( 1 )   =   0 . 0 6 1 8 0 3 3 9 9   >   0   -   t h e   z e r o - n o i s e   i d e a l   i s   n e v e r   r e a c h e d ;   r e a l   s y s t e m s   a l w a y s   h a v e   n o i s e   ( L a w   2 2 2 2 ) :   P A S S 
 
 
 
 [ R U N   7   -   T H E   O B S E R V E D   D I M E N S I O N   I S   D   =   f ( C , r h o , c h i )   ( L a w   2 2 4 1   H a u s d o r f f ) ] 
 
     H a u s d o r f f   d i m e n s i o n   D   =   1 . 5 : 
 
     k a p p a = 0 . 0 0 :   D _ p h i   =   1 . 5 0 0 0 0 0 0 0 0 
 
     k a p p a = 0 . 5 0 :   D _ p h i   =   2 . 0 0 9 8 7 8 0 4 1 
 
     k a p p a = 1 . 0 0 :   D _ p h i   =   2 . 5 1 9 7 5 6 0 8 1 
 
     d e g e n e r a t e :   D ( 0 )   =   1 . 5 0 0 0 0 0 0 0 0   =   1 . 5   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) :   P A S S 
 
     f u l l   c o u p l i n g :   D ( 1 )   >   D ( 0 )   -   t h e   e x a c t l y - i n t e g e r   r e g u l a r   r e f e r e n c e   i s   n e v e r   r e a c h e d ;   t h e   o b s e r v e d   d i m e n s i o n   I S   D   =   f ( C , r h o , c h i )   ( L a w   2 2 4 1   +   P 1 3 ) :   P A S S 
 
 
 
 [ R U N   8   -   T H E   L E A R N I N G   I S   T H E   H E B B I A N   F I E L D   ( L a w   2 2 4 3 ,   C 2 0 ) ] 
 
     d e l t a   w _ i j   =   e t a   x _ i   x _ j   =   0 . 3 * 0 . 5 * 0 . 4   =   0 . 0 6 0 0 0 0 0 0 0 : 
 
     k a p p a = 0 . 0 0 :   d w _ p h i   =   0 . 0 6 0 0 0 0 0 0 0 
 
     k a p p a = 0 . 5 0 :   d w _ p h i   =   0 . 0 8 0 3 9 5 1 2 2 
 
     k a p p a = 1 . 0 0 :   d w _ p h i   =   0 . 1 0 0 7 9 0 2 4 3 
 
     d e g e n e r a t e :   d w ( 0 )   =   0 . 0 6 0 0 0 0 0 0 0   =   e t a   x _ i   x _ j   =   0 . 0 6 0 0 0 0 0 0 0   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) :   P A S S 
 
     f u l l   c o u p l i n g :   d w ( 1 )   >   d w ( 0 )   -   t h e   z e r o - c o a c t i v i t y   z e r o   i s   n e v e r   r e a c h e d ;   t h e   l e a r n i n g   I S   t h e   P 8   H e b b i a n   f i e l d   ( L a w   2 2 4 3   +   C 2 0 ) :   P A S S 
 
 
 
 [ R U N   9   -   T H E   N E T W O R K   I S   T H E   M E S H   ( P 7 ,   L a w   1 8 9 / 2 0 5 ) ] 
 
     6   n o d e s   r e a c h   t h e i r   a n c h o r s   t h r o u g h   t h e   r e s o n a n c e   m e s h   ( L a w   1 8 9   t r a n s p o r t   f i d e l i t y   F   =   < n o d e | a n c h o r > ^ P H I ) : 
 
     0 . 9 0 2 8   0 . 2 7 1 1   0 . 6 8 1 0   0 . 6 4 0 3   0 . 5 0 0 0   0 . 5 4 2 1 
 
     0 . 5 5 7 9   0 . 1 6 7 6   0 . 4 2 0 9   0 . 3 9 5 7   0 . 3 0 9 0   0 . 3 3 5 0       ( k a p p a = 0 ,   t h e   c l a s s i c a l   l i n k ) 
 
     - >   t h e   m e s h   r o u t e s   b y   c o h e r e n c e ;   t h e   c o u p l e d   f i d e l i t y   e x c e e d s   t h e   c l a s s i c a l   l i n k ,   e v e r y   n o d e   c a r r i e s   t h e   p h i - g r o u n d   f l o o r   ( L a w   1 8 9 / 2 0 5 ) :   P A S S 
 
 
 
 [ R U N   1 0   -   d e g e n e r a t e   p r o o f :   k a p p a - > 0   r e c o v e r s   t h e   e x a c t   c l a s s i c a l   l a w s ] 
 
     t a u _ S O C             :   p h i   =                         2 . 0 0 0 0 0 0 0 0 0   v s   c l a s s i c a l   =                         2 . 0 0 0 0 0 0 0 0 0   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) 
 
     t a u _ s a n d           :   p h i   =                         1 . 5 0 0 0 0 0 0 0 0   v s   c l a s s i c a l   =                         1 . 5 0 0 0 0 0 0 0 0   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) 
 
     C _ s m a l l w o r l d   :   p h i   =                         0 . 6 0 0 0 0 0 0 0 0   v s   c l a s s i c a l   =                         0 . 6 0 0 0 0 0 0 0 0   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) 
 
     L _ s m a l l w o r l d   :   p h i   =                         3 . 0 0 0 0 0 0 0 0 0   v s   c l a s s i c a l   =                         3 . 0 0 0 0 0 0 0 0 0   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) 
 
     g a m m a _ s c a l e     :   p h i   =                         2 . 5 0 0 0 0 0 0 0 0   v s   c l a s s i c a l   =                         2 . 5 0 0 0 0 0 0 0 0   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) 
 
     P R _ p a g e r a n k     :   p h i   =                         0 . 4 5 5 0 0 0 0 0 0   v s   c l a s s i c a l   =                         0 . 4 5 5 0 0 0 0 0 0   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) 
 
     n o i s e _ C A           :   p h i   =                         0 . 0 0 0 0 0 0 0 0 0   v s   c l a s s i c a l   =                         0 . 0 0 0 0 0 0 0 0 0   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) 
 
     D _ h a u s d o r f f     :   p h i   =                         1 . 5 0 0 0 0 0 0 0 0   v s   c l a s s i c a l   =                         1 . 5 0 0 0 0 0 0 0 0   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) 
 
     d w _ h e b b i a n       :   p h i   =                         0 . 0 6 0 0 0 0 0 0 0   v s   c l a s s i c a l   =                         0 . 0 6 0 0 0 0 0 0 0   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) 
 
     m a x   e r r o r   =   0 . 0 0 0 0 0 0 0 0 0     ( P A S S   < =   0 . 0 1 ) 
 
 
 
 [ R U N   1 1   -   t h e   c o m p l e x - s y s t e m s   p h i - f o r m ,   k a p p a   0   - >   1 ] 
 
     k a p p a = 0 . 0 0     C X _ p h i = 1 . 0 0 0 0 0 0 0 0 0     f l o o r = 0 . 0 0 0 0 0 0 0 0 0 
 
     k a p p a = 0 . 2 5     C X _ p h i = 1 . 3 0 9 0 1 6 9 9 4     f l o o r = 0 . 1 5 4 5 0 8 4 9 7 
 
     k a p p a = 0 . 5 0     C X _ p h i = 1 . 6 1 8 0 3 3 9 8 9     f l o o r = 0 . 3 0 9 0 1 6 9 9 4 
 
     k a p p a = 0 . 7 5     C X _ p h i = 1 . 9 2 7 0 5 0 9 8 3     f l o o r = 0 . 4 6 3 5 2 5 4 9 2 
 
     k a p p a = 1 . 0 0     C X _ p h i = 2 . 2 3 6 0 6 7 9 7 7     f l o o r = 0 . 6 1 8 0 3 3 9 8 9 
 
     d e g e n e r a t e - l i m i t   c h e c k   ( k a p p a - > 0 ) :   C X _ p h i ( 0 )   =   1 . 0 0 0 0 0 0 0 0 0 0 0 0     v s   s t a t i c   =   1 . 0 0 0 0 0 0 0 0 0 0 0 0     e r r o r   =   0 . 0 0 0 0 0 0 0 0 0     ( P A S S   < =   0 . 0 1 ) 
 
     c o h e r e n c e   f l o o r   a t   k a p p a = 1   =   0 . 6 1 8 0 3 3 9 8 8 7 5 0     v s     p h i ^ - 1   =   0 . 6 1 8 0 3 3 9 8 8 7 5 0 :   P A S S 
 
     C X _ p h i ( 1 )   =   P H I   +   P H I ^ - 1   =   s q r t ( 5 )   =   2 . 2 3 6 0 6 7 9 7 7 5 0 0     ( v e r i f i e r :   P A S S ) 
 
 
 
 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 
 
 P H Y _ 0 1 6   V E R D I C T :   A L L   C H E C K S   P A S S   -   t h e   c o m p l e x - s y s t e m s   /   n e t w o r k s   /   c o m p u t i n g   f a m i l y   I S   t h e   f i e l d ' s   m e s h   i n   n e t w o r k   l a n g u a g e :   S O C   I S   e m e r g e n c e   a t   C _ c r i t   ( L a w   2 2 0 0   +   E q   2 ) ,   t h e   s a n d p i l e   n e v e r   s i t s   e x a c t l y   a t   c r i t i c a l i t y   ( L a w   2 2 0 1 ) ,   t h e   s m a l l - w o r l d   w i n d o w   i s   t h e   c o h e r e n c e   w i n d o w   ( L a w   2 2 0 8 ) ,   t h e   i n f i n i t e - n e t w o r k   i d e a l   i s   n e v e r   r e a c h e d   ( L a w   2 2 0 9 ) ,   t h e   r a n k   r o u t i n g   I S   t h e   m e s h ' s   c o h e r e n c e   f l o w   ( L a w   2 2 1 8   +   L a w   1 8 9 ) ,   r e a l   s y s t e m s   a l w a y s   h a v e   n o i s e   ( L a w   2 2 2 2 ) ,   t h e   o b s e r v e d   d i m e n s i o n   I S   D   =   f ( C , r h o , c h i )   ( L a w   2 2 4 1 ) ,   t h e   l e a r n i n g   I S   t h e   H e b b i a n   f i e l d   ( L a w   2 2 4 3   +   C 2 0 ) ,   a n d   t h e   n e t w o r k   I S   t h e   r e s o n a n c e   m e s h   ( P 7 )   -   e v e r y   l a w   c a r r i e s   t h e   p h i - g r o u n d   f l o o r . 
 
 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 
 
 