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