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