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