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