Diamond Bracket Forms and How to Count to Two; 7 of 10

From "Cybernetics & Human Knowing", Vol. 24 (2017), No. 3-4, pp. 161-188

6. Diffraction

Recall “but”, the “junction” operator:

x / y     =     (x & i) V  (y & j)     =     (x V  j) & (y V  i)

Junction defines function diffraction:

f_L(a;b)   =   f(a/b) / f(b/a)

f_R(a;b)   =   f(b/a) / f(a/b)  =      f_L(b;a)   

Diffracted functions obey these rules: 

f_L(a;b) / f_R(a;b)   =   f(a/b)

f_R(a;b) / f_L(a;b)   =   f(b/a)

f_L( a/b; b/a )   =   f(a) / f(b)

f_R( a/b; b/a )   =   f(b) / f(a)

f_L and f_R display diamond’s phase-weaving:

if f is a positive function, then:

f_L(x;y)  =  f(x)     ;           f_R(x;y)  =  f(y)

if f = (~ g), then:

f_L(x;y)  =   ~ g_R(x;y)  ;   f_R(x;y)  =   ~ g_L(x;y) 

so positives preserve phase while negation reverses it.

Diffraction can be defined by the intermix function:

J(a,b) =       ( a/b , b/a )

Note:  J(J(a,b)) = (a,b)

 ~ J ( ~ a , ~ b )  =   ( b , a )

In diamond wiring, J is a simple shuffle gate:

 x     x/y        

      \ \  / /     made purely of wires

       \_\/ /          

          \/       ergo conserves information

        __/\

       / /\ \      also resembles Feynman diagram

      / /  \ \

      y     y/x       

J is its own inverse; therefore we can dualize with J,thus:

J o (f,f) o J  ( a ,b )   =    ( f_L(a;b) ; f_R(a;b) )

Phase separation:

a   f   f_L(a;b)

 \ / \ /

  J   J

 / \ / \

b   f   f_R(a;b)

Dual to this, by conjunction algebra, is:

J o (f_L,f_R) o J  ( a ,b )         =    (  f(a)  ,  f(b)  )

Phase recombination:

a   f_L(a;b)   f(a)

 \ /       \ /

  J         J

 / \       / \

b   f_R(a;b)   f(b)

Phase separation resembles a 2-slit diffraction experiment, with J as the half-silvered mirror, and f as the filter. Similarly, phase recombination resembles a hologram, with phase data reshuffled to retrieve local data.