## Tuesday, May 22, 2018

### 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:
fL(a;b)   =   f(a/b) / f(b/a)
fR(a;b)   =   f(b/a) / f(a/b)  =      fL(b;a)
Diffracted functions obey these rules:
fL(a;b) / fR(a;b)   =   f(a/b)
fR(a;b) / fL(a;b)   =   f(b/a)
fL( a/b; b/a )   =   f(a) / f(b)
fR( a/b; b/a )   =   f(b) / f(a)

fL and fR display diamond’s phase-weaving:
if f is a positive function, then:
fL(x;y)  =  f(x)     ;           fR(x;y)  =  f(y)
if f = (~ g), then:
fL(x;y)  =   ~ gR(x;y)  ;   fR(x;y)  =   ~ gL(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 )   =    ( fL(a;b) ; fR(a;b) )

Phase separation:
a   f   fL(a;b)
\ / \ /
J   J
/ \ / \
b   f   fR(a;b)

Dual to this, by conjunction algebra, is:
J o (fL,fR) o J  ( a ,b )         =    (  f(a)  ,  f(b)  )

Phase recombination:
a   fL(a;b)   f(a)
\ /       \ /
J         J
/ \       / \
b   fR(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.