Felix-is-not-Klein conjecture

Biradical numbers are those numbers that are obtained recursively from 1 by addition x+y , multiplication x.y  , negativation -x , reciprocation 1/x for x not 0 and biradication sr(x) of nonnegative numbers, meaning square root (contrary to the Bavarian notation of the included publications here conventional notation is used for functions).

Felix-numbers are the biradical numbers inside the interval from -1 to 1 (without the limits).

Klein-numbers are constructed on the basis of the Euclidean Klein-model of Lobachevsky geometry. Klein-numbers are biradical numbers that are obtained recursively:

starting from sr(2sr(2)-2)   by successive application of

annition   (x+y)/(1+x.y)

menition  -x

evition     x.y/(y+z)   where (x.x+y.y)<1 and not y+z=0

riation     sr(x.x+y.y-x.x.y.y)

clarition  sr((x.x-y.y)/(1-y.y))  where not x.x<y.y

jacition   (1-sr((1-y.y.(1-x.x))/(1-z.z)))/x  where not x=0 , else equal 0

Klein-numbers lie  inside the interval from -1 to 1 (without the limits) by construction.

Klein-numbers are Felix-numbers.

Conjecture: not all Felix-numbers are Klein-numbers

It looks quite obvious, however, a proof is needed. It is  sufficient to show that 1/2 is not a Klein-number. If you have a proof, please contact the author !

With two Felix-numbers as coordinates one gets the points of the Felix-circle

With two Klein-numbers as coordinates one gets the points of the Klein-circle.