Universal Computing Machine for Number Genetics in the Proportiones Perfectus Theory

Abstract

The proportiones perfectus law states that σ_x^y=(x+√(x^2+4y))/2 is a proportione perfectus if 1≤y≤x such that for an arbitrary positive integer , there exists an integer sequence defined simultaneously by the quasigeometric relation h_(n+1)=round(σ_x^y h_n ),n≥1 and the arithmetic relation h_(n+2)=〖xh〗_(n+1)+〖yh〗_n,n≥1.        When x=y=1 σ_x^y  is the golden mean. When x=2,y=1,σ_x^y  is the silver mean. In previous works we introduced the theory of number genetics – a framework of logic within which the golden section is studied. In this work we apply the concept to all proportiones perfectus. Let  be defined as above. Furthermore, let h1 satisfy {█(round(h_1/(σ_x^y ))=w@round(wσ_x^y)≠h_1 ). ┤Now let H'nbe Hn  with h1=1 Again let  be defined as  A robust universal computing machine is herein developed for the purpose of establishing the relationship  h_i=g_(n+i-1)±h_i^',i,n≥1,  which relationship is key to the logic protocol. It is clear therefore that this system of logic presents  as the building block of every  (including itself) within a particular σ_x^y regime. Clearly number genetics takes central place in the proportiones perfectus theory.