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1 decimal integer ring cycle of many

Quantum Field Fractal Polarization Math Constants

nemeth braille printable arx calc

pronounced why phi prime quotients

ᐱ Y φ Θ P Q Ψ

condensed matter

Y Phi Theta Prime Q Quotients Base Numerals 1dir 2dir 3dir cdir

numer nu mer numerical nomenclature & arcs

ACQFPS

Advanced Complex Quantum Fractal Polarization Sets

When calculating for variables Y  φ  Θ  P  Q  A  M  V  W  E  F  I  H  D B O G L K U J R Z T S Ψ ᐱ  ᗑ  ∘⧊°  ∘∇° the ratio of the fractals must first be factored before using those quantity numeral values in equations with math formulas and other calculus. An infinite number of variables to each path is in a potential and with that fact extremely complex equations of precision can be factored. 

A numeral of one scale set in a consecutive string will produce a base logic(s) library of the variables mentioned, however shifting just the base factors will produce a path sequence and set of completely different ratios.

if Yn1 is divided with Yn2 then both φ and Θ are quotients determined by the path of noted ⅄ using consecutive numerals of Y scale.

if Pn1 is divided with Pn2 then both 1⅄Q and 2⅄Q are quotients determined by the path of noted ⅄ using consecutive numerals of P scale.

A potential of the path being that 1⅄ and 2⅄ and 3⅄ is not limited to anything aside from the variables from consecutive string orders is noted means that the numbered variable of Y or P is simply noted being a numeral from the consecutive numbers defined as prime or fibonacci count. Entirely different sets of ratio variables can be factored in library arrays of a much different scale between the two factors of Y scale or the two factors of P scale.

Examples

Path 2⅄(Yn1/Yn2) is to Θ just as is (Yn1/Yn101) and so on.

Path 1⅄(Yn2/Yn1) is to φ just as is (Yn101/Yn1) and so on.

and 

Path 1⅄(Pn2/Pn1) is to 1⅄Q just as is (Pn101/Pn1) and so on.

Path 2⅄(Pn1/Pn2) is to 2⅄Q just as is (Pn1/Pn101) and so on.

if 3⅄ Path is not applicable to bases Y and P because neither contains a decimal value of potential cycle stem in variable then it is reserved for variables on same path strings that do have potential for change in the decimal stem where a cycle is factorable. Just as the numbered variable of a scale in the examples is noted being Θ φ 1⅄Q and 2⅄Q applied to (Nn1/Nn101) or (Nn101/Nn1), the path 3⅄cn is applicable to variable equations containing Θ φ 1⅄Q and 2⅄Q that have that decimal stem cycle variable potential.

While equations potentials are infinite and this description space is limited to a computer display shown through a limited communications of that description and its potential variables, a base library of factored numerals is still able to be shown. 

While the base scale factors are able to shift in alignments of matrice array combinations, a basic foundation of factoring can be built based on no shift and shift of 1 left or right so to say.

Examples

Path 2⅄(Yn1/Yn2) is to Θ just as is (Yn1/Yn3) and so on of set ∈2⅄Yn1

Path 1⅄(Yn2/Yn1) is to φ just as is (Yn1/Yn3) and so on of set ∈1⅄Yn1

and 

Path 1⅄(Pn2/Pn1) is to 1⅄Q just as is (Pn3/Pn1) and so on of set ∈1⅄Pn1

Path 2⅄(Pn1/Pn2) is to 2⅄Q just as is (Pn1/Pn3) and so on of set ∈2⅄Pn1

Variables A  M  V  W are derived from a path set ∈⅄ of Y and P for equations

A=∈1⅄(φ/Q)cn

M=∈2⅄(φ/Q)cn

V= ∈1⅄(Q/φ)cn

W=∈2⅄(Q/φ)cn

3⅄A of A=∈1⅄(φ/Q)cn

3⅄M of M=∈2⅄(φ/Q)cn

3⅄V of V= ∈1⅄(Q/φ)cn

3⅄W of W=∈2⅄(Q/φ)cn

Variables E  F  I  H are derived from a path set ∈⅄ of Y and P for equations

E=∈1⅄(Θ/Q)cn

F=∈2⅄(Θ/Q)cn

I=∈1⅄(Q/Θ)cn

H=∈2⅄(Q/Θ)cn

3⅄E of E=∈1⅄(Θ/Q)cn

3⅄F of F=∈2⅄(Θ/Q)cn

3⅄I of I=∈1⅄(Q/Θ)cn

3⅄H of H=∈2⅄(Q/Θ)cn

Then DB and OG paths in wave functions of cycle variants 1⅄, 2⅄, 3⅄

D=∈1⅄(φ/Θ)cn

B=∈2⅄(φ/Θ)cn

O=∈1⅄(Θ/φ)cn

G=∈2⅄(Θ/φ)cn

Then LK and UJ paths in wave functions of cycle variants 1⅄, 2⅄, 3⅄

L=∈1⅄(1⅄Qn2/2⅄Qn1)cn

K=∈2⅄(1⅄Qn1/2⅄Qn2)cn

U=∈1⅄(2⅄Qn2/1⅄Qn1)cn

J=∈2⅄(2⅄Qn1/1⅄Qn2)cn

Then RZ and TS paths in wave functions of cycle variants

R=∈(φ/P)cn  that paths  1⅄, 2⅄, 3⅄ are then applicable to the variables φ and P of ∈R

Z=∈(P/φ)cn  that paths  1⅄, 2⅄, 3⅄ are then applicable to the variables P and φ of ∈Z

T=∈(Θ/P)cn  that paths  1⅄, 2⅄, 3⅄ are then applicable to the variables Θ and P of ∈T

S=∈(P/Θ)cn  that paths  1⅄, 2⅄, 3⅄ are then applicable to the variables Θ and P of ∈S

AMVWEFIHDBOGLKUJRZTS variables are all ratio sets that are then also applicable to division with Y in paths  1⅄, 2⅄, 3⅄ and more such as every other or every third of the consecutive numbers in Y Θ φ P Q scale to degrees in stem cycle variants of Ncn applicable in functions with AMVWEFIHDBOGLKUJRZTS that have such potential.

Variables A  M  V  W  E  F  I  H  then have the potential of a basic foundation of factoring on no shift and shift of 1 or more and so on. Let N be a number that represents A  M  V  W  E  F  I  H variables

Path 1⅄(Nn2/Nn1) is to 1⅄N just as is (Nn3/Nn1) and so on of set ∈1⅄Nn1

Path 2⅄(Nn1/Nn2) is to 2⅄N just as is (Nn1/Nn3) and so on of set ∈2⅄Nn1

Path 3⅄(Nncn/Nncn) is to 3⅄Ncn just as is (Nncn/Nncn) and so on of set ∈3⅄Nncn

The Shift in the base scale consecutive numbers will factor quotients that are real numbers and ratios that are definable and those variables are not errors and are precise calculations of advanced complex quantum fractal polarization sets.

To begin

Define variable factors of no shift in Y P φ  Θ  1⅄Q and 2⅄Q that sets ∈3⅄Nncn each have a library defined to path of ⅄N.

1dir 

A=∈1⅄(φ/Q)cn then (φn1/Qn1) and (φn2/Qn1) for ∈1⅄A 

so that variables of ∈1⅄A can be factored as defined variables of cn in other functions and equations that have use with the factor from set ∈1⅄Ancn such as paths 1⅄2A, 2⅄2A and 3⅄2A or more complex uses of the variable defined. Paths (φ/P) and (φ/Y) differ from A=∈1⅄(φ/Q)

M=∈2⅄(φ/Q)cn then (φn1/Qn1) and (φn1/Qn2) for ∈2⅄M

so that variables of ∈2⅄M can be factored as defined variables of cn in other functions and equations that have use with the factor from set ∈2⅄Mncn such as paths 1⅄2M, 2⅄2M and 3⅄2M or more complex uses of the variable defined. Paths (φ/P) and (φ/Y) differ from M=∈2⅄(φ/Q)

V= ∈1⅄(Q/φ)cn then (Qn1/φn1) and (Qn2/φn1) for ∈1⅄V

so that variables of ∈1⅄V can be factored as defined variables of cn in other functions and equations that have use with the factor from set ∈1⅄Vncn such as paths 1⅄2V, 2⅄2V and 3⅄2V or more complex uses of the variable defined. Paths (Q/P) and (Q/Y) differ from V=∈1⅄(Q/φ)

W=∈2⅄(Q/φ)cn then (Qn1/φn1) and (Qn1/φn2) for ∈2⅄W

so that variables of ∈2⅄W can be factored as defined variables of cn in other functions and equations that have use with the factor from set ∈2⅄Wncn such as paths 1⅄2W, 2⅄2W and 3⅄2W or more complex uses of the variable defined. Paths (Q/P) and (Q/Y) differ from W=∈2⅄(Q/φ)

E=∈1⅄(Θ/Q)cn then (Θn1/Qn1) and (Θn2/Qn1) for ∈1⅄E

so that variables of ∈1⅄E can be factored as defined variables of cn in other functions and equations that have use with the factor from set ∈1⅄Encn such as paths 1⅄2E, 2⅄2E and 3⅄2E or more complex uses of the variable defined. Paths (Θ/P) and (Θ/Y) differ from E=∈1⅄(Θ/Q)

F=∈2⅄(Θ/Q)cn then (Θn1/Qn1) and (Θn1/Qn2) for ∈2⅄F

so that variables of ∈2⅄F can be factored as defined variables of cn in other functions and equations that have use with the factor from set ∈2⅄Fncn such as paths 1⅄2F, 2⅄2F and 3⅄2F or more complex uses of the variable defined. Paths (Θ/P) and (Θ/Y) differ from F=∈2⅄(Θ/Q)

I=∈1⅄(Q/Θ)cn then (Qn1/Θn1) and (Qn2/Θn1) for ∈1⅄I

so that variables of ∈1⅄I can be factored as defined variables of cn in other functions and equations that have use with the factor from set ∈1⅄Incn such as paths 1⅄2I, 2⅄2I and 3⅄2I or more complex uses of the variable defined. Paths (P/Θ) and (Y/Θ) differ from I=∈1⅄(Q/Θ)

H=∈2⅄(Q/Θ)cn then (Qn1/Θn1) and (Qn1/Θn2) for ∈2⅄H

so that variables of ∈2⅄H can be factored as defined variables of cn in other functions and equations that have use with the factor from set ∈2⅄Hncn such as paths 1⅄2H, 2⅄2H and 3⅄2H or more complex uses of the variable defined. Paths (P/Θ) and (Y/Θ) differ from H=∈2⅄(Q/Θ)

Later functions of  1X, 1+⅄cn, 1-⅄, 2-⅄, ⅄n, ⅄ncn are then applicable to variables from sets 

Y  φ  Θ  P  Q  A  M  V  W  E  F  I  H  D B O G L K U J R Z T S ᐱ  ᗑ  ∘⧊°  ∘∇°

And arrays are structurable to the same base shift and paths described of those variable ordinal consecutive scales.

Basic Library of Quantum Fractal Polarization Sets  2dir  3dir  cdir  1dir

Numeral Array logic symbols to use of next tier alt paths and stem cell cycle notations 

ᐱ for ∈(⅄ᐱ)

∈1ᐱ is a ratio of divided variables of A  B  D  E  F  G  H  I  J  K  L  M  N  O  P  Q  R  S  T  U  V  W  Y  Z  φ  Θ using the later divided by previous path 1⅄ for ᐱ

∈(ᐱ)=∈1(⅄ᐱ) and ∈2(⅄ᐱ) and and ∈3(⅄ᐱ)of variables ⅄A , ⅄M, ⅄V, ⅄W, A, M, V, W, φ, Θ, Q, Y, P and whole numbers or a variable factors noted that ∈(ᐱ) is applicable with in quanta such that 1⅄, 2⅄, and 3⅄ paths differ variables of ᐱ

∈(ᐱ)=sets like

∈(A/A) then

Example

if ∈(ᐱ) of A/A=∈(A/A)

then ᐱ of A/A needs a definition of the path of variables of A.

as ∈(ᐱ) of 1⅄(A/A) and ∈(ᐱ) of 2⅄(A/A) and ∈(ᐱ) of 3⅄(A/A) differ in path of Nn/Nn

∈(ᐱ) of 1⅄(A/A)=(An2/An1)=(1⅄(φn2/1⅄Qn1)n2 /1⅄(φn2/1⅄Qn1)n1) of set ∈A=1⅄(φn2/1⅄Qn1)cn 

and

∈(ᐱ) of 1⅄(A/A)=(An2/An1)=(1⅄(φn2/2⅄Qn1)n2 /1⅄(φn2/2⅄Qn1)n1) of set A=∈1⅄(φn2/2⅄Qn1)cn

and the defined path of Q in the base of A is essential to the definition of an ∈(ᐱ) of 1⅄(A/A)ncn

then as 2⅄(A/A) would differ in path of its set ∈(ᐱ)

∈(ᐱ) of 2⅄(A/A)=(An1/An2) dependent again on the path of Q to the A

and so on for 

∈(ᐱ) of 3⅄(A/A)=(Ancn/Ancn) that is division path of like terms with a difference in variant stem decimal cycle of Nncn

∈(ᐱ)=sets like ∈(A/A) that ∈1⅄ᐱ(An2/An1) and ∈2⅄ᐱ(An1/An2) ∈3⅄ᐱ(An1cn/An1cn) for all variables A,B,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z,φ,Θ that provide variables for factoring complex ᗑ variables.

∈n⅄ᐱ(An/An), ∈n⅄ᐱ(An/Bn), ∈n⅄ᐱ(An/Dn), ∈n⅄ᐱ(An/En), ∈n⅄ᐱ(An/Fn), ∈n⅄ᐱ(An/Gn), ∈n⅄ᐱ(An/Hn), ∈n⅄ᐱ(An/In), ∈n⅄ᐱ(An/Jn), ∈n⅄ᐱ(An/Kn), ∈n⅄ᐱ(An/Ln), ∈n⅄ᐱ(An/Mn), ∈n⅄ᐱ(An/Nn), ∈n⅄ᐱ(An/On), ∈n⅄ᐱ(An/Pn), ∈n⅄ᐱ(An/Qn), ∈n⅄ᐱ(An/Rn), ∈n⅄ᐱ(An/Sn), ∈n⅄ᐱ(An/Tn),∈n⅄ᐱ(An/Un),  ∈n⅄ᐱ(An/Vn), ∈n⅄ᐱ(An/Wn), ∈n⅄ᐱ(An/Yn), ∈n⅄ᐱ(An/Zn), ∈n⅄ᐱ(An/φn), ∈n⅄ᐱ(An/Θn)

∈n⅄ᐱ(Bn/An), ∈n⅄ᐱ(Bn/Bn), ∈n⅄ᐱ(Bn/Dn), ∈n⅄ᐱ(Bn/En), ∈n⅄ᐱ(Bn/Fn), ∈n⅄ᐱ(Bn/Gn), ∈n⅄ᐱ(Bn/Hn), ∈n⅄ᐱ(Bn/In), ∈n⅄ᐱ(Bn/Jn), ∈n⅄ᐱ(Bn/Kn), ∈n⅄ᐱ(Bn/Ln), ∈n⅄ᐱ(Bn/Mn), ∈n⅄ᐱ(Bn/Nn), ∈n⅄ᐱ(Bn/On), ∈n⅄ᐱ(Bn/Pn), ∈n⅄ᐱ(Bn/Qn), ∈n⅄ᐱ(Bn/Rn), ∈n⅄ᐱ(Bn/Sn), ∈n⅄ᐱ(Bn/Tn),∈n⅄ᐱ(Bn/Un),  ∈n⅄ᐱ(Bn/Vn), ∈n⅄ᐱ(Bn/Wn), ∈n⅄ᐱ(Bn/Yn), ∈n⅄ᐱ(Bn/Zn), ∈n⅄ᐱ(Bn/φn), ∈n⅄ᐱ(Bn/Θn)

∈n⅄ᐱ(Dn/An), ∈n⅄ᐱ(Dn/Bn), ∈n⅄ᐱ(Dn/Dn), ∈n⅄ᐱ(Dn/En), ∈n⅄ᐱ(Dn/Fn), ∈n⅄ᐱ(Dn/Gn), ∈n⅄ᐱ(Dn/Hn), ∈n⅄ᐱ(Dn/In), ∈n⅄ᐱ(Dn/Jn), ∈n⅄ᐱ(Dn/Kn), ∈n⅄ᐱ(Dn/Ln), ∈n⅄ᐱ(Dn/Mn), ∈n⅄ᐱ(Dn/Nn), ∈n⅄ᐱ(Dn/On), ∈n⅄ᐱ(Dn/Pn), ∈n⅄ᐱ(Dn/Qn), ∈n⅄ᐱ(Dn/Rn), ∈n⅄ᐱ(Dn/Sn), ∈n⅄ᐱ(Dn/Tn),∈n⅄ᐱ(Dn/Un),  ∈n⅄ᐱ(Dn/Vn), ∈n⅄ᐱ(Dn/Wn), ∈n⅄ᐱ(Dn/Yn), ∈n⅄ᐱ(Dn/Zn), ∈n⅄ᐱ(Dn/φn), ∈n⅄ᐱ(Dn/Θn)

∈n⅄ᐱ(En/An), ∈n⅄ᐱ(En/Bn), ∈n⅄ᐱ(En/Dn), ∈n⅄ᐱ(En/En), ∈n⅄ᐱ(En/Fn), ∈n⅄ᐱ(En/Gn), ∈n⅄ᐱ(En/Hn), ∈n⅄ᐱ(En/In), ∈n⅄ᐱ(En/Jn), ∈n⅄ᐱ(En/Kn), ∈n⅄ᐱ(En/Ln), ∈n⅄ᐱ(En/Mn), ∈n⅄ᐱ(En/Nn), ∈n⅄ᐱ(En/On), ∈n⅄ᐱ(En/Pn), ∈n⅄ᐱ(En/Qn), ∈n⅄ᐱ(En/Rn), ∈n⅄ᐱ(En/Sn), ∈n⅄ᐱ(En/Tn),∈n⅄ᐱ(En/Un),  ∈n⅄ᐱ(En/Vn), ∈n⅄ᐱ(En/Wn), ∈n⅄ᐱ(En/Yn), ∈n⅄ᐱ(En/Zn), ∈n⅄ᐱ(En/φn), ∈n⅄ᐱ(En/Θn)

∈n⅄ᐱ(Fn/An), ∈n⅄ᐱ(Fn/Bn), ∈n⅄ᐱ(Fn/Dn), ∈n⅄ᐱ(Fn/En), ∈n⅄ᐱ(Fn/Fn), ∈n⅄ᐱ(Fn/Gn), ∈n⅄ᐱ(Fn/Hn), ∈n⅄ᐱ(Fn/In), ∈n⅄ᐱ(Fn/Jn), ∈n⅄ᐱ(Fn/Kn), ∈n⅄ᐱ(Fn/Ln), ∈n⅄ᐱ(Fn/Mn), ∈n⅄ᐱ(Fn/Nn), ∈n⅄ᐱ(Fn/On), ∈n⅄ᐱ(Fn/Pn), ∈n⅄ᐱ(Fn/Qn), ∈n⅄ᐱ(Fn/Rn), ∈n⅄ᐱ(Fn/Sn), ∈n⅄ᐱ(Fn/Tn),∈n⅄ᐱ(Fn/Un),  ∈n⅄ᐱ(Fn/Vn), ∈n⅄ᐱ(Fn/Wn), ∈n⅄ᐱ(Fn/Yn), ∈n⅄ᐱ(Fn/Zn), ∈n⅄ᐱ(Fn/φn), ∈n⅄ᐱ(Fn/Θn)

∈n⅄ᐱ(Gn/An), ∈n⅄ᐱ(Gn/Bn), ∈n⅄ᐱ(Gn/Dn), ∈n⅄ᐱ(Gn/En), ∈n⅄ᐱ(Gn/Fn), ∈n⅄ᐱ(Gn/Gn), ∈n⅄ᐱ(Gn/Hn), ∈n⅄ᐱ(Gn/In), ∈n⅄ᐱ(Gn/Jn), ∈n⅄ᐱ(Gn/Kn), ∈n⅄ᐱ(Gn/Ln), ∈n⅄ᐱ(Gn/Mn), ∈n⅄ᐱ(Gn/Nn), ∈n⅄ᐱ(Gn/On), ∈n⅄ᐱ(Gn/Pn), ∈n⅄ᐱ(Gn/Qn), ∈n⅄ᐱ(Gn/Rn), ∈n⅄ᐱ(Gn/Sn), ∈n⅄ᐱ(Gn/Tn),∈n⅄ᐱ(Gn/Un),  ∈n⅄ᐱ(Gn/Vn), ∈n⅄ᐱ(Gn/Wn), ∈n⅄ᐱ(Gn/Yn), ∈n⅄ᐱ(Gn/Zn), ∈n⅄ᐱ(Gn/φn), ∈n⅄ᐱ(Gn/Θn)

∈n⅄ᐱ(Hn/An), ∈n⅄ᐱ(Hn/Bn), ∈n⅄ᐱ(Hn/Dn), ∈n⅄ᐱ(Hn/En), ∈n⅄ᐱ(Hn/Fn), ∈n⅄ᐱ(Hn/Gn), ∈n⅄ᐱ(Hn/Hn), ∈n⅄ᐱ(Hn/In), ∈n⅄ᐱ(Hn/Jn), ∈n⅄ᐱ(Hn/Kn), ∈n⅄ᐱ(Hn/Ln), ∈n⅄ᐱ(Hn/Mn), ∈n⅄ᐱ(Hn/Nn), ∈n⅄ᐱ(Hn/On), ∈n⅄ᐱ(Hn/Pn), ∈n⅄ᐱ(Hn/Qn), ∈n⅄ᐱ(Hn/Rn), ∈n⅄ᐱ(Hn/Sn), ∈n⅄ᐱ(Hn/Tn),∈n⅄ᐱ(Hn/Un),  ∈n⅄ᐱ(Hn/Vn), ∈n⅄ᐱ(Hn/Wn), ∈n⅄ᐱ(Hn/Yn), ∈n⅄ᐱ(Hn/Zn), ∈n⅄ᐱ(Hn/φn), ∈n⅄ᐱ(Hn/Θn)

∈n⅄ᐱ(In/An), ∈n⅄ᐱ(In/Bn), ∈n⅄ᐱ(In/Dn), ∈n⅄ᐱ(In/En), ∈n⅄ᐱ(In/Fn), ∈n⅄ᐱ(In/Gn), ∈n⅄ᐱ(In/Hn), ∈n⅄ᐱ(In/In), ∈n⅄ᐱ(In/Jn), ∈n⅄ᐱ(In/Kn), ∈n⅄ᐱ(In/Ln), ∈n⅄ᐱ(In/Mn), ∈n⅄ᐱ(In/Nn), ∈n⅄ᐱ(In/On), ∈n⅄ᐱ(In/Pn), ∈n⅄ᐱ(In/Qn), ∈n⅄ᐱ(In/Rn), ∈n⅄ᐱ(In/Sn), ∈n⅄ᐱ(In/Tn),∈n⅄ᐱ(In/Un),  ∈n⅄ᐱ(In/Vn), ∈n⅄ᐱ(In/Wn), ∈n⅄ᐱ(In/Yn), ∈n⅄ᐱ(In/Zn), ∈n⅄ᐱ(In/φn), ∈n⅄ᐱ(In/Θn)

∈n⅄ᐱ(Jn/An), ∈n⅄ᐱ(Jn/Bn), ∈n⅄ᐱ(Jn/Dn), ∈n⅄ᐱ(Jn/En), ∈n⅄ᐱ(Jn/Fn), ∈n⅄ᐱ(Jn/Gn), ∈n⅄ᐱ(Jn/Hn), ∈n⅄ᐱ(Jn/In), ∈n⅄ᐱ(Jn/Jn), ∈n⅄ᐱ(Jn/Kn), ∈n⅄ᐱ(Jn/Ln), ∈n⅄ᐱ(Jn/Mn), ∈n⅄ᐱ(Jn/Nn), ∈n⅄ᐱ(Jn/On), ∈n⅄ᐱ(Jn/Pn), ∈n⅄ᐱ(Jn/Qn), ∈n⅄ᐱ(Jn/Rn), ∈n⅄ᐱ(Jn/Sn), ∈n⅄ᐱ(Jn/Tn),∈n⅄ᐱ(Jn/Un),  ∈n⅄ᐱ(Jn/Vn), ∈n⅄ᐱ(Jn/Wn), ∈n⅄ᐱ(Jn/Yn), ∈n⅄ᐱ(Jn/Zn), ∈n⅄ᐱ(Jn/φn), ∈n⅄ᐱ(Jn/Θn)

∈n⅄ᐱ(Kn/An), ∈n⅄ᐱ(Kn/Bn), ∈n⅄ᐱ(Kn/Dn), ∈n⅄ᐱ(Kn/En), ∈n⅄ᐱ(Kn/Fn), ∈n⅄ᐱ(Kn/Gn), ∈n⅄ᐱ(Kn/Hn), ∈n⅄ᐱ(Kn/In), ∈n⅄ᐱ(Kn/Jn), ∈n⅄ᐱ(Kn/Kn), ∈n⅄ᐱ(Kn/Ln), ∈n⅄ᐱ(Kn/Mn), ∈n⅄ᐱ(Kn/Nn), ∈n⅄ᐱ(Kn/On), ∈n⅄ᐱ(Kn/Pn), ∈n⅄ᐱ(Kn/Qn), ∈n⅄ᐱ(Kn/Rn), ∈n⅄ᐱ(Kn/Sn), ∈n⅄ᐱ(Kn/Tn),∈n⅄ᐱ(Kn/Un),  ∈n⅄ᐱ(Kn/Vn), ∈n⅄ᐱ(Kn/Wn), ∈n⅄ᐱ(Kn/Yn), ∈n⅄ᐱ(Kn/Zn), ∈n⅄ᐱ(Kn/φn), ∈n⅄ᐱ(Kn/Θn)

∈n⅄ᐱ(Ln/An), ∈n⅄ᐱ(Ln/Bn), ∈n⅄ᐱ(Ln/Dn), ∈n⅄ᐱ(Ln/En), ∈n⅄ᐱ(Ln/Fn), ∈n⅄ᐱ(Ln/Gn), ∈n⅄ᐱ(Ln/Hn), ∈n⅄ᐱ(Ln/In), ∈n⅄ᐱ(Ln/Jn), ∈n⅄ᐱ(Ln/Kn), ∈n⅄ᐱ(Ln/Ln), ∈n⅄ᐱ(Ln/Mn), ∈n⅄ᐱ(Ln/Nn), ∈n⅄ᐱ(Ln/On), ∈n⅄ᐱ(Ln/Pn), ∈n⅄ᐱ(Ln/Qn), ∈n⅄ᐱ(Ln/Rn), ∈n⅄ᐱ(Ln/Sn), ∈n⅄ᐱ(Ln/Tn),∈n⅄ᐱ(Ln/Un),  ∈n⅄ᐱ(Ln/Vn), ∈n⅄ᐱ(Ln/Wn), ∈n⅄ᐱ(Ln/Yn), ∈n⅄ᐱ(Ln/Zn), ∈n⅄ᐱ(Ln/φn), ∈n⅄ᐱ(Ln/Θn)

∈n⅄ᐱ(Mn/An), ∈n⅄ᐱ(Mn/Bn), ∈n⅄ᐱ(Mn/Dn), ∈n⅄ᐱ(Mn/En), ∈n⅄ᐱ(Mn/Fn), ∈n⅄ᐱ(Mn/Gn), ∈n⅄ᐱ(Mn/Hn), ∈n⅄ᐱ(Mn/In), ∈n⅄ᐱ(Mn/Jn), ∈n⅄ᐱ(Mn/Kn), ∈n⅄ᐱ(Mn/Ln), ∈n⅄ᐱ(Mn/Mn), ∈n⅄ᐱ(Mn/Nn), ∈n⅄ᐱ(Mn/On), ∈n⅄ᐱ(Mn/Pn), ∈n⅄ᐱ(Mn/Qn), ∈n⅄ᐱ(Mn/Rn), ∈n⅄ᐱ(Mn/Sn), ∈n⅄ᐱ(Mn/Tn),∈n⅄ᐱ(Mn/Un),  ∈n⅄ᐱ(Mn/Vn), ∈n⅄ᐱ(Mn/Wn), ∈n⅄ᐱ(Mn/Yn), ∈n⅄ᐱ(Mn/Zn), ∈n⅄ᐱ(Mn/φn), ∈n⅄ᐱ(Mn/Θn)

∈n⅄ᐱ(Nn/An), ∈n⅄ᐱ(Nn/Bn), ∈n⅄ᐱ(Nn/Dn), ∈n⅄ᐱ(Nn/En), ∈n⅄ᐱ(Nn/Fn), ∈n⅄ᐱ(Nn/Gn), ∈n⅄ᐱ(Nn/Hn), ∈n⅄ᐱ(Nn/In), ∈n⅄ᐱ(Nn/Jn), ∈n⅄ᐱ(Nn/Kn), ∈n⅄ᐱ(Nn/Ln), ∈n⅄ᐱ(Nn/Mn), ∈n⅄ᐱ(Nn/Nn), ∈n⅄ᐱ(Nn/On), ∈n⅄ᐱ(Nn/Pn), ∈n⅄ᐱ(Nn/Qn), ∈n⅄ᐱ(Nn/Rn), ∈n⅄ᐱ(Nn/Sn), ∈n⅄ᐱ(Nn/Tn),∈n⅄ᐱ(Nn/Un),  ∈n⅄ᐱ(Nn/Vn), ∈n⅄ᐱ(Nn/Wn), ∈n⅄ᐱ(Nn/Yn), ∈n⅄ᐱ(Nn/Zn), ∈n⅄ᐱ(Nn/φn), ∈n⅄ᐱ(Nn/Θn)

∈n⅄ᐱ(On/An), ∈n⅄ᐱ(On/Bn), ∈n⅄ᐱ(On/Dn), ∈n⅄ᐱ(On/En), ∈n⅄ᐱ(On/Fn), ∈n⅄ᐱ(On/Gn), ∈n⅄ᐱ(On/Hn), ∈n⅄ᐱ(On/In), ∈n⅄ᐱ(On/Jn), ∈n⅄ᐱ(On/Kn), ∈n⅄ᐱ(On/Ln), ∈n⅄ᐱ(On/Mn), ∈n⅄ᐱ(On/Nn), ∈n⅄ᐱ(On/On), ∈n⅄ᐱ(On/Pn), ∈n⅄ᐱ(On/Qn), ∈n⅄ᐱ(On/Rn), ∈n⅄ᐱ(On/Sn), ∈n⅄ᐱ(On/Tn),∈n⅄ᐱ(On/Un),  ∈n⅄ᐱ(On/Vn), ∈n⅄ᐱ(On/Wn), ∈n⅄ᐱ(On/Yn), ∈n⅄ᐱ(On/Zn), ∈n⅄ᐱ(On/φn), ∈n⅄ᐱ(On/Θn)

∈n⅄ᐱ(Pn/An), ∈n⅄ᐱ(Pn/Bn), ∈n⅄ᐱ(Pn/Dn), ∈n⅄ᐱ(Pn/En), ∈n⅄ᐱ(Pn/Fn), ∈n⅄ᐱ(Pn/Gn), ∈n⅄ᐱ(Pn/Hn), ∈n⅄ᐱ(Pn/In), ∈n⅄ᐱ(Pn/Jn), ∈n⅄ᐱ(Pn/Kn), ∈n⅄ᐱ(Pn/Ln), ∈n⅄ᐱ(Pn/Mn), ∈n⅄ᐱ(Pn/Nn), ∈n⅄ᐱ(Pn/On), ∈n⅄ᐱ(Pn/Pn), ∈n⅄ᐱ(Pn/Qn), ∈n⅄ᐱ(Pn/Rn), ∈n⅄ᐱ(Pn/Sn), ∈n⅄ᐱ(Pn/Tn),∈n⅄ᐱ(Pn/Un),  ∈n⅄ᐱ(Pn/Vn), ∈n⅄ᐱ(Pn/Wn), ∈n⅄ᐱ(Pn/Yn), ∈n⅄ᐱ(Pn/Zn), ∈n⅄ᐱ(Pn/φn), ∈n⅄ᐱ(Pn/Θn)

∈n⅄ᐱ(Qn/An), ∈n⅄ᐱ(Qn/Bn), ∈n⅄ᐱ(Qn/Dn), ∈n⅄ᐱ(Qn/En), ∈n⅄ᐱ(Qn/Fn), ∈n⅄ᐱ(Qn/Gn), ∈n⅄ᐱ(Qn/Hn), ∈n⅄ᐱ(Qn/In), ∈n⅄ᐱ(Qn/Jn), ∈n⅄ᐱ(Qn/Kn), ∈n⅄ᐱ(Qn/Ln), ∈n⅄ᐱ(Qn/Mn), ∈n⅄ᐱ(Qn/Nn), ∈n⅄ᐱ(Qn/On), ∈n⅄ᐱ(Qn/Pn), ∈n⅄ᐱ(Qn/Qn), ∈n⅄ᐱ(Qn/Rn), ∈n⅄ᐱ(Qn/Sn), ∈n⅄ᐱ(Qn/Tn),∈n⅄ᐱ(Qn/Un),  ∈n⅄ᐱ(Qn/Vn), ∈n⅄ᐱ(Qn/Wn), ∈n⅄ᐱ(Qn/Yn), ∈n⅄ᐱ(Qn/Zn), ∈n⅄ᐱ(Qn/φn), ∈n⅄ᐱ(Qn/Θn)

∈n⅄ᐱ(Rn/An), ∈n⅄ᐱ(Rn/Bn), ∈n⅄ᐱ(Rn/Dn), ∈n⅄ᐱ(Rn/En), ∈n⅄ᐱ(Rn/Fn), ∈n⅄ᐱ(Rn/Gn), ∈n⅄ᐱ(Rn/Hn), ∈n⅄ᐱ(Rn/In), ∈n⅄ᐱ(Rn/Jn), ∈n⅄ᐱ(Rn/Kn), ∈n⅄ᐱ(Rn/Ln), ∈n⅄ᐱ(Rn/Mn), ∈n⅄ᐱ(Rn/Nn), ∈n⅄ᐱ(Rn/On), ∈n⅄ᐱ(Rn/Pn), ∈n⅄ᐱ(Rn/Qn), ∈n⅄ᐱ(Rn/Rn), ∈n⅄ᐱ(Rn/Sn), ∈n⅄ᐱ(Rn/Tn),∈n⅄ᐱ(Rn/Un),  ∈n⅄ᐱ(Rn/Vn), ∈n⅄ᐱ(Rn/Wn), ∈n⅄ᐱ(Rn/Yn), ∈n⅄ᐱ(Rn/Zn), ∈n⅄ᐱ(Rn/φn), ∈n⅄ᐱ(Rn/Θn)

∈n⅄ᐱ(Sn/An), ∈n⅄ᐱ(Sn/Bn), ∈n⅄ᐱ(Sn/Dn), ∈n⅄ᐱ(Sn/En), ∈n⅄ᐱ(Sn/Fn), ∈n⅄ᐱ(Sn/Gn), ∈n⅄ᐱ(Sn/Hn), ∈n⅄ᐱ(Sn/In), ∈n⅄ᐱ(Sn/Jn), ∈n⅄ᐱ(Sn/Kn), ∈n⅄ᐱ(Sn/Ln), ∈n⅄ᐱ(Sn/Mn), ∈n⅄ᐱ(Sn/Nn), ∈n⅄ᐱ(Sn/On), ∈n⅄ᐱ(Sn/Pn), ∈n⅄ᐱ(Sn/Qn), ∈n⅄ᐱ(Sn/Rn), ∈n⅄ᐱ(Sn/Sn), ∈n⅄ᐱ(Sn/Tn),∈n⅄ᐱ(Sn/Un),  ∈n⅄ᐱ(Sn/Vn), ∈n⅄ᐱ(Sn/Wn), ∈n⅄ᐱ(Sn/Yn), ∈n⅄ᐱ(Sn/Zn), ∈n⅄ᐱ(Sn/φn), ∈n⅄ᐱ(Sn/Θn)

∈n⅄ᐱ(Tn/An), ∈n⅄ᐱ(Tn/Bn), ∈n⅄ᐱ(Tn/Dn), ∈n⅄ᐱ(Tn/En), ∈n⅄ᐱ(Tn/Fn), ∈n⅄ᐱ(Tn/Gn), ∈n⅄ᐱ(Tn/Hn), ∈n⅄ᐱ(Tn/In), ∈n⅄ᐱ(Tn/Jn), ∈n⅄ᐱ(Tn/Kn), ∈n⅄ᐱ(Tn/Ln), ∈n⅄ᐱ(Tn/Mn), ∈n⅄ᐱ(Tn/Nn), ∈n⅄ᐱ(Tn/On), ∈n⅄ᐱ(Tn/Pn), ∈n⅄ᐱ(Tn/Qn), ∈n⅄ᐱ(Tn/Rn), ∈n⅄ᐱ(Tn/Sn), ∈n⅄ᐱ(Tn/Tn),∈n⅄ᐱ(Tn/Un),  ∈n⅄ᐱ(Tn/Vn), ∈n⅄ᐱ(Tn/Wn), ∈n⅄ᐱ(Tn/Yn), ∈n⅄ᐱ(Tn/Zn), ∈n⅄ᐱ(Tn/φn), ∈n⅄ᐱ(Tn/Θn)

∈n⅄ᐱ(Un/An), ∈n⅄ᐱ(Un/Bn), ∈n⅄ᐱ(Un/Dn), ∈n⅄ᐱ(Un/En), ∈n⅄ᐱ(Un/Fn), ∈n⅄ᐱ(Un/Gn), ∈n⅄ᐱ(Un/Hn), ∈n⅄ᐱ(Un/In), ∈n⅄ᐱ(Un/Jn), ∈n⅄ᐱ(Un/Kn), ∈n⅄ᐱ(Un/Ln), ∈n⅄ᐱ(Un/Mn), ∈n⅄ᐱ(Un/Nn), ∈n⅄ᐱ(Un/On), ∈n⅄ᐱ(Un/Pn), ∈n⅄ᐱ(Un/Qn), ∈n⅄ᐱ(Un/Rn), ∈n⅄ᐱ(Un/Sn), ∈n⅄ᐱ(Un/Tn),∈n⅄ᐱ(Un/Un),  ∈n⅄ᐱ(Un/Vn), ∈n⅄ᐱ(Un/Wn), ∈n⅄ᐱ(Un/Yn), ∈n⅄ᐱ(Un/Zn), ∈n⅄ᐱ(Un/φn), ∈n⅄ᐱ(Un/Θn)

∈n⅄ᐱ(Vn/An), ∈n⅄ᐱ(Vn/Bn), ∈n⅄ᐱ(Vn/Dn), ∈n⅄ᐱ(Vn/En), ∈n⅄ᐱ(Vn/Fn), ∈n⅄ᐱ(Vn/Gn), ∈n⅄ᐱ(Vn/Hn), ∈n⅄ᐱ(Vn/In), ∈n⅄ᐱ(Vn/Jn), ∈n⅄ᐱ(Vn/Kn), ∈n⅄ᐱ(Vn/Ln), ∈n⅄ᐱ(Vn/Mn), ∈n⅄ᐱ(Vn/Nn), ∈n⅄ᐱ(Vn/On), ∈n⅄ᐱ(Vn/Pn), ∈n⅄ᐱ(Vn/Qn), ∈n⅄ᐱ(Vn/Rn), ∈n⅄ᐱ(Vn/Sn), ∈n⅄ᐱ(Vn/Tn),∈n⅄ᐱ(Vn/Un),  ∈n⅄ᐱ(Vn/Vn), ∈n⅄ᐱ(Vn/Wn), ∈n⅄ᐱ(Vn/Yn), ∈n⅄ᐱ(Vn/Zn), ∈n⅄ᐱ(Vn/φn), ∈n⅄ᐱ(Vn/Θn)

∈n⅄ᐱ(Wn/An), ∈n⅄ᐱ(Wn/Bn), ∈n⅄ᐱ(Wn/Dn), ∈n⅄ᐱ(Wn/En), ∈n⅄ᐱ(Wn/Fn), ∈n⅄ᐱ(Wn/Gn), ∈n⅄ᐱ(Wn/Hn), ∈n⅄ᐱ(Wn/In), ∈n⅄ᐱ(Wn/Jn), ∈n⅄ᐱ(Wn/Kn), ∈n⅄ᐱ(Wn/Ln), ∈n⅄ᐱ(Wn/Mn), ∈n⅄ᐱ(Wn/Nn), ∈n⅄ᐱ(Wn/On), ∈n⅄ᐱ(Wn/Pn), ∈n⅄ᐱ(Wn/Qn), ∈n⅄ᐱ(Wn/Rn), ∈n⅄ᐱ(Wn/Sn), ∈n⅄ᐱ(Wn/Tn),∈n⅄ᐱ(Wn/Un),  ∈n⅄ᐱ(Wn/Vn), ∈n⅄ᐱ(Wn/Wn), ∈n⅄ᐱ(Wn/Yn), ∈n⅄ᐱ(Wn/Zn), ∈n⅄ᐱ(Wn/φn), ∈n⅄ᐱ(Wn/Θn)

∈n⅄ᐱ(Yn/An), ∈n⅄ᐱ(Yn/Bn), ∈n⅄ᐱ(Yn/Dn), ∈n⅄ᐱ(Yn/En), ∈n⅄ᐱ(Yn/Fn), ∈n⅄ᐱ(Yn/Gn), ∈n⅄ᐱ(Yn/Hn), ∈n⅄ᐱ(Yn/In), ∈n⅄ᐱ(Yn/Jn), ∈n⅄ᐱ(Yn/Kn), ∈n⅄ᐱ(Yn/Ln), ∈n⅄ᐱ(Yn/Mn), ∈n⅄ᐱ(Yn/Nn), ∈n⅄ᐱ(Yn/On), ∈n⅄ᐱ(Yn/Pn), ∈n⅄ᐱ(Yn/Qn), ∈n⅄ᐱ(Yn/Rn), ∈n⅄ᐱ(Yn/Sn), ∈n⅄ᐱ(Yn/Tn),∈n⅄ᐱ(Yn/Un),  ∈n⅄ᐱ(Yn/Vn), ∈n⅄ᐱ(Yn/Wn), ∈n⅄ᐱ(Yn/Yn), ∈n⅄ᐱ(Yn/Zn), ∈n⅄ᐱ(Yn/φn), ∈n⅄ᐱ(Yn/Θn)

∈n⅄ᐱ(Zn/An), ∈n⅄ᐱ(Zn/Bn), ∈n⅄ᐱ(Zn/Dn), ∈n⅄ᐱ(Zn/En), ∈n⅄ᐱ(Zn/Fn), ∈n⅄ᐱ(Zn/Gn), ∈n⅄ᐱ(Zn/Hn), ∈n⅄ᐱ(Zn/In), ∈n⅄ᐱ(Zn/Jn), ∈n⅄ᐱ(Zn/Kn), ∈n⅄ᐱ(Zn/Ln), ∈n⅄ᐱ(Zn/Mn), ∈n⅄ᐱ(Zn/Nn), ∈n⅄ᐱ(Zn/On), ∈n⅄ᐱ(Zn/Pn), ∈n⅄ᐱ(Zn/Qn), ∈n⅄ᐱ(Zn/Rn), ∈n⅄ᐱ(Zn/Sn), ∈n⅄ᐱ(Zn/Tn),∈n⅄ᐱ(Zn/Un),  ∈n⅄ᐱ(Zn/Vn), ∈n⅄ᐱ(Zn/Wn), ∈n⅄ᐱ(Zn/Yn), ∈n⅄ᐱ(Zn/Zn), ∈n⅄ᐱ(Zn/φn), ∈n⅄ᐱ(Zn/Θn)

∈n⅄ᐱ(φn/An), ∈n⅄ᐱ(φn/Bn), ∈n⅄ᐱ(φn/Dn), ∈n⅄ᐱ(φn/En), ∈n⅄ᐱ(φn/Fn), ∈n⅄ᐱ(φn/Gn), ∈n⅄ᐱ(φn/Hn), ∈n⅄ᐱ(φn/In), ∈n⅄ᐱ(φn/Jn), ∈n⅄ᐱ(φn/Kn), ∈n⅄ᐱ(φn/Ln), ∈n⅄ᐱ(φn/Mn), ∈n⅄ᐱ(φn/Nn), ∈n⅄ᐱ(φn/On), ∈n⅄ᐱ(φn/Pn), ∈n⅄ᐱ(φn/Qn), ∈n⅄ᐱ(φn/Rn), ∈n⅄ᐱ(φn/Sn), ∈n⅄ᐱ(φn/Tn),∈n⅄ᐱ(φn/Un),  ∈n⅄ᐱ(φn/Vn), ∈n⅄ᐱ(φn/Wn), ∈n⅄ᐱ(φn/Yn), ∈n⅄ᐱ(φn/Zn), ∈n⅄ᐱ(φn/φn), ∈n⅄ᐱ(φn/Θn)

∈n⅄ᐱ(Θn/An), ∈n⅄ᐱ(Θn/Bn), ∈n⅄ᐱ(Θn/Dn), ∈n⅄ᐱ(Θn/En), ∈n⅄ᐱ(Θn/Fn), ∈n⅄ᐱ(Θn/Gn), ∈n⅄ᐱ(Θn/Hn), ∈n⅄ᐱ(Θn/In), ∈n⅄ᐱ(Θn/Jn), ∈n⅄ᐱ(Θn/Kn), ∈n⅄ᐱ(Θn/Ln), ∈n⅄ᐱ(Θn/Mn), ∈n⅄ᐱ(Θn/Nn), ∈n⅄ᐱ(Θn/On), ∈n⅄ᐱ(Θn/Pn), ∈n⅄ᐱ(Θn/Qn), ∈n⅄ᐱ(Θn/Rn), ∈n⅄ᐱ(Θn/Sn), ∈n⅄ᐱ(Θn/Tn),∈n⅄ᐱ(Θn/Un),  ∈n⅄ᐱ(Θn/Vn), ∈n⅄ᐱ(Θn/Wn), ∈n⅄ᐱ(Θn/Yn), ∈n⅄ᐱ(Θn/Zn), ∈n⅄ᐱ(Θn/φn), ∈n⅄ᐱ(Θn/Θn)

With 27 base sets to each set of ᐱ have 1⅄, 2⅄, 3⅄ paths occur with infinite library potential each that then 81 basic sets have other potential functions of X, 1+⅄, 1-⅄, 2-⅄, ⅄n from Y, P, N, whole number fractals in quantum field fractal polarization.

27 libraries multiplied by 8 basic function potentials multiplied by 27 library variables equals 5832 base libraries each with infinite numeral library stacking to infinite variables more give infinite cycles of a cn in those ratios. 

5,832 library base paths of 27 sets ᐱ with 8 functions applicable to  1⅄, 2⅄, 3⅄, X, 1+⅄, 1-⅄, 2-⅄, ⅄n , 3 libraries that all 5,832 libraries align to are ∈ᐱ of (N) and ∈ᐱ of (Y) and ∈ᐱ of (P).

∈ᐱ of (P) is a function of a complex ratio with a prime number.

∈ᐱ of (Y) is a function of a complex ratio with a fibonacci number.

∈ᐱ of (N) is a function of a complex ratio with a number.

Another array of libraries are factorable of variables such as 2φ from 1⅄1φ for example to all sets and paths of the 5,832 library bases and more just as are array sets of variant paths for 3φ from 1⅄2φ and so on for all sets ∈ᐱ of A,B,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z,φ,Θ variables and paths 1⅄, 2⅄, 3⅄, X, 1+⅄, 1-⅄, 2-⅄, ⅄n and so on.

This library is a base logic of sets ∈ᐱ prior to factoring variable sets for a library of ∈(ᗑ) variables.

Numeral Array logic symbols to use of next tier alt paths and stem cell cycle notations 

ᗑ for ∈(⅄ᗑ ) applicable to variable sets ᐱ,A,B,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z,φ,Θ

ᗑ=(ᐱ/ᐱ) for paths 1⅄, 2⅄, and 3⅄ of ᗑ variables

∈(ᗑ)=∈1(⅄ᗑ) and ∈2(⅄ᗑ) and ∈3(⅄ᗑ) of variables ⅄ᐱ, ⅄A , ⅄M, ⅄V, ⅄W, A, M, V, W, φ, Θ, Q, Y, P  and whole numbers or a variable factors noted that ∈(ᐱ) is applicable with in quanta such that ∈(ᗑ)=(ᐱ/ᐱ) variables defined before a variable 

n⅄ncn=(⅄ᗑ)(ncn)ncn 

or 

n⅄ncn=(⅄ᐱ)(ncn)ncn 

are able to be calculated dependent on ncn definitions to φ, Θ, Q variables.

Given that these symbols represent a base of 10 variable tiers a categorizing library could then be devised for continued factoring.

The order of these from base numeral whole numbers is Y and φ path and P and Q path then A and M and V and W then E and F and I and H then ᐱ then ᗑ paths ⅄.

⅄ represents alternate path factoring of the variables 1⅄, 2⅄, 3⅄ncn.

Logical Complex Variable Notation

1⅄(⅄ᗑncn)/(⅄ᐱncn)

2⅄(⅄ᐱncn)/(⅄ᗑncn)

3⅄(1⅄ᐱncn)/(1⅄ᐱncn)

3⅄(2⅄ᐱncn)/(2⅄ᐱncn)

3⅄(3⅄ᐱncn)/(3⅄ᐱncn)

3⅄(1⅄ᗑncn) /(1⅄ᗑncn)

3⅄(2⅄ᗑncn) /(2⅄ᗑncn) 

3⅄(3⅄ᗑncn) /(3⅄ᗑncn) 

∀ refers to all or for any and when applied with notation for path ⅄ of consecutive variables of sets ∈ and variables of not same sets ∉.

Given ∀ represents for any and ⅄ represents a function of sequential variables of sets ∈ variable number N or n.

then

1⅄∀2nd=(n3/n1)

2⅄∀2nd=(n1/n3)

and

1⅄∀3rd=(n4/n1)

2⅄∀3rd=(n1/n4)

and so on for functions of 1⅄∀ and 2⅄∀ variables.

Path functions +⅄, 1-⅄, 2-⅄, and X of ∀ function to set ∈ consecutive variables Nncn

+⅄∀ then are

+⅄∀2nd=(n3+n1)

+⅄∀2nd=(n1+n3)

and

+⅄∀3rd=(n4+n1)

+⅄∀3rd=(n1+n4)

and so on for functions of +⅄∀ variables.

1-⅄∀ then are

1-⅄∀2nd=(n3-n1)

and

1-⅄∀3rd=(n4-n1)

and so on for functions of 1-⅄∀ variables.

2-⅄∀ then are

2-⅄∀2nd=(n1-n3)

and

2-⅄∀3rd=(n1-n4)