Some  elementary  initial  conditions  for  Francom Adjacency

 We define the Euler-Riemann Summation, which defines the 'Mixing of the Count' in 
linking  Arithmetic Progression to the multiplicative Factorial Function '!'.

Define Eo=0 as the singularity (interval), then for any integer n, we find for the Harmonic Form of
Riemann's Zeta-Function  (z=k=constant): 
                                                                    z(z)=S(1/n^z)=1/1^k+1/2^k+1/3^k+1/4^k +...+1/n^k

This Sum diverges for [ 0<k<1], i.e.  for k=1/2:  {1+Sqrt(2)/2+Sqrt(3)/3+...+Sqrt(n)/n}  increases without limit.

For k>1, we have convergence, however:
Formally, let:  S1/n^p=1^-p+2^-p+3^-p+...
For even terms   2.2^-p >= 2^-p+3^-p  for a geometric series  1^1-p+2^1-p+4^1-p+...+(2^n-1)^1-p

This Geometric Progression sums to:  [1-(2^1-p)^n]/[1-2^1-p]=1/[1-2^1-p]
So for p=2, this limit maximises in 1/(1-1/2)=2 , and for p=3 it becomes 4/3 converging towards
1 for increasing p.

We consider the special case for p=1 applied to the Singularity Interval Eo.

Define: for a nth term (numerator):                                     T^k(En) = n^k.T^k(En-1) + [(n-1)!]^k             
               for the nth sum per n (denominator [n!]^k):           S^k(En) = T^k(En)/(n!)^k 
                           
 {Here the suffices  n,n-1 etc. denote subscript}.

T^1(E1)=1/1=1.0+0!=1=S^1(E1)=1/1!=1;   T^2(E2)=2.T^1(E1)+1!=2+1=3  with  S^2(E2)=T^2(E2)/2!=3/2=1+1/2;

T^3(E3)=3.T^2(E2)+2!=9+2=11  with  S^3(E3)=T^3(E3)/3!=11/6=1+1/2+1/3=1+5/6  and so on.

Further Example:  T^1(4)=4.11+3!=50;   S^1(4)=50/4!=25/12 for the nesting:  4{3(2+1!)2!}3!  with [4!]^1=24.

For 4 terms, the Euler-Riemann Summation so is:      S^1(4)=1+1/2+1/3+1/4=25/12=2+1/12.

For 7 terms, S^1(7)=T^1(7)/7!=(7.T^1(6)+6!)/7!=13068/5040=363.36/(140.36)=2+83/140=1+1/2+1/3+1/4+1/5+1/6+1/7.

 Project the Numberline with the Positive Integers mapping the Factorial-Function and the Negative Integers 
remaining invariant in Feyman Summation T(n)=n(n+1)/2 as absolute value, mirroring the positive integers.

   (n!)<---4...3...2...[Eo]...1...2...3---> (n);   where Integer 1 maps 2! in suppression of -1=2* and in algoradius eo=1.

Similarly, Integer 2 maps 3! in suppression of -2=3* and algoradius e1=2=2eo, etc. etc.

The singularity so mixes the interval [0!-1!]=[-1,0] with Functional-Riemann-Bound (FRB=-1/2) becoming 'real'
in its mapping (FRB'=1/2) in [0,1] and the central limit or pole, about which the Zero's of the Riemann-Zeta-
Function propagate.

The first annulus in the Riemann-Euler-Harmonic so phasemixes the numbers 2 and 1 and the nth number is 
mixed with (n+1) as crystallised in the Feynman-Path-Integral or T(n)=1 in n(n+1), as a summation for all pos-
sible particular histories in quantum mechanics.

This also maps the series:  SEps=Fibonacci#1=0,1,1,2,3,5,8,.....for a nth Term: Tn=[]-Y^n - X^n[]/Sqrt(5), for
absolute value []...[] and obtained say via MacLaurin-Expansion of the coefficients (Experience-Factors)
in the power series:

                                f(x)=1+x+2x^2+3x^3+...= STn.x^n-1

Set    x.f(x) + x^2.f(x) = f(x) -1,  then by (a+b)(a-b)  f(x)=a/(x-X) + b/(x-Y)  for a=-b=1/(Y-X) and (Y-X)=-Sqrt(5).

SuperSEps=Fibonacci#2=2,1,3,4,7,11,18,29,....  for a nth Term:  STn=[]-Y^2n - X^2n[]/[]-Y^n - X^n[]=[]T2n/Tn[]

                                                                                 for n=1,2,3,...; T(2n=0)=2 mapping T(n=0)=0.

The combined SEps-SuperSEps  (T-ST)-sequence of experience factors {from the triplet propagation of
[OldState, Experience, NewState]} can then be written as:
{Tn,STn}={(So=0,STo=2=S3); (S1=1=ST1=S2); (S2,S4=3=ST2); (S3,ST3=4); (S4=ST2,ST4=7); (S5,ST5)..(Sn,STn)..} 

{Tn,STn}={(0,2), (1,1), (1,3), (2,4), (3,7), (5,11),...} containing integerset:  {0,1,2,3,4,5,7,8,11,13,18,21,29,....}

We now represent the mappings in matrix form denoted as F-M-C, where the 'well behaved' terms 
for the mapping (from {T5,ST5}) sets algorithmic C-Space and  the preceding elements the
initialisation for the former.

Note we define Cantorian Denumerability Aleph-Null in Cardinality Aleph-All in the form:
Aleph-Null:  limit{n=>Infinity}[T(n)]=Infinity
Aleph-All:   limit{n=>X}[T(n)]=1     and counting Infinities as mapped one-to-one onto the positive Integer set.

                       SEps=Fibonacci#1                    maps                      SuperSEps=Fibonacci#2
                  .       .     .                                                                    .      .      .       .
                  .       .     0                                                                   .      .      .       .
                  0      0    0*                                                                 -4     7     3     n=-2=2i^2         
FSpace     0*    0     1     n=Infinity via 0+0=Infinity=1*=0*=1          3   -4    -1      n=-1=i^2    [a+ib complex upwards]
MSpace     1      0*   1     n=0  via  (1,1,1)                                    -1     3      2*    n=0
                  1      1     2     n=1  via  (1,1,10=2*=0/0=1*)                    2*  -1     1      n=0      (Reflection-Interval)
                  2      1     3     n=2   well behaved                                            1     2*   3       n=0                                             
CSpace    3      2     5     n=3   well behaved                                             3     1     4      n=1       well behaved
                  5      3     8     n=4   well behaved                                             4     3     7      n=2 
                  .       .      .      n=5   continue downwards                               .       .      .     n=3

The linearity of the generating triplet configurations is extended in a complexification into a 2D symmetry.
SEps propagates the Experience Factors in an adjacent displacement of 1, in moving from one configuration
state to the next - this is termed Francom Adjacency.
[0*,1,1,2,3,5,...] as OldStates  transfigure in Experiences [0,0*,1,1,2,3,5,8,...] into NewStates [1,1,2,3,5,8,...].

This algorithmic configuration space is however broken in the mapping onto SuperSEps.
Here the matching 'good behaviour' of the n-count is delayed in a factor of 2 in a 'reflection interval'.

  Algorithmic modelling for this Francom Adjacency must generate the mapping of SEps onto SuperSEps in
an  geometry of the pentagonal symmetries intrisic to the two series.
Hence a  synthesis between linear propagation about an internal spiralling form is necessitated.
A longrange rotational- and a longrange translational order for the Experienc-Factors is indicated in
the geometry of say Penrosian Tiling Patterns and the Schechtmanite Quasicrystals  of empirical
form (Mg(32)[Al,Zn](49)).
The general form,  (physically{ akin to the propagation of magnetic fields) is the reduction of physical
parameters to a state of information transmission, say  in the data transfer between two neighbouring
cells in mitosis and neuronal-synaptic processing.

 A general modality for the cosmogenetic reproduction on all levels must crystallise, should the
matrices above become sufficiently deciphered from their algorithmic encoding.

Derivation of SuperSeps

The relative primeness of the Fibonacci Numbers allows a one-to-one mapping between the SEps-Set and
other such sets derived from it.

A logical derived set of such a nature is the sequence: 2,1,3,4,7,11,18,29,....
All adjacent members  of this set are relatively prime to each other.
7 is relatively prime to both 4 and 11 (no common divisors except 1) and 11 is relativelyprime to both 7 and 18.

We now tabulate the sums and differences in our nth-term definition for SEps, so recalling the propagation for
the natural numbers in counter n:

n         Tn            (X)^n                                 (-Y)^n                         {[]-Y[]^n  +  []X[]^n}                 {[]-Y[]^n  -  []X[]^n}  
 0            0          1                                      +1                                       +2                                       0
 1            1          0.6180339885               -1.618033989                 +2.236067978                            +1
 2            1          0.3819660109              +2.618033989                 +3                                     +2.236067978
 3            2          0.2360679772               -4.236067979                 +4.472135956                            +4
 4            3          0.1458980335              +6.854101970                 +7                                     +6.708203937
 5            5          0.0901699436             -11.09016995                 +11.18033989                            +11
 6            8          0.0557280899              17.94427193                 +18                                    +17.88854384
 7          13          0.0344418537             -29.03444189                 +29.06888374                            +29  
 8          21          0.0212862362            +46.97871382                 +47                                    +46.95742758 
 9          34          0.0131556175             -76.01315572                 +76.02631134                             +76
10         55          0.0081306187          +122.9918696                 +123                                  +122.983739
...           ..            ....................                  ....................                        ....................                                  ...................
20    6765          0.0000661070           +15126.99998            +15127                                +15126.99991
...       .......            .........................           ..............................          .....................................                     ....................................

We see that for increasing n, the absolute magnitude for Y converges to an integral value in the Sum {+},
but only for even n.
For odd n, the difference Sum {-} gives a specific integer for specific n.

The product of the two sums is:   {+}.{-} = []-Y[]^2n - []X[]^2n=Sqrt(5).T2n .

The sum of the two sums is:         {+}+{-}= 2[]-Y[]^n , with STn ={+}+{-} - Tn.Sqrt(5) = [](-Y)^n +(X)^n[]

Multiplying each term as:  Sqrt(5).({+}+{-}), we can form the alternating series:

(0+2.Sqrt(5)), (5+1.Sqrt(5)), (5+3.Sqrt(5)), (10+4.Sqrt(5)), (15+7.Sqrt(5)), .....as the alternating form of
SuperSEps, given in the term:
                                                   [5.Tn + Sqrt(5).T'n],  

But for even n, we have:   T'n ={+} and for odd n, we have T'n ={-};  then by (a-b)(a+b)=a^2-b^2 :

STn.Sqrt(5).Tn={+}.{-}=Sqrt(5).T2n  &  STn=T2n/Tn  = []-Y^2n  - X^2n[]/[]-Y^n - X^n[]   (quod erat demonstrandum). 

The significance of this result is that STn, T2n and Tn are all integers.
We so have a primary  extension for SEps with elements 1, 2 and 3 duplicated and resulting in the mappings
as  previously specified.
The Null-Initialisation (OSj, EXj, NSj) as the Fibonacci-Triplet (An-1, An, An+1) then reflects STn about n=0*
to define the complex number set as negative STn's mapped in a 0=>1=>Infinity correspondence to Tn.

This is the mathematical mapping of Cantorian Enumerability as previously indicated.

Pacificap@hotmail.com                              Tony  Bermanseder