Quantum Gravity Via Expansion-Contraction 22.1 More About Cyclotomic and Umbral "Observer"/Perception

From Osher Doctorow mdoctorow@xxxxxxxxxxx

Let's look at the equation:

1) y^n - x^n = (y - x)(y^(n-1) + y^(n-2)x + ... + x^(n-2)y + x^(n-1))

for positive integer n greater than 1 (regarding y^2 - x^2 = (y - x)(y
+ x) as a special case) from the viewpoint mentioned in Section 20.0
(typed 20 there) that additive sums of positive quantities express
"and/or" while alternating positive and negative quantities express
(probable) cause-effect with negative-signed quantities causes and
positive-signed quantities effects in the latter scenario.

The long factor on the right hand side of (1), from this viewpoint,
means that y^k x^j for k and j between 0 and n - 1 (except that one of
y^k or x^j always has an exponent of 1 or more) goes through various
"and-or" combinations in generating the result. But the expression
that we're examining, y^k x^j, is multiplicative, and earlier in this
thread I pointed out that real multiplication reduces in
probability-statistics to "independence". So (1) says that to generate
y^n - x^n, we generate an influence of x on y (which is y - x) and
multiply it by the "and/or" sum of independent y^k x^j terms with lower
than n positive integer exponents.

This is arguably deserving of a "summation convention" of its own,
since as written in (1) we lose track of the underlying independence in
the general term y^k x^j and the "and/or" relationship between
different finite "realizations" of this general term. But I won't
pursue this just now.

Let me emphasize that algebraic expressions which select different
"realizations" or "instantiations" of statistically independent
variables correspond to selecting different finite paths in a graph,
and that the question still remains about infinite quantities that
don't decompose into finite numbers of paths. For example:

2) exp(x) = sum x^n /n!

goes through all the all the 0 to n - 1 paths of x^k for k = 0 to n - 1
and lets n --> infinity, but the variable x^n is divided by n! which is
1 times 2 times 3 times...times n. This doesn't decompose into finite
paths and the role of the 1/n! factor violates the above "and/or" idea.
In fact, the limit itself violates the "and/or" idea for finite paths,
leading one to ask in vain "where are the infinitely many paths?"

This is, however, OK. The quantity exp(kx) is the "kernel" or
"essence" of the Riccati Differential equation for rational
non-periodic real solutions. The quantity y^n - x^n, which Fermat was
one of the first to look at seriously, gives us insight into the
"independent finite-choices" scenarios. We need arguably to keep the
two pictures in mind.

Osher Doctorow

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