John G said:
Lots of terms via that constraint get zeroed out usually in the natural numbers equation and that enables me to have no problem when doing my hyperfinite thought experiment of sorts. Not zeroing out a bunch of terms is what I have a problem with when doing a one to one matchup.
By didn't like I mean Ark didn't understand what you were doing with the divergent series set equal. He seemed to think you might have some special procedure (for me the special procedure would require zeroing out some terms).
Ark's comments from the Some Comments on Information Theory thread:
I am saying that I do not know the concept of "infinite sum series". Please define the concept for me. Or give a link. I can't find it online... Or, more exactly, what the equal sign "=" between two such formal expressions means for you.
In your case both sides are divergent. They produce both infinity. What do you mean by the fact that the two infinities are "equal"?... You probably mean (I am guessing), that with an appropriate procedure on the right hand side (infinite product) you can recover the elements of the left hand side one-by-one? Is that what you mean? Not only "elements" but also partial sums. Is this the case?
Your response was:
You are right Ark. I will try to clean up the nonsense statements like 'infinite sum series', it really makes a mess of everything. I need to go back to something much more simple and basic.
I know a lot of time has passed since you posted your response above, but I will try to pick this up again.
I think if I had set a better/clearer expectation from the start, that my conversation with Ark may have been easier. What I am trying to show does not require any calculus. It is simpler than that. We don't need to know if anything is divergent or convergent. Then when I said something stupid, everything kind of went sideways quickly.
A basic review of what I wanted to do.
1) Get the reader to realize that Euler's infinite product from Theorem 7 is actually an infinite product of prime geometric series.
Why that was important is because there are two clues in Euler's Theorem 7 that point to the origin of the sieving process Euler uses. One is that each term of the infinite product is a prime geometric series and clue two is the actual terms of the sub series removed in each iteraation of his sieving process, along with the remainder sub series of each iteration. Those two clues show us that sieving process is the Sieve of Eratosthenes.
2) Show that the same sieving logic works in the natural numbers and the rational numbers and also results in similar equalities in those scopes. (The series of all natural numbers is equal to an infinite product of prime geometric series and the series of all rational numbers is also equal to an infinite product where each term is a prime geometric series.
There are two major observations that come out of the information I provided in the Sieve of Eratosthenes.
The first is that the sieved sub series and the remainder series have a relationship to one another that is directly due to Euclid's Unique Factorization Theorem. That relationship is that the sieved sub series has an equality, it is always 100% of the time the sieved prime number's geometric series multiplied times the remainder sub series. This holds true whether you are working in the inverse naturals, the naturals, or the rationals. (I have attached a simple PHP script that demonstrates this).
The second observation that comes out of the Sieve of Eratosthenes information is that the special procedure for recovering each and every term of the infinite series is simply performing the operations of infinite product in its prime geometric series form.
The infinite product is the generator of every term of the infinite series.
3) The next step in the process I wanted to present was to just examine one of those infinite product expressions and ask a question. The Sieve of Eratosthenes shows us that just performing the operations of the infinite product creates the infinite series, but can we perform the operations of the infinite product in such a way that the terms of the infinite series that it creates, that each term of the infinite series is expressed as an infinite product itself (meaning countably infinite [limitless] terms)? I think the answer to that is Yes. But if the answer to that question is yes, then the infinite product expression also generates/creates a second infinite series. Each term of the first infinite series has countably infinite terms where only finitely many terms have non-zero exponents and each term of the second infinite series has countably infinite terms where there are infinitely many terms that have non-zero exponents. Both infinite series arise simultaneously from the infinite product's prime geometric series terms.
4) The fourth level arises by just including a real valued variable in the exponent of the infinite products prime geometric series terms.
The fifth and sixth levels arise by simply expanding the real valued variable to include all of the numbers that are created that have countably infinite terms where the exponent is non-zero.
It is very simple. It only requires the Sieve of Eratosthenes, the Unique Factorization Theorem, multiplication and addition.
When complete, the construction of the six levels fits dozens of the C's excerpts regarding the densities. In my April 2022 paper, I cataloged around 28 of those excerpts that it satisifies. Since then I have found probably another dozen excerpts that are satisfied.
I will mention two additional C's excerpts here.
** First quote **
Session 22 July 2000
========================
A: To an extent, but you may not yet understand what exactly a "soul unit" is in that sense. And of course, there is more than one sense for this as well. The "trick" that 3rd density STS life forms will learn, either prior to transition to 4th density, or at the exact juncture, is to think in absolutely limitless terms. The first and most solid step in this process is to not anticipate at all. This is most difficult for you. We understand this, but this as also why we keep reiterating this point. For example, imagine if one of your past lives is also a future life?
The "trick" that 3rd density STS life forms will learn, either prior to transition to 4th density, or at the exact juncture, is to think in
absolutely limitless terms.
Notice that the C's have said "3rd density STS life forms will learn', with the emphasis on "will", not might or may, but will. The only variable in that is that it WILL happen either prior to the wave/transition or at the exact junction. Whether it is prior or at the exact juncture will probably determine how plentiful the STO harvest will be. In all of the C's talk of thinking in unlimited ways and infinite possibilities, etc. this is the only excerpt where they used a phrase that is general "
absolutely limitless terms", but it is also specifc to mathematics,
"limitless terms".
** Second quote **
I have mentioned that my construction of the densities satisfies upwards of 40 or so of the C's excerpts and if I am correct then we can add 40 or so excerpts to the thread "Hits for the Cs" and this may fit a Cs quote, "if you want to reveal "many beautiful and amazing things."
Session 19 July 1997
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A: Laura, my dear, if you really want to reveal "many beautiful and amazing things," all you need to do is remember the triad, the trilogy, the trinity, and look always for the triplicative connecting clue profile. Connect the threes... do not rest until you have found three beautifully balancing meanings!!
Q: So, in everything there are three aspects?
A: And why? Because it is the realm of the three that you occupy. In order to possess the keys to the next level, just master the Third Man Theme, then move on with grace and anticipation.
What does the "triplicative connecting clue profile" and the "Third Man Theme" have to do with this?
Well, I am not positive about this but it may be possible that my name is a "triplicative connecting clue profile".
I would ask Laura, Ark, members of the Fellowship, Forum members what they think?
My full name, "Donald Roy Christmann."
Donald:
From my Irish heritage:
Name meaning: From the Scottish Gaelic name Dòmhnall meaning "ruler of the world", composed of the Old Irish elements domun "world" and fal "rule".
Roy:
From my English heritage:
Name meaning: From the Anglo-Norman language, derived from the word 'roi' meaning 'king' or 'regal one.' Roy evolved from being primarily a surname or nickname to a standalone given name, gaining popularity in English-speaking countries during the late 19th and early 20th centuries.
Christmann:
From my German heritage:
Name meaning: "Christian man" or "follower of Christ," derived from the name Christian (meaning "follower of Christ") and the German word "Mann" (man).
So we have: "ruler of the world", "king", and "follower of Christ".
In light of what we know about true Christianity, Julius Caesar, Paul and his cosmology, and Paleo-Christianity, it seems it may fit as a "triplicative connecting clue profile".
Then my last name can also be examined for its word roots.
Christos: Greek, "anointed one"
Mann: Old English, either human being (person) or All humanity.
Using the root word meanings we have anointed person or anointed humanity.
So we could have: "ruler of the world", "king", and "anointed humanity".
So, if I am correct about the "triplicative connecting clue profile," then I am guessing that I am probably correct in my mathematical construction of the densities. If I am wrong about the "triplicative connecting clue profile," Then I guess I have just made a bad Meat Loaf joke, "Cuz Two Out of Three Ain’t Bad."
, k
to indicate the basis vectors of R3: it is being thought of as the purely imaginary quaternions. From the perspective of geometric algebra, the even subalgebra of the Space Time Algebra is isomorphic to the GA of 3D Euclidean space and quaternions are isomorphic to the even subalgebra of the GA of 3D Euclidean space, which unifies the three approaches.
