Conclusion

Math­e­mat­i­cal­ly, form and anti-form rep­re­sent a per­pet­u­al dance of re­spec­tive in­vers­es. Their in­verse re­la­tion­ship is mul­ti­plica­tive in the ex­po­nen­ti­at­ed group form and ad­di­tive in the Lie al­ge­bra form. Any physics treat­ment of the uni­verse that presents a sin­gu­lar matrix with­out an in­verse is in­cor­rect right out of the gate.

Philo­soph­i­cal­ly, we must ask our­selves; can we phys­i­cal­ly know which space we reside in? Do we exist in both spaces equal­ly and in tandem? The meta­phys­i­cal im­pli­ca­tions of the math­e­mat­ics pre­sent­ed in this book and the books to come will lead to the ne­ces­si­ty for rig­or­ous dis­course. The ul­ti­mate volume in the Arith­moí series will re­volve around the meta­phys­i­cal im­pli­ca­tions of this entire math­e­mat­i­cal ex­plo­ration that the mind has pi­lot­ed.

The Emer­ald Tablet was a Her­met­ic text, whose second verse read

"Quod est su­perius est sicut quod in­ferius,

et quod in­ferius est sicut quod est su­perius.",

which, trans­lat­ed from Latin to Eng­lish, is

"That which is above is like to that which is below,

and that which is below is like to that which is above."

These words do not miss the mark when ap­plied to the math­e­mat­ics of form and anti-form, and we will soon re­al­ize that the essence of what is em­bod­ied in these words rep­re­sents a re­cur­ring uni­ver­sal theme.

The an­cients did not have access to the modern math­e­mat­ics pre­sent­ed in this book, but they had in­tu­ition at their dis­pos­al. Groups like the Pythagore­ans used geom­e­try in the at­tempt to sub­stan­ti­ate the eter­nal truth they in­tu­it­ed, which was that "all things are number." They had al­ready found the arche, the first prin­ci­ple that formed the foun­da­tion of every­thing. Σεβασμός.

In the book, we have made strides in un­cov­er­ing the math­e­mat­ics of the arche and the Uni­fied Field it causes. From the atom­istic view­point, each unit sin­gu­lar­i­ty is one of its build­ing blocks. Each one of these unit sin­gu­lar­i­ties has the po­ten­tial to ac­tu­al­ize into ab­solute­ly all ob­serv­able things. It is part of the whole, while and at the same time, the whole is con­tained within it. As Above, So Below.

In the up­com­ing book of the Arith­moí series, "Math­e­mat­i­cal Foun­da­tions - Beyond the Bound­ary," we ex­plore non-lo­cal­i­ty by math­e­mat­i­cal­ly let­ting the mind become in­tro­spec­tive upon itself. Its depths are un­fath­omable, and their nav­i­ga­tion brings us to beau­ti­ful har­mon­ic forms, which exist lo­cal­ly and non-lo­cal­ly and in tandem with the Uni­fied Field. The pre­sen­ta­tion will be heav­i­ly based on math­e­mat­i­cal analy­sis, but it is pur­pose­ful­ly ex­ten­sive in de­scrib­ing the de­riva­tions and con­tent. The sub­tan­gent space pre­sent­ed in this book is our point of em­bark­ment, and we will find its clear form rooted in the ex­po­nen­tial log­a­rithm func­tion of non-com­mu­ta­tive al­ge­bras. The path ahead, en­abled by the mind, will lead us di­rect­ly to the heart of prime number theory and the ori­gins of the ex­po­nen­tial func­tion. Math­e­mat­i­cal co­he­sive­ness will remain the pri­or­i­ty in the pre­sen­ta­tion on lo­cal­i­ty and non-lo­cal­i­ty, the dy­nam­ics of light, and the math­e­mat­i­cal frame­work of con­scious­ness.

A dis­course that sets out to ad­vance our un­der­stand­ing the nature of being must be math­e­mat­i­cal in form. It needs to bring non-lo­cal­i­ty to light, while at the same time, there should be a clear-cut path that leads di­rect­ly to the math­e­mat­i­cal­ly based sci­ences. It fails to ad­vance the di­alect if it cannot con­nect the sen­so­ry and non-sen­so­ry phe­nom­e­nol­o­gy. The con­nec­tion that the math­e­mat­i­cal con­tent pre­sent­ed in this book has with physics is touched upon in this chap­ter and aims to show that the work is rooted in a co­he­sive con­nec­tion. To the astute reader versed in math­e­mat­ics and physics that might chance upon this book, you have the lim­it­less abil­i­ty to use its foun­da­tion and take this con­nec­tion fur­ther. The second volume of books in the Arith­moí series will ex­ten­sive­ly con­nect the parts.

Di­chotomies are a per­ma­nent fix­ture of the uni­verse, both in the sense of phe­nom­e­nol­o­gy and an­thro­po­mor­phic

char­ac­ter­i­za­tions, taking forms such as light and dark, fem­i­nine and mas­cu­line, truth and lie, as well as left brain and right brain. Math­e­mat­i­cal­ly, the di­choto­my of the lo­cal­i­ty of the ex­tend­ed spaces takes on this du­al­i­ty by way of the trigono­met­ric forms of the sine and cosine func­tions and the anal­o­gous hy­per­bol­ic sine and cosine func­tions. In con­trast, the un­der­ly­ing math­e­mat­ics of the geo­met­ric num­bers inside the sin­gu­lar­i­ty are dif­fer­ent par­ti­tions of the ex­po­nen­tial func­tion.

The cyclic trigono­met­ric func­tions that form the basis of the equi­lat­er­al tri­an­gle's dy­nam­ics are not two-way trigono­met­ric func­tions, in­stead they are the three-way func­tions, $\mathfrak{O}_{k}^{3}(X)$. These were de­fined in Chap­ter 2 for an imag­i­nary-valued $X$, as equal to

for which all three func­tions are a mix of even and odd pow­ered el­e­ments. The $\mathfrak{O}_{k}^{3}(X)$ func­tions rep­re­sent the trigono­met­ric dy­nam­ics of light inside the sin­gu­lar­i­ty and define the math­e­mat­i­cal trin­i­ty united in odd and even­ness.

The ul­ti­mate theory of every­thing will math­e­mat­i­cal­ly unify the mind with the sci­ences. It will rep­re­sent the syn­the­sis of the fun­da­men­tal di­choto­my, the du­al­i­ty of the mental form and the phys­i­cal form. The true re­li­gion will have stepped into the spot­light when it ar­rives, and it will be fun­da­men­tal­ly math­e­mat­i­cal.

What is your mo­ti­va­tion, as a reader? Is your mo­ti­va­tion mired with the con­cerns of world­ly ex­is­tence, or does it go beyond that?

If you seek any moniker of uni­ver­sal truth, you will find it em­bod­ied in the math­e­mat­i­cal form. Those ready to cast aside ex­cus­es and pro­cras­ti­na­tion, and who will devote the time nec­es­sary to learn math­e­mat­ics at the level where it be­comes co­he­sive, de­serve the reward that comes with it.

Who am I?... I am nobody. I came into this world as every­body else did; naked, cold, ig­no­rant, and in­cred­i­bly de­pen­dent. I am the prod­uct of the Chica­go Public School system, and as a result of this, I am happy to say that I was not af­ford­ed any ad­van­tage over the next person.

Was I chosen to do this? ... No, but I did choose to listen.

I have seen the best and the worst of the human con­di­tion through med­ical school and a dual res­i­den­cy in Emer­gency Med­i­cine/In­ter­nal Med­i­cine. I have had my hands in the chest of some­one I have never met before, trying to cross-clamp an aorta and ex­pe­di­tious­ly close the hole in their heart caused by the bullet fired with little thought from the gun of an­oth­er in­di­vid­ual. Con­sis­tent with the du­al­ism one comes to expect from the uni­verse, I also have had the priv­i­lege to be the first to hold scores of babies leav­ing the com­fort of the womb, making their world­ly en­trance.

The human ex­pe­ri­ence helps one find their pur­pose, as it helped reveal mine, which is to write this book and the books to come.

In Greek mythol­o­gy, Athena, the god­dess of wisdom, was ac­com­pa­nied by a little owl of the Athene noctua species. Owls were revered by the Athe­ni­ans given their abil­i­ty to see in the dark, and the owl became a symbol of knowl­edge and wisdom in the West­ern world. The owl of Athena would sit atop her shoul­der and reveal truths to the god­dess of wisdom.

Tat­tooed on my left shoul­der is an Athene noctua stand­ing on a book, put there as a re­minder my pur­pose. If one choos­es to listen, the uni­verse whis­pers truths in the mind's ear. In its raw form it is opaque but comes to take shape in the mind that learns to tune into the lan­guage it speaks.

I am the type that lets few people into those deeper spaces that lay within; in­tro­ver­sion is my de­fault. Put me alone in a win­dow­less room with blank walls and a closed-door; I will still find beauty. Now, I am let­ting the world in by writ­ing a book based on my thoughts; the irony is pal­pa­ble. If you want to know nobody, they are found etched through the pages of this book's num­bers, equa­tions, proofs, and ideas. Im­pact­ful books for the reader in­te­grate and become the new pieces that form who they are. If this book im­pact­ed you as its reader, what was nobody's is now yours.

Ἰάσων

Per Me

Caeci Vident