1.6.10 Summary

In Chap­ter 5, the sphere of tan­gent spaces was made con­tin­u­ous, which led to the de­f­i­n­i­tion of the in­fi­nite sym­me­try group $S_{3\mathbb{C}}^{\infty}$, and its Lie al­ge­bra $\mathfrak{s}_{3\mathbb{C}}^{\infty}$, and this smooth space is caused by a sub­tan­gent space un­der­ly­ing the Lie al­ge­bra of the in­fi­nite sym­me­try group. As we have men­tioned, one of the most re­mark­able facts about Lie theory is that the el­e­ments of the Lie al­ge­bra con­tain all the in­for­ma­tion about the group, that or­di­nar­i­ly might be ob­scured by the ex­po­nen­tial map­ping. The in­fi­nite Lie al­ge­bra of the $S_{3\mathbb{C}}^{\infty}$ group re­sides on the bound­ary of in­fi­nite sphere of tan­gent spaces $\mathfrak{T}\left( S_{3\mathbb{C}}^{\infty} \right)$, shown in Figure 1.6.28 to rep­re­sent a hand­ful of the in­fi­nite number of its Lie al­ge­bras. $\mathfrak{T}\left( S_{3\mathbb{C}}^{\infty} \right)$ is the sin­gu­lar­i­ty at the heart of the whole $S_{3\mathbb{C}}^{\infty}$ in­fi­nite sym­me­try group, en­cod­ing holo­graph­i­cal­ly all the group's in­for­ma­tion on its sur­face.

This chap­ter rep­re­sents the syn­thet­ic cul­mi­na­tion of many dis­ci­plines in math­e­mat­ics, and re­sults in an in-depth treat­ment of the sin­gu­lar­i­ty and its dy­nam­ics. Math­e­mat­i­cal­ly, the sin­gu­lar­i­ty can be en­vi­sioned as the kernel that drives all the phe­nom­e­nol­o­gy in the ex­po­nen­ti­at­ed spaces around it. It may be hard to con­tem­plate from the stand­point of the an­thro­po­mor­phic bias, but any in­ter­ac­tion that occurs in the ex­po­nen­ti­at­ed space under the aus­pices of the sin­gu­lar­i­ty is en­grained in the Lie al­ge­bras on its sur­face. The Lie al­ge­bra trans­form in rep­re­sen­ta­tion theory is syn­ony­mous with the Fouri­er trans­form from the fre­quen­cy domain for wave-like groups, and syn­ony­mous with Laplace trans­forms for com­plex-valued Lie al­ge­bra ar­gu­ments.

The sur­face of the sin­gu­lar­i­ty is the bound­ary where the log­a­rithm and ex­po­nen­tial func­tions meet, and we have found these func­tions are dis­tinct meth­ods of in­te­gra­tion and dif­fer­en­ti­a­tion. The study of the in­fi­nite sym­me­try group $S_{3\mathbb{C}}^{\infty}$, with its con­tin­u­ous tan­gent space, re­veals math­e­mat­i­cal­ly, the non-ex­tend­ed sin­gu­lar­i­ty exists as the sine qua non for which ex­tend­ed phe­nom­e­non is con­tin­gent. Non-ex­tend­ed forms move freely in the sin­gu­lar­i­ty and all other ex­tend­ed or con­tract­ed spaces through the space of angles, which are not lim­it­ed to the geo­met­ric con­straints of move­ment in ex­po­nen­tial sub­group ex­ten­sion. The outer prod­uct ap­proach to the two in­fi­nite series that com­prise Euler's for­mu­la shows that the nu­mer­ic in­ter­ac­tion of non-ex­tend­ed forms with an energy state below $\left( \frac{radius}{2\pi} \right)$, stays within the bound­ary of the sin­gu­lar­i­ty and the energy states above this thresh­old man­i­fest in the ex­tend­ed state.