Nine central
Chps one and nine share similar characteristics as they are the only chps without the opening statement - the tetron (see Glossary), the unnumbered verse that precedes all other chps. The number of their verses are both prime numbers too. The fact that they form the digits of 19 suggests a sort of symmetry between 1 and 9, that they are treated in a way on an equal footing. We also notice a similar symmetry between 9 and 27. This in turn implies a deeper relationships with other chps as well. In particular, we see several links between chps 1, 9, and 27, since, the missing opening in chp 9 is compensated in 27:30.
When we look at the free verses in chp 9 we see that there are 27 of them. This is also the number of chps in the G:9 set. Then, we notice in chp 27 that number 9 occurs in verses 12 and 48, which are positioned symmetrically with respect to verse 30. From 12 to 30 there are 19 verses, as are from 30 to 48. Furthermore, 912 = 19 × 48! So, in a way, the first occurrence of 9 in verse 12 actually predicts the second occurence in verse 48! 912 is also the ALR counts of chp 15 which is of ord 9, and 15th ord is 29 which itself is the verses of chp 48. Additionally, the initial letter "T" in chp 27 has value 9 and occurs 27 times. We also notice the emphasis on verse 30 in relation to 9, as it is the only verse other than 1:1 that contains the tetron, compensating for the missing one proceeding chp 9. The accumulative norm at 27:30 is 1815 which points back to 9, as both 18 and 15 are of identical ord 9. As another indirect link between 9 and 27, we note that 927 = 9 × 103 and 279 = 9 × 31, where 103 happens to be the 27th prime and 127 (verses of 9) is the 31st prime.
The number of verses from 1 to 9 and the accumulative norm (label GN) at the end-verse 9:127 are 1362 and 1273 = 19 × 67 respectively as shown by the top curved arrow in the Fig. These two numbers are then reproduced as we traverse verses from 1:1 to 2:83, and then to 9:127. Verse 2:83 is the 31st (pointing back to 127) GV from 1:1 with the GV sum 1362! What's even more striking is that from 2:83 to 9:127 there are 1273 verses, the norm at 9:127! as shown by the straight arrows. The prime factors 19 and 67 point indirectly back to 9 in that from 19 to 67 (the concave-down EVC arrow) there are 26 EV's with sum 1131. And 1131st prime is 9127! ;-) Furthermore, 31 itself is the 11th prime. Both 19 and 67 happen to be the sum and the difference of 43 with 24, where one would recognize that 43 is of ord 24. Furthermore, 24 is of ord 13, and 13 itself has 43 verses (the straight V arrow). Additionally, from 9 to 13 there are 5 chps and 513 = 19 × 27 verses (the curved arrow on the right) where, 27 itself is of ord 13. The fact that 2413 = 19 × 127, re-enforces the relationship between 9 and 13.
Both 127 and 67 are primes, 31st and 19th respectively. The 67th ord is chp 96 with 19 verses which is also the verses of chp 56. And chp 31 has 34 verses. It is interesting that chps 31 and 56 which so far are only indirectly related to chp 9 can provide the following relationship between the three chps and their verses: 3431 + 5696 = 9127. As a side note, we notice that 3134 is 2688th composite. Here, we see the reference to chp 88 and its 26 verses. We also notice that 88 is of ord 59, which itself is of ord 31! Additionally, 2688 + 3134 = 5822, where we see the connection with chp 58 which happens to have 22 verses.
We noticed 26 appeared twice before. Once in connection to EV's and then via the verses of chp 88. This link to 26 becomes stronger when we consider the sum of the verses of chps 1, 9, and 27: 7 + 127 + 93 = 227 which is the verses of chp 26. It's noteworthy that the number of GV's and FV's in those three chps add up to 127 and 100 respectively, which, at the same time are the verses and GV's of chp 9! Both 26 and 88 provide links to 9 and 19 as follows. The sets G:19 and G:26 exhibit a form of mutual symmetry between them in that they have 26 and 19 chps in them respectively! And 19 × 88 is the ALM counts of chp 29. From 1 to 29 there are 1273 GV's as illustrated in the Fig. Also 29 is the verses of chp 48 (curved V arrow) and 2948 is the number of un-init verses from 9 to 114 = 88 + 26. Verses 9:1 and 9:114 are both G1's. The G1 sum from 9:1 to 9:114 is also 2948!
Chp 29 is an ALM initialed chp and the 6 ALM chps add up to 127, the verses of chp 9 which itself is of ord 6. From 9 to 29 there are 12 OV's and their verses add up to 1324. Apart from the fact that 13 and 24 are related as mentioned before, we'll see later how this goes back to GV's and chp 9 again. Chp 2 is also an ALM chp and its verse 83 is already intricately connected with chp 9 as shown in the Fig. This may serve as a hint that there exists something more fundamental between them. This correspondence between 2 and 9 turns out to involve the primes. The 1362nd prime is 11261 and 11261 − 1362 = 9899, which is the ALM counts of chp 2 !;-) The link between chp 9 and the init-chps is further strengthened by the fact that the init verses from 19 to 68 (ords 10 and 29 respectively) is 1514 and this happen to be the 1273rd composite!
The normal and free verses in chp 9 provide additional means which help to pin down several intricate relationships with other chps and their verses. We noticed above how 58 came about in connection with the primes and composites. We notice that the 58th GV in chp 9 is verse 74 corresponding to norm 104. Chp 104 happense to have 9 verses. The GV sum up to 74 is 2147 = 19 × 113, which, at the same time is the HM counts of the 7 init-chps from 40 to 46 with ords 21 through 27. If we include the 209 counts of ASQ, the 2nd set of initials in chp 42, the sum would be 2147 + 209 = 2356 = 76 × 31. Here, we see the reference to chp 76 and its 31 verses. The 23rd FV is 9:113, and 56 is the verses of chp 74. Verse 56 itself is the 45th GV, and 74 is of ord 45. The norm at 9:56 is 1183 and 1183rd composite is 1406 = 19 × 74! The norm 1974 occurs at 1324th GV and 1406 + 1324 = 2730 !;-) We mentioned above the 12 OV's from 9 to 29. The 12th ord is 26, and 26th ord is 45 which points back to 74 again.
So far we've seen several deep relationships between chp 9 and others. We saw the connection to 74 above. Since, 74 is the only chp where 19 is mentioned, we would expect strong ties between 9 and 19 as is the case. From 9 to 19 there are 11 chps and 1111 = 101 × 11 verses between them. We see the reference between 101 (the 26th prime) and its 11 verses. This however, points indirectly to the verse where the tetron is compensated. The total norms of the sets from G:11 to G:101 add up to 1719. This is also the number of init verses upto 27:30. Chp 101 is of ord 72, which itself is of ord 43 (solid and dotted curved U-arrows). The norm 101 in chp 9 happens to be at verse 72! The number of verses in those 43 un-init chps turns out to be 2730, a reference to the special nature of verse 27:30 for the missing opening. The intervening init verses then, is 2743 as shown by the curved arrow. The importance of links to 27 and 30 is reflected not only with the missing opening, but, also with the even-odd parity of all the chps. If we partition the chps into disjoint sets based on the parity of chps and verses, we notice that there are 27 EO's, 27 OO's, 30 EE's, and 30 OO's.
Verse 9:72 is also the 57th GV. Chp 57 has 29 verses and is of ord 29. And the fact that the GV count at the end of chp 29 is 1273 would close the link back to chp 9. Additionally, the norm of the G:9 set is also 29. As a side note, from 29 to 57 there are 1764 verses. This is also the sum of the 27 FV's in chp 9 ;-) There are 68 FV's in chp 27 and 68 is of ord 29. The remaining 100 GV's in chp 9 add up to 6364 which is a simple permutation of 6346, the total number of verses. We notice that the 63rd GV from verse 127 is verse 46! And 46 is of ord 27. If instead, we do a weighted sum of those GV's in chp 9 based on their individual norms - the dot product of Gvec and Nvec, we get 10327!!! We see here the reference to 103, which is the 27th prime! ;-) It's also noteworthy that chp 103 is the 27th chp with prime verses!
We have mentioned the G:9 set and its relationship with 27 and 29. Among its members are chps 49, 57, and 64 which are positioned at locations 18th, 19th, and 24th in the set respectively. We see how 49 and 64 are pinned down in the set according to their verses, and 57 = 27 + 30 at position 19 has already been linked to GV's in chp 9 as previously discussed. The weighted sum of the GV's in chp 29 is 1425. This is also the number of free verses from 48 to 114. We've already mentioned that the count of GV's up to 29:69 is 1273. The sum 1273 + 1425 = 2698 is the total accumulative norm. We see several interesting properties in the proceeding sum: 12th ord is 26, 73 is of ord 44, which itself is of ord 25, and the latter is of ord 14. Chp 98 is of ord 69, which itself is the verses of chp 29! We see how their components are intricately connected.
The counts of G1's and GH's in chp 9 are 54 and 46 respectively. The 46th and 54th GV's in chp 9 are verses 59 and 68 respectively. 68 − 59 = 9, 68 + 59 = 127, and 6859 = 193! Moreover, the 31st odd G1 is verse 127. Since, 98 is the verses of chp 19, we may take that as a hint that a more direct link with chp 9 is in place. The GV count and the norm at the end of chp 98 are 1815 and 2693 respectively. Now, 1815 is the norm at 27:30 as we saw earlier. Interestingly, 2693 also points to 27 and 30 indirectly, as we look at its components 26 and 93. We already know 93 is the verses of chp 27. Chp 30 is ALM initialed with counts 1254. Here, we see how 12 and 54 point to 26: the 12th ord is 26, and the 26th ord is 54! Furthermore, chp 30 is of ord 16 and the norm from 16 to 98 is 1204 which is the GV count at 27:30!
Verse 9:31 is the 25th GV and 31st G1 is verse 77. Here, we realize the indirect hint to 25 and its 77 verses. Additionally, chp 77 has 50 verses, and 77 + 50 = 127, 77 − 50 = 27. Verse 77 is also the 60th GV. From chp 31 to 60 there are 1273 FV's! (bottom right arrow), as 60 itself has 13 verses (the right corner up arrow). Furthermore, chp 77 is of ord 48 and from 31 to 48 there are 1143 = 9 × 127 verses (bottom middle arrow), confirming the verses of chp 9! The 60th ord is 89 which is also the verses of chp 43 as shown in the Fig. From 48 to 89 (curved right arrow) there are 1343 un-init verses! And as was discussed above, the 43rd ord is 72 and 72nd ord is 101. From 48 to 101 there are 1362 FV's.
The prime factor 67 of 1273 as illustrated in the EVC arrow is linked separately to 9 as shown by the concaved-up arrow from below. From 9 to 67 there are 59 chps spanning 2948 = 44 × 67 FV's between them. We recognize the connection to chp 44 and its 59 verses, and of course, 48 and its 29 verses. Additionally, 48 × 29 = 1392, where chp 92 happens to be the 13th FC. The norm 1392 occurs at verse 12:73! Chp 92 itself points back to 9 via its 21 verses, in that 9221 happens to be the 1143rd prime!!! It is interesting that the sum of the GV's up to 11:43 is 81974. We already know the relationship between 8, 19, and 74. An important permutation of the proceeding number is 19874 which happens to be the total ALM counts in the 6 ALM initialed chps! There's an interesting link between 19 and 874. When we look at all the GV's we note that the 874th GV is in verse 19! The relationship between 9 and the ALM chps was pointed out earlier: the sum of the ALM chps is 127 and 1143 = 9 × 127.
The weighted sum of GV's (the dot product of Gvec and Nvec) up to 11:43 is 134359. We notice here the components 13, 43, and 59 and their properties; 13 has 43 verses and that 13th ord is 24 which itself is the verses of chp 59. Since, 19 is the 10th ord and 9 and 10 add up to 19, it may suggest some links between 9 and 10. As it turns out, there are 49 GV's in chp 10 with sum 2413 = 19 × 127. And the norm from 10 to 49 is 1131 ! The corresponding GV count between 10 and 49 is 841. Now, the sum 1131 + 841 = 1972 = 29 × 68, where we see the pointer to 68 as the 29th ord. Furthermore, among 114 chps there are 77 base verses and 127 is the 68th such verse in ascending order.