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Half 360 days = 180 and half 354 = 177 (= 6 * 29½), but June 21 is not day 354 / 2 + 1. The method to find the June solstice is instead to subtract 183 (= ca 365¼ / 2) days from day 355 (the December solstice). The Gregorian calendar does not begin with the northern winter solstice and there are 11 days from December 21 to the end of the year.

Akleel 3 (212)   4
December 14 (348) 15
Ca10-13 (268) Ca10-14
kua tupu te mea - i te inoino ka tupu te toromiro - i te inoino
June 15 (166) 16
Albatain 3 (30) 4
Akleel 5 6 7 (216) 8 9
December 16 17 18 19 20 (354)
Ca10-15 Ca10-16 Ca10-17 (272) Ca10-18 Ca10-19
rima heu ki te vai te moko oho mai te marama te kava manu rere
June 17 18 19 20 21 (172)
Albatain 5 6 7 8 9 (36)
Akleel 10   11 (220)
December 21 (355) 22
Ca10-20 (275) Ca10-21
te inoino erua
June 22   23
Albatain 10 11 (38)
Akleel 12 13 (222) Qalb al Akraab 1 2 3 4
December 23 24 25 26 (360) 27 28
Ca10-22 (277) Ca10-23 Ca10-24 Ca10-25 Ca10-26 Ca10-27
tupu te raau i vai oka hia te moko te marama te kava manu kara etahi te mauga pu hia
June 24 25 26 27 28 29 (180)
Albatain 12 13 (40) Al Tuwaibe' 1 2 3 4

At Ca10-13 we can count 12 months with 29 days in each = 348 days from January 1 = 268 days from March 21.

Possibly my perceived gap between the glyphs for December 14 and 15 could be connected with the distance from January 1.

6 * 29 = 174 = 94 days from March 21 = June 23 could then, maybe, be expected to be followed by a similar gap, but such is not the case. Neither is there any obvious gap at day 348 - 183 = 165 (June 14):

Albatain 1 (28) 2 3 4 5
June 13 14 (165) 15 16 17 (168)
Ca4-8 Ca4-9 (85) Ca4-10 Ca4-11 Ca4-12
manu te rau hei te hokohuki te moko te hokohuki
Lesath (264.7), Shaula (265.3) Sargas (266.3)   Girtab (267.6) Apollyon (268.9)
Hatysa (83.5), φ² Orionis (83.6), Alnilam (83.7) Alnitak, (84.7)   Saiph (86.5) Betelgeuze (88.3)
Albatain 6 (33) 7 (399) 8 9
June 18 19 20 21 (172)
Ca4-13 Ca4-14 Ca4-15 Ca4-16 (92)
kua tuu tona mea te henua te hau tea mauga hua - te henua
η Leporis (89.0), Praja-pāti, Menkalinan, Mahashim, and γ Columbae (89.3)  η Columbae (89.7), μ Orionis (90.3) 6h (91.3) ξ Orionis (92.5)
χ² Orionis (90.5), ν Orionis (91.4)
Albatain 10 11 (403) 12  13 (40)
June 22 23 (St John's Eve) 24 (St John's Day)  25
Ca4-17 Ca4-18 (94) Ca4-19 Ca4-20
te hau tea tupu te rakau - te henua te hau tea tupu te rakau
 Tejat Prior (93.4)  κ Aurigae (93.6) Furud (94.9), Tejat Posterior, Mirzam (95.4) Canopus (95.6), ψ1 Aurigae (95.9)

Instead of 12 * 29 = 348 days the cycle could be 13 * 29 = 377 days, with the halfway point 377 / 2 - 80 = 108½ days from March 21. But there is a gap 107 days from March 21:

Al Tuwaibe' 10 11 12 13 (53)
July 5 (186) 6 7 8
Ca5-1 Ca5-2 (107) Ca5-3 Ca5-4
E hua ki te henua - ka huki te hau tea - te henua kua hua te henua
 7h (106.5) Wezen (107.1) no stars listed
Alzirr (105.7), Muliphein (105.8)

348 - 80 - 107 = 161 is not very convincing. The distance from the heliacal rising of Wezen to the December solstice, however, was 355 - 80 - 107 = 168 days.

2 * 168 = 336 = 348 - 12. If there was a cycle with 13 * 29 = 377 nights, then the difference beetween this cycle and the Gregorian year would also be 12 nights.

In other words, given a basic structure defined from the Gregorian calendar there will be an overlap of 12 days and Ca10-1 could be the beginning of a cycle with 377 nights.

Az Zubana 3 (199)
December 1 (335)
Ca9-27 (255)
etoru gagata hakaariki kia raua
Denebakrab (254.7), ι Ophiuchi (255.3), Grafias (255.4)
June 2 (153)
Pleione 3 (17)
no star listed
Az Zubana. 4 (200) 5 6
December 2 (336) 3 4
Ca10-1 (256) Ca10-2 Ca10-3
Erua inoino kua hua te vai
κ Ophiuchi (256.2) Cujam (256.9) no star listed
June 3 4 5 (156)
Pleione 4 5 6 (20)
Hassaleh (73.6) Almaaz (74.7), Haedus I (74.8) 5h (76.1)
Haedus II (75.9), ε Leporis (76.0), Cursa (76.4)

There are 29 days from December 2 to the end of the Gregorian year. 365 - 336 = 29.

There are 12 days from December 2 to December 14. 348 - 336 = 12.

Counting 377 / 2 = 188½ ahead from December 336 leads to day 336 + 188½ = 524½ = ca 525 = 365 + 160:

Pleione 7 (21) 8 9
June 6 (157) 7 8
Ca4-1 (77) Ca4-2 Ca4-3
kua tupu te rakau kua tupu - te kihikihi te hau tea
 λ Eridani (76.7)   Rigel (78.1), Capella (78.4) η Scorpii (259.9)
Pleione 10 11 12 (26) 13
June 9 (160) 10 11 12 (528)
Ca4-4 Ca4-5 Ca4-6 Ca4-7 (83)
tagata - te rau hei te hokohuki i te moko te rau hei e gagata hakaariki
no star listed Bellatrix, Saif al Jabbar (80.7) Mintaka (82.4)
φ¹ Orionis (83.1), Heka (83.2)

This is rather convincing. There are 528 - 335 = 193 days between the hakaariki glyphs. The hakaariki type of glyph could mean 'a dark night when a new king will be alighted'.

December 1 is 255 days after the March equinox. This number could signify a night similar to day 355.

The Gregorian year is not a 'bivalve' cycle, therefore there should be no need to have a hakaariki at its end - the new year 'baby' is the same 'person' as the old year:

... And this is when a maiden heard of it, the daughter of a lord. Blood Gatherer is the name of her father, and Blood Moon is the name of the maiden. And when he heard the account of the fruit of the tree, her father retold it. And she was amazed at the account: I'm not acquainted with that tree they talk about. It's fruit is truly sweet! they say, I hear, she said. Next, she went all alone and arrived where the tree stood. It stood at the Place of Ball Game Sacrifice.

What? Well! What's the fruit of this tree? Shouldn't this tree bear something sweet? They shouldn't die, they shouldn't be wasted. Should I pick one? said the maiden. And then the bone spoke; it was there in the fork of the tree: Why do you want a mere bone, a round thing in the branches of a tree? said the head of One Hunaphu when it spoke to the maiden. You don't want it, she was told. I do want it, said the maiden. Very well. Stretch out your right hand here, so I can see it, said the bone. Yes, said the maiden. She stretched out her right hand, up there in front of the bone. And then the bone spit out its saliva, which landed squarely in the hand of the maiden.

And then she looked in her hand, she inspected it right away, but the bone's saliva wasn't in her hand. It is just a sign I have given you, my saliva, my spittle. This, my head, has nothing on it - just bone, nothing of meat. It's just the same with the head of a great lord: it's just the flesh that makes his face look good. And when he dies, people get frightened by his bones. After that, his son is like his saliva, his spittle, in his being, whether it be the son of a lord or the son of a craftsman, an orator. The father does not disappear, but goes on being fulfilled. Neither dimmed nor destroyed is the face of a lord, a warrior, craftsman, an orator. Rather, he will leave his daughters and sons. So it is that I have done likewise through you. Now go up there on the face of the earth; you will not die. Keep the word. So be it, said the head of One and Seven Hunaphu - they were of one mind when they did it ...