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There were 11 days from Hyades II to Haedus II and there were also 11 days from Anunitum to Sheratan. From December 21 (solstice) to January 1 (366) there were 11 dark nights. Sirius would have risen 26 + 11 = 37 days earlier than Spica at the time of Hyades II. Ga5-28 (138) - 37 = 101 (Ga4-18):

●JUNE 17 18 19 (¤354) 20 (171) 29
Ga4-16 Ga4-17 (100) Ga4-18 Ga4-19
JUNE 28 29 SIRIUS (181) JULY 1 (¤1)
ν Hydrae (163.1) no star listed (164)

Altair

Wings-27 ANA-TIPU
η Oct. (165.4), ALKES (165.6) Merak (166.2), DUBHE (166.7)
August 31 September 1 2 3 (246)
NAKSHATRA DATES:
●DECEMBER 17 18 19 (¤172) (12 * 29½)
DECEMBER 28 29 30 (364) 31 (¤184)
ι Cephei (346.0), λ Aquarii, γ Piscis Austrini, σ Pegasi (346.5) Scheat Aquarii (347.0), ρ Pegasi (347.2), δ Piscis Austrini (347.4), FOMALHAUT, τ Gruis (347.8) Fum al Samakah (348.3), ζ Gruis (348.5), ο Andromedae (348.9) Al Fargh al Mukdim-24 / Purva Bhādrapadā-26 / House-13
Scheat Pegasi, π Piscis Austrini (349.3), κ Gruis (349.4), MARKAB PEGASI (349.5)
March 2 3 (427) 4 5 (64)
●JULY 20 (*121) 21 7-22 (ф184) 23 (204) 24 25 26 (¤26)
Ga5-22 Ga5-23 Ga5-24 Ga5-25 Ga5-26 (136) Ga5-27 Ga5-28
JULY 31 (ф193) AUGUST 1 2 3 (*135) 4 (216) 5 6 (¤37)
δ Muscae (196.5), Vindemiatrix (196.8) 13h (197.8) APAMI-ATSA, ψ Hydrae (198.5) Al Dafīrah (199.4) σ Virginis (200.4) γ Hydrae (201.0), ι Centauri (201.4) Al Simāk-12 / Chitra-14 / Horn-1 / Sa-Sha-Shirū-19

ANA-ROTO

ξ¹ Centauri (197.1), ξ² Centauri (197.9) Mizar (202.4), SPICA, Alcor (202.7)

Sadalmelik

October 3 4 5 6 (ф260) 7 (280) 8 9
'September 6 7 8 9 10 11 (254) 12 (*175)
NAKSHATRA DATES:
●JAN 19 (ф0) 20 (386 = 2 * 193) 22 23 24 25
JANUARY 30 31 FEBRUARY 1 2 3 4 (400) 5 (36)
no star listed (14) 1h (15.2) Al Batn Al Hūt-26 / Revati-28 / 1-iku ν Phoenicis (17.4), κ Tucanae (17.6) no star listed (18) Adhil (19.3), θ Ceti (19.7) Ksora (20.1), ω Andromedae (20.6), γ Phoenicis (20.8)
β Phoenicis (15.1), υ Phoenicis, ι Tucanae (15.6), ζ Phoenicis, η Ceti (15.7) MIRACH, Keun Nan Mun (16.0), ANUNITUM (16.5), REVATI (16.9)

Regulus

April 4 5 6 (ф77) 7 8 (463) 9 10 (100)
'March 8 9 10 (ф50) 11 12 (436) 13 (72) π

Possibly Bayer thought the empty drinking vessel at Alkes (α Crateris) motivated η as zero for an almost invisible star in Octans.

It should be close to the line for 11h and the little star at the line for 12h cannot be η. Wikipedia has 6.19 for its visibility. 11h / 24h * 365.25 = 167.4.

After θ came ζ and then η:

Manacle ziqq Phoenician zayin Greek zeta Ζ (ζ)

... Zeta (uppercase Ζ, lowercase ζ; Greek: ζήτα ... is the sixth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 7. It was derived from the Phoenician letter Zayin. Letters that arose from zeta include the Roman Z and Cyrillic З ...

Zayin (also spelled Zain or Zayn or simply Zay) is the seventh letter of many Semitic abjads ... It represents the sound [z]. The Phoenician letter appears to be named after a sword or other weapon. (In Biblical Hebrew, 'Zayin'  means sword, and the verb 'Lezayen' means to arm. In modern Hebrew, 'zayin' means penis and 'lezayen' is a vulgar term which generally means to perform sexual intercourse and is used in a similar fashion to the English word fuck, although the older meaning survives in 'maavak mezuyan' (armed struggle) and 'beton mezuyan' (armed, i.e., reinforced concrete). The Proto-Sinaitic glyph according to Brian Colless may have been called ziqq, based on a hieroglyph depicting a 'manacle'.

The role of Sirius cannot be ignored, and it was probably this star who made the Sea Water return after Canopus had digested it all and laid bare the enemies. Perhaps another name for Sirius was Rigi:

... Old-Spider then took the snail, placed it in the west of the shell, and made it into the moon. Then there was a little light, which allowed Old-Spider to see a big worm. At her request he opened the shell a little wider, and from the body of the worm flowed a salted sweat which collected in the lower half-shell and became the sea. Then he raised the upper half-shell very high, and it became the sky. Rigi, the worm, exhausted by this great effort, then died. Old-Spider then made the sun from the second snail, and placed it beside the lower half-shell, which became the earth ...

27 Wings α Crateris (Alkes) Snake (165.6) Sep 2 (245) 245 = 237 + 8
28 Chariot γ Corvi (Gienah) Worm (185.1) Sep 22 (265) 265 = 245 + 20
September equinox