I would like to understand the math behind this...

Simply and objectively!In decimal for ease.

All calculations:
p = 115792089237316195423570985008687907853269984665640564039457584007908834671663
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Private key : 1

Public key :

x = 55066263022277343669578718895168534326250603453777594175500187360389116729240

55066263022277343669578718895168534326250603453777594175500187360389116729240**3 + 7 =

32748224938747404814623910738487752935528512903530129802856995983256684603122**28948022309329048855892746252171976963317496166410141009864396001977208667916=

y = 32670510020758816978083085130507043184471273380659243275938904335757337482424

OKAY
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Private key : 2

Public key :

x = 89565891926547004231252920425935692360644145829622209833684329913297188986597

89565891926547004231252920425935692360644145829622209833684329913297188986597**3 + 7 =

57199941671890039290383617934355424684258807805258215939368959893591666662646**28948022309329048855892746252171976963317496166410141009864396001977208667916=

y = 12158399299693830322967808612713398636155367887041628176798871954788371653930

OKAY
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Private key : 3

Public key :

x = 112711660439710606056748659173929673102114977341539408544630613555209775888121

112711660439710606056748659173929673102114977341539408544630613555209775888121**3 + 7 =

104193826873522593991639737736096919049125888873761064059040146529970392609905**28948022309329048855892746252171976963317496166410141009864396001977208667916=

y = 90209061256745311731914079131285931446821116410824268969537695047367247992253

NO OKAY
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Why did the y value of private key 3 need to be inverted?

What math or method is used to know whether or not I will need to invert the y value?