Is Lord Ganesh an avatar of Lord Vishnu?

The Maha Bhagwat Purana states the Vishnu himself is Ganesha. The story goes that when Vishnu saw Kartikeya cuddling in the lap of Parvati, he desired to experience the bliss of being the son of Jagadamba and drink her nectar milk. The Goddess sensed this desire of Vishnu and granted him the boon that he'll surely become her son. Thus, the Goddess gave birth to Ganesha who was none other than Sri Hari himself. She embraced him and made him drink her milk. After Shiva had beheaded Ganesha, he apologized to Vishnu that just as he has beheaded him in the form of Ganesha, a time will come when Vishnu will incarnate as Sri Krishna. As Krishna, he'll come to battle with Shiva in Shonitpura, and there, the weapon of Krishna will benumb Shiva so that he'll be unable to further fight in the battle. He will equate himself with Vishnu in knowledge, culture, wisdom, prowess.this is the reason why Ganesh is 2nd most intelligent being after Mahavishnu.

But in some other puran it is written that before creating Ganesh, Goddess Ma Parvati prayed to Prabhu Vishnu so some consider Shree Ganesh as an avatar of Shree Hari that's why some people say that this is a reason Prabhu Ganesh and Goddess Ma Laxmi are worshipped together on Deepawali. Well understanding puranic description is sometimes hard to understand. There is also a story that when the world was being created then Prabhu Mahadev Shiv think about a Shakti (power/deity) whom will be for clear hurdles (Vighnaharta) so Prabhu Mahadev think about a deity with elephant face(some people will say that was elephant created till that time the answer is no but Lord Prabhu think about a deity of Shree Ganesh's Figure/Shape not elephant shape to be exact).

So all the story of Prabhu Shiv beheaded Ganesh was to make him that deity and also Prabhu Shiv knew he was his son but for prabhu everyone is equal entire universe is his child. Prabhu Shiv tried to tell him that he is his father but Shree Ganesh was in so proud of his powers given by Ma Parvati so to kill his proud (Ahankaar) Prabhu Shiv had to behead him and then given elephant head to him also some people may say Supreme Lord killed a innocent animal for his son answer is no that baby elephant had life of only one day as per his previous karmas but as he became a mean to fulfill one of the main God's thought so Prabhu gave him moksha.

Lord Vishnu has non-countable incarnations. If we see from the eternal point of view then Lord Vishnu is everything himself. Some scriptures of Sanatana dharma says that it was Goddess Parvati who did penance to Lord Vishnu and asked for a son like Brahma w.r.t. knowledge, strong and intelligent like Lord Vishnu and innocent like Lord Shiva. After impressing by Parvati, the supreme Godhead Vishnu gave the respective boon to her. Moreover it is not just about intelligence like Lord Vishnu but Lord Ganesh shares the same satvik tattva as Lord Vishnu and as his mother Goddess Parvati. You can see Lord Ganesh relaxing on the snake just like how Lord Vishnu does. You can see him with the flute. You can see him with the peacock feather. Even his divine play is similar to Lord Vishnu. He is perfectly closest to Lord Vishnu because of his activity.

As Goddess Parvati and Lord Vishnu is the same tattva so Lord Ganesh has these qualities because of his mother and some because of this boon. So Lord Ganesh can be called as an incarnation of Lord Vishnu, no doubt in that. And also he is the son of Lord Vishnu as he was further adopted by Goddess Lakshmi. Ganesha is the formless Divinity - encapsulated in a magnificent form, for the benefit of the devotee. Gan means group. The universe is a group of atoms and different energies. This universe would be in chaos if there was no supreme law governing these diverse groups of entities. The Lord of all these groups of atoms and energies is Ganesha. He is the supreme consciousness that pervades all and brings order in this universe.

Lord Ganesha is not Vishnu Tattva but a Shakti Tattva. Ganesha is the same energy which is the reason for this universe. It is the energy from which everything manifests and into which everything will dissolve. Moreover, Parvati is not the sister of Lord Vishnu, she is the external Maya energy of Lord Vishnu. They are not material Gods and Goddesses. Some fake tamasic puranas have written like this, but don't believe such fake tamasic puranas, and none of the gods and goddesses have sisters or brothers, they are only part of Lord Vishnu. So, Ganesha is Lord Vishnu's child like Prahlad but not Lord Vishnu's avatar.

The Tripuram galaxy has short flare of star streams like elephant trunk. The three big nearby galaxies created by gods Shiva,Vishnu and Brahma are Andromeda, Milky way and Triangulum or Tripuram. Goddess Tripurasundari or Parvati belongs to Tripuram galaxy and she created Ganesha from her own energy and mass not dirt. The Tripuram orbits Andromeda galaxy which is Shivalogam. While orbiting the Shivalogam also moved resulting in the Tripuram passing through the center of Shivalogam. The Tripuram also was very close to Milky way galaxy of god Vishnu during orbit which exerted gravitational pull on Tripuram.

Tripuram galaxy has different shape than the Andromeda and Milky way galaxies which have spiral star streams in coil shape and called spiral galaxies. Tripuram galaxy star streams are short and flaring like the trunk of an elephant. when the Tripuram entered the Shivalogam this projecting head was first pulled by gravity and got cut off. This was given in the story as God Brahma was punished for desiring His own creation Swaroopa. The Milky way galaxy pulled the edge of the main body of Tripuram and the other head was pulled by Shivas galaxy by gravitation in opposite directions and this head also got cutoff. This head belongs to Ganesha and got destroyed. Since Milky way held the main body edge of Tripuram it was given in the story as god Vishnu held the hands of Ganesha. Very surprisingly the outer ring of the Milky way galaxy is disturbed in two places close by as if two hands pulled them out. This is on the side where the inner ring accommodates our solar system.

The Tripuram suffered extensively and some stars and the Swaroopa star garland shifted around God Shiva. Goddess Tripurasundari also shifted next to god Shiva. God Shiva fixed another star stream of short flare like elephant trunk to Tripuram galaxy. The god Brahma Black hole merged with god Shivas black hole in a continuous rotation which we call the dance of god Nataraja. God Ganesha become the creator of Tripuram galaxy and also became the first full god formed from pure energy unlike the other gods who formed from mass called Parabrahma.