We begin this journey remembering the Mangal Murti Ganapati. (गणपति)

'गणशब्दः समूहस्य वाचकः परिकीर्तितः'

Word Gana indicates 'group'
Ganapati is lord of groups.

Lord of group of Devas, Rishis, Pitris Manavas, Asuras is Ganpati.
ॐ नमस्ते गणपतये त्वमेव केवलं कर्तासि, त्वमेव केवलं धर्तासि, त्वमेव केवलं हर्तासि,त्वमेव सर्वं खल्विदं ब्रह्मासि। etc गणपत्यथर्वशीर्ष(1) sing the reality as गणपति।
The above mantra also says त्वमेव प्रत्यक्षं तत्वमसि। Hence mahatma say Ganapati with visage of a गज (elephant) and body of नर (man) establishes oneness of two different entities-ईश्वर and जीव। तत् is ईश्वर and त्वम is जीव।
His 4 hands has- पाश, अंकुश, वरद and अभय मुद्रा।
पाश binds delusions and illusions of Sadhaka,
अंकुश controls this entire creation. With अभय and वर he protects the devotees and fulfills their desires.
His वाहन is मूषक। मूषक is अन्तर्यामी in the den called हृदय।

मूष स्तेये- this is origin of मूषक। Just as mouse steals things in a house, same way अंतर्यामी steals भोग hence पुण्य and पाप
The same ineffable reality manifests itself in various Yugas as Son of शिव and पार्वती।

It's well known हरि अनंत हरि कथा अनंता।
The reality is beyond measures hence his legends are infinite. That's why परात्पर तत्व is born in different ways in different Kalpas
In श्वेतकल्प - गणपति was created by Parvati out her remains of a paste on her body. He was beheaded by Shiva after a fierce battle, and was revived with a head of an elephant. He was younger brother of Skanda.

This story comes in Shiva Purana
In another Kalpa, Devtas had been defeated by Asuras. Then for their welfare elephant faced विनायक emerged from his mouth. He created obstacles in acts of Asuras and made Devtas victorious. Here he was elder brother of Skanda. Story is in Linga Purana
In Brahmavaivarta Purana, Parvati performs the arduous पुण्यक व्रत and Shri Krishna was born as her son.

Via वक्र दृष्टि of शनि, the new born was beheaded and was revived with an elephant Head.
In Ramcharitmanas it was shown that गणपति was worshipped even in शिव पार्वती विवाह, because गणपति is सनातन तत्व।

मुनि अनुसासन गनपतिहि पूजेउ संभु भवानि।
कोउ सुनि संसय करै जनि सुर अनादि जियं जानि।।
गज regarding गणपति has been described in Shastras thus- समाधिना योगिनो यत्र गच्छंति इति गः- that where yogis go via समाधि is ग।
यस्माद् बिम्बप्रतिबिम्बतया प्रणवात्मकं जगज्जायते इति जः- just as reflection emerges from an image, same way from whom this creation arises is ज।
Thus one परात्पर तत्व has extolled as गणपति गजानन by Vedas and Puranas.

We bow down to Auspicious विनायक and pray to be him that he illuminates the intellect of Hindus via his kripa
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