Chai Elul (the 18th of Elul) is the birthday of the Baal Shem Tov in the year 5458, and the Alter Rebbe in the year 5505.
It is well known that the Alter Rebbe would call the Baal Shem Tov zeide (grandfather), Simply, this was because the Alter Rebbe was a student of the Baal Shem Tov's student. And our Sages say: "Whoever teaches Torah to the son of another person is considered as if he fathered him."
Our Sages say "as if," with a letter Chaf that signifies resemblance, and not that he actually gave birth to him. As Chassidus also teaches, that every effluence from intellect is only a ray of the teacher, and not his essential influence, as is conveyed by birth. And therefore it is only “as if” he gave birth to him.
Yet the Alter Rebbe called the Baal Shem Tov his grandfather without this qualification (“as if he is my grandfather.") Moreover: it is known that he told R. Baruch of Mezibush, who was the Baal Shem Tov's actual grandson (and he was the son of his daughter, which is a closer connection than a son's son, that he - R. Baruch – is a grandson in the material sense, but that “I” -- the Alter Rebbe referring to himself – am the Baal Shem Tov's grandson in a spiritual sense. And he said this as an more advantageous level. Namely. That the Alter Rebbe received the Pnimiyut (inner dimension) of the Baal Shem Tov. not just an intellectual influence which is a ray, but even that level of Pnimiyut which is not transferred in the essential effluence of birth.
To explain briefly: " Tzadikkim (The righteous) resemble their Creator”. Just as G-d says: "I wrote down and gave over My soul (Anochi)" namely that G-d communicated His very essence in the Torah, so too Tzadikkim communicate their essence in Torah. In Toras Chassidus, the Baal Shem Tov gave over his essence, something which was not given over even when fathering a child.
And since the Alter Rebbe took the inner dimensions of the Baal Shem Tov's teachings, he also, in so doing so, took the Pnimiyut and essence of the Baal Shem Tov himself, and therefore is the true grandson.
So much so, that the inner and essential connection between the Alter Rebbe and the Baal Shem Tov was expressed in the fact that their birthdays, in a physical body, where the body is connected with Atzmut – both of their birthdays – occurred on the same day (of the year) - Chai Elul.
2. The relationship between a grandfather and a grandson is two-sided: .The grandson receives from his grandfather, but he also complements his grandfather. As Chazal state on the verse: "This Torah scroll will not depart from your children, and your grandchildren," our Sages comment: "The Torah returns to its [familiar] abode." And it is written: "Grandchildren are the crown of the aged."
Since the Alter Rebbe considered himself the Baal Shem Tov's grandchild, is understood that he possessed both qualities: a) he received the Baal Shem Tov's essence, as above; and b) he complements the Baal Shem Tov.
The intent of the Baal Shem Tov's teachings was brought to consummate perfection by the teachings of Chabad. The intellectual teachings of Chabad give an elevation also in Chagas and Niyam (emotional and instinctual characteristics) . And as explained elsewhere, Chabad Chassidus accomplished, that even simple people, who seemingly are not suited to understanding and perceiving [Chassidus], could also be uplifted through Chabad Chassidus, even higher than through the general Chassidic approach.
Besides that which the Alter Rebbe and the nesi'im who followed him, revealed, drawing down and enclothing the teachings of the Baal Shem Tov in holy thought, so much so that everyone can understand these spiritual concepts with mundane wisdom;
In addition, they accomplished a completeness in the Baal Shem Tov's teachings.
For in order to draw down, the teachings of the Baal Shem Tov, which transcend mortal intellect, into mundane intellect, requires a very strong (deep) power, through this a higher light is drawn down, which brings into it completenes
According to the above, we can understand the statement of the Tzemach Tzedekconcerning the Alter Rebbe: "Also from my grandfather, we heard prophecies that were fulfilled in hair-splitting detail."
On the surface, this is not understood. Is the Tzemach Tzedek praising the Alter Rebbe in superficial matters, the ability to perform wonders? Is this an appropriate expression of praise for the Alter Rebbe?
But the explanation is: This comes after the Tzemach Tzedek's previous statement that: "He [The Baal Shem Tov] and his disciple, the Maggid, would actually see from one end of the world to the other with a visual sight . . . namely, [ they were able to see] through having revealed to them, the light which was created on the first day of creation that G-d hid in the Torah.
And on this statement, the Tzemach Tzedek said: "Also from my grandfather, we heard prophecies etc." From the word “Also” it is proven that the Alter Rebbe's “prophecies” were of the same nature as the miracles performed by the Baal Shem Tov and the Maggid: the revelation of the “light that was created on the first day”. But by the Alter Rebbe they were fulfilled - in “hair-splitting detail”.
The “light that was created on the first day” is above the limitations of Seder HaHishtalshelus, (the chain of creative stages) and even higher - the light which existed before the tzimtzum, and even there, the light intended to reveal G-d's essence.
Even though, that for the Baal Shem Tov and the Maggid, this light also illuminated regarding physicality, so that they would actually see with physical sight; For the from one end of the world to the other" with this light; i.e., they were able to perceive how this light was reflected within material reality.
And the Alter Rebbe reached an even greater peak;for him, the light shone in a settled matter, enabling his predictions to be fulfilled "in precise detail." This indicates the influence of G-d's essence itself. For to have an unlimited light revealed within this world in a settled manner requires a manifestation of G-d's essence, which fuses opposites together.
This anticipates the revelations of the Era of the Redemption, when G-d's desire for a dwelling in the lower worlds will be fulfilled. At that time, G-d's essence, which fuses together the spiritual and the physical, will become manifest: "The glory of G-d will be revealed, and all flesh will see...." Physical flesh will see G-dliness, and this will be in a settled manner.
This also was alluded to by the Tzemach Tzedek, for the Hebrew word he used for predictions was asidos, referring to the asid lavo, "the future time," the Era of the Redemption.
Through the study of Chassidus Chabad, we can perceive the deepest spiritual lights, including even the light which was hidden. Moreover, this can be comprehended by the intellect of the G-dly soul, and even with the intellect of the animal soul. This will lead to the time when "the glory of G-d will be revealed, and all flesh will see..." with the coming of Mashiach; may this be in the immediate future.