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The following injunction is given in the Hari-bhakti-viläsa (1.194):

täntrikeñu ca mant
dékñäyäà yoñitäm api
sädhvénäm adhikäro ’sti
çüdrädénäà ca sad-dhiyäm

“Çüdras and women who are chaste and sincerely interested in understanding the Absolute Truth are qualified to be initiated with the päïcarätrika-mantras.”
This is also confirmed in Bhagavad-gétä (9.32):

mäà hi pärtha vyapäçr
ye ’pi syuù päpa-yonayaù
striyo vaiçyäs tathä çüdräs
te ’pi yänti paräà gatim

“O son of Påthä, those who take shelter in Me, though they be of lower birth—women, vaiçyas [merchants], as well as çüdras [workers]—can approach the supreme destination.”

If one actually wants to serve Kåñëa, it doesn’t maner whether one is a çüdra, vaiçya, or even a woman. If one is sincerely eager to chant the Hare Kåñëa mantra or dékñä-mantra, he is qualified to be initiated according to the päïcarätrika process. According to Vedic principles, only a brähmaëa who is fully engaged in his occupational duties can be initiated. Çüdras and women are not admitted to a vaidika initiation. Unless one is fit according to the estimation of the spiritual master, one cannot accept a mantra from the päïcarätrika-vidhi or the vaidika-vidhi. When one is fit to accept the mantra, he is initiated by the päïcarätrika-vidhi or the vaidika-vidhi.

Sudras are not entitled to the study of the Vedas

Brahma-Sutras  1-3-36:
Because purificatory ceremonies are mentioned (in the case of the twice born) and their absence are declared (in the case of the Sudras).
[Note: Purificatory ceremonies like Upanayana (Sacred Thread) etc. are declared by the scriptures to be a necessary condition of the study of all kinds of knowledge or Vidya; but these are meant only for the higher castes (Brahmins, Kshatriyas and Vaisyas). Their absence in the case of the Sudras is repeatedly declared in the scriptures. "Sudras do not incur sin (by eating prohibited food), nor have they any purificatory rights " etc. (Manusmrti  10-12-6). Consequently they are not entitled to the study of the Vedas.]

Braham-Sutras 1-3-37
And because the inclination (on the part of Gautama to impart Knowledge is seen only) on the ascertainment of the absence of Sudra-hood (in Jabala Satyakama from the Chandogya Upanishad of the Sama Veda, 4-49.)
[Note: That Sudras are not qualified is known also from the fact that great teachers like Gautama made sure before imparting Knowledge that disciples like Jabala Satyakama were not Sudras.]

Brahma-Sutras 1-3-38
And because of the prohibition in the Smrti of hearing and studying (the Vedas) and knowing their meaning and performing their Vedic rites to Sudras, they are not entitled to the Knowledge of Brahman.
[Note: Sutras 34-38 of Brahma-Sutras disqualify the Sudras for the Knowledge of Brahman (Supreme Reality) through the study of the Vedas. But it is possible for them to attain that Knowledge through the Puranas and the epics (Ramayana and the Mahabharata).

Then what is required is a transformation, what kind of transformation? The fact is that material juggling of elements and castes even is not evough, what is required is that the living entity become spiritualised. Just as an iron rod when placed into the fire takes on the attributes of fire, so in the same way when one associates with genuine spiritual practices through the system of samashrayanam what takes place is a genuine transformation in the willing recipriant:

kià janmabhis tribhir veha
karmabhir vä trayé-proktaiù
puàso ’pi vibudhäyuñä

kim—what is the use; janmabhiù—of births; tribhiù—three; vä—or; iha—in this world; çaukra—by semen; sävitra—by initiation; yäjïikaiù—by becoming a perfect brähmaëa; karmabhiù—by activities; vä—or; trayé—in the Vedas; proktaiù—instructed; puàsaù—of a human being; api—even; vibudha—of the demigods; äyuñä—with a duration of life.

A civilized human being has three kinds of births. The first birth is by a pure father and mother, and this birth is called birth by semen. The next birth takes place when one is initiated by the spiritual master, and this birth is called sävitra. The third birth, called yäjïika, takes place when one is given the opportunity to worship Lord Viñëu. Despite the opportunities for attaining such births, even if one gets the life-span of a demigod, if one does not actually engage in the service of the Lord, everything is useless. Similarly, one’s activities may be mundane or spiritual, but they are useless if they are not meant for satisfying the Lord.

PURPORT by HDG Srila A.C. Bhaktivedanta Swami Prabhupada:
The word çaukra janma means “taking birth by seminal discharge.” Animals can take their birth in this way too. However, a human being can be reformed from the çaukra janma, as recommended in the Vedic civilization. Before the birth takes place, or before father and mother unite, there is a ceremony called garbhädhäna-saàskära, which must be adopted. This garbhädhäna-saàskära is especially recommended for higher castes, especially the brähmaëa caste. It is said in the çästras that if the garbhädhäna-saàskära is not practiced among the higher castes, the entire family becomes çüdra. It is also stated that in this age of Kali, everyone is çüdra due to the absence of the garbhädhäna-saàskära. This is the Vedic system. According to the päïcarätrika system, however, even though everyone is a çüdra due to the absence of the garbhädhäna-saàskära, if a person has but a little tendency to become Kåñëa conscious, he should be given the chance to elevate himself to the transcendental platform of devotional service. Our Kåñëa consciousness movement adopts this päïcarätrika-vidhi, as advised by Çréla Sanätana Gosvämé, who says:

yathä käïcanatäà yäti
käàsyaà rasa-vidhänataù
tathä dékñä-vidhänena
dvijatvaà jäyate nåëäm

“As bell metal, when mixed with mercury, is transformed to gold, a person, even though not golden pure, can be transformed into a brähmaëa, or dvija, simply by the initiation process.” (Hari-bhakti-viläsa 2.12) Thus if one is initiated by a proper person, he can be accepted as twice-born immediately. In our Kåñëa consciousness movement, we therefore offer the student his first initiation and allow him to chant the Hare Kåñëa mahä-mantra. By chanting the Hare Kåñëa mahä-mantra regularly and following the regulative principles, one becomes qualified to be initiated as a brähmaëa, because unless one is a qualified brähmaëa he cannot be allowed to worship Lord Viñëu. This is called yäjïika janma. In our Kåñëa consciousness society, unless one is twice initiated—first by chanting Hare Kåñëa and second by the Gäyatré mantra—he is not allowed to enter the kitchen or Deity room to execute duties. However, when one is elevated to the platform on which he can worship the Deity, his previous birth does not matter.

caëòälo ’pi dvija-çreñöho
hari-bhakti-vihénaç ca
dvijo ’pi çvapacädhamaù

“Even if one is born in the family of a caëòäla, if one engages in the devotional service of the Lord, he becomes the best of brähmaëas. But even a brähmaëa who is devoid of devotional service is on the level of the lowest dog-eater.” If a person is advanced in devotional service, it does not matter whether he was born in a caëòäla family. He becomes purified. As Çré Prahläda Mahäräja said:

vipräd dviñaò-guëa-yutäd aravinda-näbha-
pädäravinda-vimukhäc chvapacaà variñöham
(SB 7.9.10)

Even if one is a brähmaëa and is qualified with all the brahminical qualifications, he is considered degraded if he is averse to worshiping the Supreme Personality of Godhead. But if a person is attached to the service of the Lord, he becomes glorified even if he is born in a caëòäla family. Indeed, such a caëòäla can deliver not only himself but all his family predecessors. Without devotional service, even a proud brähmaëa cannot deliver himself, and what to speak of his family. In many instances in the çästras it is seen that even a brähmaëa has become a kñatriya, vaiçya, çüdra, mleccha or non-brähmaëa. And there are many instances of one’s being born a kñatriya or vaiçya or even lower and, in the eighteenth year, attaining elevation to the brahminical platform by the process of initiation. Therefore Närada Muni says:

yasya yal lakñaëaà proktaà
puàso varëäbhivyaïjakam
yad anyaträpi dåçyeta
tat tenaiva vinirdiçet

It is not a fact that because one is born in a brähmaëa family he is automatically a brähmaëa. He has a better chance to become a brähmaëa, but unless he meets all the brahminical qualifications, he cannot be accepted as such. On the other hand, if the brahminical qualifications are found in the person of a çüdra, he should immediately be accepted as a brähmaëa. To substantiate this there are many quotations from Bhägavatam, Mahäbhärata, Bharadväja-saàhitä and the païcarätra, as well as many other scriptures.

As far as the duration of life of the demigods, concerning Lord Brahmä it is said:

ahar yad brahmaëo viduù
rätrià yuga-sahasräntäà
te ’ho-rätra-vido janäù
(Bg. 8.17)

The duration of one day of Brahmä is one thousand times greater than the four yugas, aggregating 4,320,000 years. Similarly, Brahmä’s one night. Brahmä lives for one hundred years of such days and nights. The word vibudhäyuñä indicates that even if one gets a long life-span, his life-span is useless if he is not a devotee. A living entity is the eternal servitor of the Supreme Lord, and unless he comes to the platform of devotional service, his life-span, good birth, glorious activities and everything else are null and void. (Srila A.C. Bhaktivedanta Swami Prabhupada. Srimad Bhagavatam 2:31:10. purport)

See more pages on the Hare Krishna maha mantra HERE

See more on Daivi Varnashram Dharma HERE