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Srimad Bhagavatam is considered natural commentary of Vedanta Sutras by Gaudiya Vaishnavas.

That is the reason Gaudiya school didn't explicitly subscribe to Vedanta school till 17th century. Then for some secondary reasons, Baladeva Vidyabhusana commented on prasthana trayi(Vedanta Sutras, Bhagavad Gita & principal upanishads).

Govinda Bhasya is the commentary of Baladeva Vidyabhusana on Vedanta Sutras. This commentary is also in the similar style as previous acharyas of other schools.

Is there any work which shows corresponding Bhagavatam verses to each Vedanta Sutra?

Eg: VS 1.1.1 -> SB 1.2.8-11 VS 1.1.2 -> SB 1.1.1 VS 1.1.3 -> SB 10.16.44 VS 4.4.22 -> SB 10.88.26, SB 2.8.6

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As far as I know, there is no ancient work that has corresponding verses of Bhagavatam mapped to each Vedanta Sutra. Jiva Goswami did that to an extent in his sandarbhas but that is only in spirit not verse by verse. He explained first 5 Vedanta Sutras using first verse of bhagavatam, but others are scattered.

There are two modern works(20th century) however by Haridasa Shastri and Ramapada Chattopadhyaya.

This is Govinda Bhasya commentary by Baladeva Vidyabhusana in Sanskrit and at the end of each Vedanta Sutra, Bhagavatam verse numbers are added based on works of Haridasa Shastri and Ramapada Chattapadhyaya.

Govinda Bhasya Adhyaya 1

Govinda Bhasya Adhyaya 2

Govinda Bhasya Adhyaya 3

Govinda Bhasya Adhyaya 4

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Bhagavata is considered natural commentary on Vedanta Sutra.

This is what Jiva Goswami says in Tattva sandarbha, text 21:

Garuḍa Purāṇa says:(quoted by Madhva in his commentary on SB 1.1.1)

pūrṇaḥ so’yam atiśayaḥ | artho’yaṁ brahma-sūtrāṇāṁ bhāratārtha-vinirṇayaḥ || gāyatrī-bhāṣya-rūpo’sau vedārtha-paribṛṁhitaḥ | purāṇānāṁ sāma-rūpaḥ sākṣād-bhagavatoditaḥ || dvādaśa-skandha-yukto’yaṁ śatavic-cheda-saṁyutaḥ | grantho’ṣṭādaśa-sāhasraḥ śrīmad-bhāgavatābhidhaḥ ||

This work is most complete. It expresses the meaning of the Brahma-sūtras and Mahābhārata. It is the explanation of gāyatrī and reinforces the meaning of the Vedas. It is the Sāma Veda among the Purāṇas and has arisen directly from the Lord. It has twelve volumes and hundred divisions. It has eighteen thousand verses. It is called Śrīmad-bhāgavatam.

Jiva Goswami explains the words 'artho'yam brahmasutranam' as follows:

It is the meaning of the Brahma-sūtras. This means it is the natural commentary on the sutras. Previously it appeared in his mind in a subtle form. Abbreviated, it appeared again as the sutras. Later it appeared as Bhāgavatam with great details. Though this is the self-evident commentary on the sūtras, other recent commentaries made by independent thinking are to be respected if they follow the Bhāgavatam.

What he says is that modern commentators on Vedanta will be taken seriously to the extent they agree with Bhagavata conclusions.

Since, the very first verse of Bhagavata contains phrases like janmadasya yatah, dhimahi. It has Vedantic import and Jiva goswami uses this verse to show how the seeds of Vedanta are there in bhagavatam. Here, he uses other commentators of Vedanta Sutras' explanations to show that they can derived from first verse of bhagavatam. He doesn't use Bhagavatam as commentary of Brahma Sutras, he uses the explanations of other commentaries wherever they are in line with Bhagavatam to show how those ideas lie in first verse of bhagavatam.

oṁ namo bhagavate vāsudevāya

janmādy asya yato ’nvayād itarataś

cārtheṣv abhijñaḥ svarāṭ tene brahma hṛdā ya ādi-kavaye muhyanti yat

sūrayaḥ tejo-vāri-mṛdāṁ yathā vinimayo yatra tri-sargo ’mṛṣā dhāmnā

svena sadā nirasta-kuhakaṁ satyaṁ paraṁ dhīmahi

Here are some things he explained on the basis of first verse of Bhagavatam.

The correlation between first 5 sutras of Brahma Sutras and first verse of Bhagavata. enter image description here

Vedantic discussions based on first verse of Bhagavatam:

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Correlation between Upanishadic passage and first verse of Bhagavatam:

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The correlation between Brahma Gayatri and first verse of Bhagavatam:

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The correlation between names of chapters of Brahma Sutra and first verse of Bhagavatam:

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This discussion he does at the end of Paramatma Sandarbha, apart from that he doesn't enter into Vedanta discussion.

As the other user pointed out, there are modern works however which try to correlate Bhagavatam verses with Vedanta Sutras based on Govinda Bhasya of Baladeva Vidyabhusana.

You may download it from Vedanta Sutra Srimad Bhagavatam Commentary. It does a good job of putting together English translations of Vedanta Sutra aphorisms along with English translations of each corresponding Bhagavata verse(s)

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  • @Pratimaputra it's Jiva goswami
    – user2612
    Mar 18, 2019 at 10:10
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Quoting from Wikipedia

Śrī Caitanya Mahāprabhu had explained that the Śrīmad Bhāgavatam is the natural commentary of the Vedanta-sūtras. But he had not explained how individual verses of the two scriptures related to each other. Sri Haridas Shastri addressed this key gap in the Gauḍīyā Vaiṣṇava literature, by writing and publishing his book Vedānta- darśanam.[6] This book explains how a verse or verses of the Śrīmad Bhāgavatam comment on a specific Vedanta-sūtra. This work is exceptional in its scholarship, and was honored with an award by the Nāgarī Pracāriṇī Sabhā of Benares

Vedanta Darshanam (Bhagavata Bhashyopetam) can be read in Hindi here

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