Sorry for the long response, there is a huge amount of information that one needs to digest to make sense of it all. So let me try to explain it to some extent.
Madhyadeśa or Madhyama-deśa (Pāli: majjhima-deśa) is not Madhya-pradeśa. Madhya-pradeśa is the modern Indian state of Madhya Pradesh.
Rājagṛha has not always been associated with modern-day Rajgir - Rājgir is a relatively modern name and it has been associated with Rājagṛha only since Alexander Cunningham’s identification of EBT-sites since circa the mid-19th century.
Before that nobody in India knew anything about such an association. There is nothing excavated or found at Rajgir or any site within a 100 mile radius that conclusively shows that the site was indeed called Rājagṛha (or that it was unambiguously associated with Magadha or the Mauryan dynasty) in antiquity. The same can be said about Patna which is a modern name (there is no conclusive evidence to show that the place was called pataliputra in antiquity that I am aware of).
Yes, and most of those sites were identified and/or renamed by British colonial archaeologists like General Cunningham in the 19th century to build a credible theory about them being located in Eastern India in the time of Ashoka. Since then, those proposals have not been challenged as far as I know, so old speculations continue to circulate as if they are incontrovertible facts. However the places so identified are based on flimsy evidence, and there is more solid evidence against them than there is for them. The Indian central and state governments do not want to lose tourist revenue from the Buddhist circuit so they keep propagating the colonial-era stories.
A few examples of where the evidence doesn’t fit :
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There is a eastern site called Sarnath that is associated with Pali Isipatana (BHS Ṛṣi-patana). The pali commentary says it is called so because it was the place where the buddhist isi-s (buddhas & pacceka-buddhas) used to launch into and descend from the skies when they were flying about. (isipataneti buddhapaccekabuddhasaṅkhātānaṃ isīnaṃ dhammacakkappavattanatthāya ceva uposathakaraṇatthāya ca āgantvā patane, sannipātaṭṭhāneti attho). However Prof. Collette Caillat has shown half a century ago that the name is a misunderstanding - the word is not the name of a place, isipatana is basically a corruption of a common noun ṛśya-vrjana (which means the same as migadāya) - you can read an English translation of her paper at https://www.academia.edu/22721043/Isipatana_Migadaya_Translation
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Most of the Jain populations (and their oldest and holiest sites are located in Western Indian states such as Gujarat, Rajasthan and Maharashtra. The literary evidence however points to the idea that Jains inhabited pretty much the same regions as the Buddhists. Girnar itself is a very important Jain pilgrimage site, while they are nowhere to be found in Patna - the ostensible capital of the Mauryans. Thus Magadha is likely located in and around Gujarat, not anywhere near Bihar. The Jains have never had any historical associations with eastern-India. Here is a map of their population-density. Jainism in India - Wikipedia
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There is also evidence that the janapadas of Kāśi, Videha & specifically Kosala (and its capital Śrāvasti which is described as being on the banks of Ajiravatī) was located closer to Punjab and not in Eastern UttarPradesh. Ajiravatī is called Hydraotes in Greek records in the accounts of Alexander, and is also mentioned by Panini as a north-western river. Further Patañjali mentions in the 2nd century BCE that the Yavanas had attacked Sāketa in his time, which could have been the invasion of Demetrius II (as Sāketa was located somewhere close to Punjab in his time) and Demetrius never ruled eastern India as per available evidence. All Indo-Greek kingdoms were based only in the North-West.
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Mauryas are called மோரியர் (Moriyar) or its lexical variant ஓரியர் (Oriyar) in early Tamil literature and they have the epithet சக்கரவாள சக்கரவர்த்திகள் (cakravāla cakravartin-s) - the cakravāla of course referring to their rule from the circular mountains at Girivraja (Girnar) which functioned as their natural fortress. [In later literature, the name cakravāla is mythologized as the name of a similar circular cosmic mountain range that surrounds the entire world but originally the Mauryans are specifically associated with the cakravāla/girivraja mountains that are naturally shaped round like a cakra]. This does not fit Rajgir, but fits Girnar
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We know for a matter of fact that Girnar was a major political center for Ashoka, he has left the best preserved copy of all his major rock edicts there, the language looks very stable and clear here and seems to be the first recorded site of those edicts. Several major Indian Emperors such as Rudradāman and Samudragupta have left their inscriptions right next to Ashoka’s edicts at Girnar - thus we see that Girnar was a prestigious/important place at least from the time of Chandragupta Maurya (4th century BCE) until Samudragupta (4th century CE) - thus spanning about 700-800 years.
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Also there are references in the Pāli canon to trade and business visits of Brahmins and others arriving from Vairantya/Verañja to Śrāvasti/Sāvatthī (MN42 for example) which makes more sense if we assume that Śrāvasti and Vairantya were 350 kms away from one another and not 1500 kilometres away from each other (i.e. if we assume Śrāvasti was in the Punjab and not in Eastern Uttar-Pradesh).
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Pāli Aciravatī (mentioned as river on whose banks stood Śrāvastī, the capital of Kośala janapada as per the Pāli canon) is Sanskrit Ajiravatī (mentioned by Pāṇini - A6.3.119, as the name of a north-western river). The greek accounts of Alexander’s invasion preserve the name of the river Ajiravatī as Hydraotes. Vitastā similarly gets recorded as Hydaspes, and Asiknī is mentioned as Acesines, Vipāś is named Hyphasis. Moreover there is a king named Ambhi or Ambisares ruling in Punjab in the Alexander accounts which reminds of a similarly named earlier king Bimbisāra. Alexander after crossing the Hydraotes(Ajiravatī) and advancing east came across the Malloi/Mallians which can be interpreted as the Malla janapada who are described as fierce warriors. The Alexander records also mention Sisicottus, ruler of the Assaceni in North-West India who met Alexander - and I believe this was perhaps Śaśigupta (a variant name for Chandragupta, śaśi and candra both mean “moon”) and perhaps he was already a ruler/governor commanding the Aśvakas/Aśvakāyanas in the time of Alexander (around 325 BCE).
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Besides, most of the excavated personal relics of the Buddha until date are from sites in the Gandhara region between Taxila and Peshawar, not from eastern-India.
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The Arthaśāstra of Kauṭilya (2.30.29) composed towards the end of the 4th century BCE, classifies horses based on the places in the western border regions from which they are imported as follows: prayogyānām uttamāḥ kāmboja.saindhava.āraṭṭa.vanāyujāḥ. madhyamā bāhlīka.pāpeyaka.sauvīraka.taitalāḥ. śeṣāḥ pratyavarāḥ
Meaning of the above: "Best horses are from : Kāmboja (realm of Cambyses i.e. Persian), Sindhu, Āraṭṭa and Vanāyu (i.e. Arabia). Middle-quality are from: Bahlika (Bactria), Pāpā, Sauvīra & Titala. Rest are poor quality.
Almost all of these are either other countries to the west & north-west - like Persia, Arabia, Bactria etc - or western & northwestern janapadas. Sindhu & sauvīra were located next to each other, and are usually mentioned together as Sindhu-sauvīra. What interested me is that apart from some locations outside India proper, like Kāmboja, Vanāyu & Bahlika, there are some relatively rare places like Pāpā that are mentioned in the Arthaśāstra. Pāpā of the Arthaśāstra is Pāvā of the Pali canon, the capital of one of the two Malla janapadas, Kusināra being the capital of the other Malla janapada. So we see that Pāpā/Pāvā was very likely a north-western town from where horses were imported by the Mauryans to other parts of their empire. -
Alexander met the Malloi/Malli (Malla-janapada) between the Hydaspes (Vitastā/Jhelum) and Acesines (Asikni/Chenab) - and found them fierce and warlike. Their neighbours the Oxidraci (i.e. Kṣudraka, in Pāṇini’s opinion (sūtra 5.3.114), were āyudhajīvis i.e. professional warriors) who joined with the Mallas to oppose Alexander. The Kṣudrakāḥ are named khuddakā in the pāli canon but are not very frequently met with there). We dont know which Mallas these are who are located between the Vitastā and Asiknī - the Pāpā mallas or the Kuśinagara mallas. But it gives further evidence of the Mallas also being a north-western janapada.
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Let me also draw your attention to Major Rock Edict 1 of Ashoka - his very first order in his very first edict is to ban animal slaughter in his kingdom (idha na kiṃci jīvaṃ…) and commit himself to vegetarianism - and the grammarian Kātyāyana in his vārttikas refers to devānāṃpriya (the title of the Mauryan kings) as an exception to Pāṇini’s sūtra 6.3.21 and calls Ashoka the vegetarian king (śāka-pārthiva) while commenting on sūtra 2.1.69. We have to see where Aśoka’s ban had the most effect - in Magadha itself surely? Do you think Bihar (or the eastern janapadas) became predominantly vegetarian - or does vegetarianism fit the description of Gujarat/Rajasthan etc? Here is a map which makes the answer easy.
- Coming to the information we can gather from other Ashokan edicts, at Dhauli (Orissa) he made two separate edicts which are not found outside the Kalinga region. In one of these edicts he makes a promise to the people of Kalinga that he will send out periodically his officers from Ujjayini and from Takshashila - to Kalinga to check about the people’s welfare. This is very important - why are these Mauryan royal officers despatched by Aśoka from the Western Janapadas i.e. Ujjayini (Avanti) and Takṣaśilā rather than coming to Kalinga from Patna or Rajgir if the Mauryans ruled from Bihar? It would rather make sense if his royal base was mainly in the Western part of the country. [[The relevant text from the edict: Ujenite pi cu kumāle etāye va aṭhāye nikhāmayisa . . . . . . hedisameva vagaṃ no ca atikāmayisati tiṃni vasāni hemeva Takhasilāte pi adā …]]
Here is what Aśoka’s Major Rock Edict 5 at Girnar and its copies at other places say:
Girnar Major Rock Edict 5 - Ashoka says in one of the lines: Pāṭalipute ca bāhiresu ca nagaresu
The same line of the same RE5 in Kalsi, Uttarakhand reads: hid[ā] bā[h]ilesu cā naga[l]esu
The same line of the same RE5 in Shahbazgarhi (near Peshwar) reads: ia bhahireśu ca nagareśu
The same line of the same RE5 in Mansehra (near Abbotabad & Muzaffarabad, Kashmir) reads: hida bahireṣu ca nagareṣ[u]
The same line of the same RE5 in Dhauli (in Odisha) reads: hida ca bāhilesu ca nagalesu
So we see that only at Girnar is the name Pataliputra mentioned (in all other places of the subcontinent, it is replaced by hida i.e. iha) - which certainly means that only the people around Girnar were expected to know the location of Pāṭaliputra (as it was close to that location). At the other places where the name Pāṭaliputra would not be recognized (as it was a new city built in the Mauryan era) unlike Girivraja, the edicts did not mention the name. But the people of Magadha living at Girivraja were expected to recognize and know Pataliputra.
So Magadha was somewhere in Gujarat, not near Patna or Bihar. Do you have a more plausible reason than mine as to why the name of Pataliputra wasnt mentioned in any of the copies and only at Girnar?
If we consider Girnar as the earlier capital of Magadha, it explains why Pali (as a western-Indo-Aryan language close to the language of the Girnar Ashokan edicts) is described by the Theravadins as being based on Magadhabhāṣā.