I don’t know about the maTha and inscription, but I know a little about sannyAsa. So, I’ll write about that here.

First of all there is no advaita-sannyAsa in this world. The sannyAsa which we take is vaidika-sannyAsa, if it is to be labeled. We label it as vaidika-sannyAsa to distinguish it from sannyAsa which follows Agama-s and tantra-s.
We maintain that sha~NkarachArya didn’t commence or devised sannyAsa. sannyAsa as Ashrama is as old as other Ashrama-s and veda-s are.
You may call us advaitin because we support advaita-philosophy, and not because our sannyAsa is according to such tradition.
Till date, I’ve not heard the term ‘kApila-sannyAsa’ being used for us in any text. We are not sA~Nkhya-s at all.
The text which is being followed now for rites of sannyAsa in kailAsa-Ashrama and all related maTha-s (which includes RK Mission, shivAnanda Ashrama, etc., i.e. tradition of daNDatyAgI paramahaMsa-s) is rudradeva’s sannyAsa-paddhati. It is mere collection of detailed rites which are scattered in veda-s and smR^iti-s.
I don’t know what daNDin-s(eka-daNDin-s to be precise, as I can’t see tridaNDin-s following advaita) follow. Though we accept that daNDa-sannyAsa exists(we ourselves shed daNDA after taking it!!!), we have doubts about origin of mudrA-s they put on daNDa-s and the use of such daNDa. vidyAraNya in jIvanmuktiviveka, while explaining paramahaMsopaniShad, explains the use of daNDa as ‘go-sarpa-adyupadrava-parihAra’, which is not accepted by present daNDin-s. They think that it is for alaukika-kArya, pUjana-tarpaNa, etc.
If we add nAgA-paraMparA(I already talked to two – daNDatyAgI paramahaMsa-s and daNDI paramahaMsa), then they are not totally different from daNDa-tyAgin-s. They have a few more steps before they get sannyAsa, which are not found in scriptures. They get the sannyAsa which includes daNDa-tyAga by similar sannyAsin-s.
BTW, nAgA-sannyAsI-s maintain that they have originated from either kapila, skanda, dattAtreya, etc. and not from sha~NkarAchArya or madhusUdana-sarasvatI. If you say that any of both person created their saMpradAya, you may get rebuked and hit.

Although, veda-s say that sannyAsI shouldn’t stay more than 3 days in village and 5 days in city; it is allowed to stay for long times in tIrtha-xetra-s and forever in kAshI. I may need access to books to exactly put the shloka-s here. I’ll do that when I get them near me.
And, while explaining upaniShad-s, shrI upaniShad-brahmendra supports it by saying that for sake of vedAntAdhyayana, a sannyAsI can stay with guru for long. There can be no serious study for ever moving man.
Therefore, it is good to conclude that 3-5 days rule is for those vidvAn-virakta-s or upAsaka-virakta-s who have no desire to study and are not interested in jana-saMsarga.
Those who stay for long nowadays, either do it for studies, or because they lack vairAgya.

You are correct when you said that sannyAsI can’t create temple, just because he is supposed to be aparigrahI and jana-saMsarga-virata.
Those who make temples and maTha-s are not aparigrahI/virakta, can be easily deduced.

He is also not supposed to do bAhya-pUjana to maintain his aparigraha, ahiMsA, etc.
Those who do so are not aparigrahI and ahiMsaka, is to be accepted.

Now, the question of texts regarding creation of temple is left.
As I said that we are vaidika-s and don’t think ourselves a separate tradition, so any text which follows veda-s and smR^iti-s and talks about temple construction, is good according to us.
We have no problem with it’s validity.
But, I’ve doubt if there is any text.
I’ve seen Agama-s and tantra-s dealing with temple construction, though.

यतिरहं श्रीमतो भगवतः शङ्कराचार्यस्य भगवत्पादाभिधां बिभ्रतः सम्प्रदायानुगतो वेदतदनुकूलसकलशास्त्रनिचये श्रद्धावान् गुरुणाऽनुल्लङ्घनीयशासनेन विदितवेदवेदान्तशैवाद्यागमतन्त्रादिरसरहस्येनाऽनुगृहीतो निश्चितवेदप्रामाण्यस्तादृशविद्वत्त्वलोभी येन जीवन्मुक्त्यादिक्रमेण विदेहकैवल्याप्तिरुपदिष्टा मात्रा श्रीमत्या स्नेहपरयोमया हैमवत्या श्रुत्या॥

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