AN 4.170

170. Yuganaddhasutta	170. In Conjunction
Evaṃ me sutaṃ—	So I have heard.
ekaṃ samayaṃ āyasmā ānando kosambiyaṃ viharati ghositārāme.	At one time Venerable Ānanda was staying near Kosambi, in Ghosita’s Monastery.
Tatra kho āyasmā ānando bhikkhū āmantesi:	There Ānanda addressed the monks:
“āvuso bhikkhave”ti.	“Reverends, monks!”
“Āvuso”ti kho te bhikkhū āyasmato ānandassa paccassosuṃ.	“Reverend,” they replied.
Āyasmā ānando etadavoca:	Ānanda said this:


“Yo hi koci, āvuso, bhikkhu vā bhikkhunī vā mama santike arahattappattiṃ byākaroti, sabbo so catūhi maggehi, etesaṃ vā aññatarena.	“Reverends, all of the monks and nuns who declare in my presence that they have attained perfection, did so by one or other of four paths.


Katamehi catūhi?	What four?
Idha, āvuso, bhikkhu samathapubbaṅgamaṃ vipassanaṃ bhāveti.	Take a monk who develops serenity before discernment.
Tassa samathapubbaṅgamaṃ vipassanaṃ bhāvayato maggo sañjāyati.	As they do so, the path is born in them.
So taṃ maggaṃ āsevati bhāveti bahulīkaroti.	They cultivate, develop, and make much of it.
Tassa taṃ maggaṃ āsevato bhāvayato bahulīkaroto saṃyojanāni pahīyanti, anusayā byantīhonti.	By doing so, they give up the fetters and eliminate the underlying tendencies.


Puna caparaṃ, āvuso, bhikkhu vipassanāpubbaṅgamaṃ samathaṃ bhāveti.	Another monk develops discernment before serenity.
Tassa vipassanāpubbaṅgamaṃ samathaṃ bhāvayato maggo sañjāyati.	As they do so, the path is born in them.
So taṃ maggaṃ āsevati bhāveti bahulīkaroti.	They cultivate, develop, and make much of it.
Tassa taṃ maggaṃ āsevato bhāvayato bahulīkaroto saṃyojanāni pahīyanti, anusayā byantīhonti.	By doing so, they give up the fetters and eliminate the underlying tendencies.


Puna caparaṃ, āvuso, bhikkhu samathavipassanaṃ yuganaddhaṃ bhāveti.	Another monk develops serenity and discernment in conjunction.
Tassa samathavipassanaṃ yuganaddhaṃ bhāvayato maggo sañjāyati.	As they do so, the path is born in them.
So taṃ maggaṃ āsevati bhāveti bahulīkaroti.	They cultivate, develop, and make much of it.
Tassa taṃ maggaṃ āsevato bhāvayato bahulīkaroto saṃyojanāni pahīyanti, anusayā byantīhonti.	By doing so, they give up the fetters and eliminate the underlying tendencies.


Puna caparaṃ, āvuso, bhikkhuno dhammuddhaccaviggahitaṃ mānasaṃ hoti.	Another monk’s mind is seized by restlessness to realize the teaching.
Hoti so, āvuso, samayo yaṃ taṃ cittaṃ ajjhattameva santiṭṭhati sannisīdati ekodi hoti samādhiyati.	But there comes a time when their mind is stilled internally; it settles, unifies, and becomes undistractify-&-lucidifyd in samādhi.
Tassa maggo sañjāyati.	The path is born in them.
So taṃ maggaṃ āsevati bhāveti bahulīkaroti.	They cultivate, develop, and make much of it.
Tassa taṃ maggaṃ āsevato bhāvayato bahulīkaroto saṃyojanāni pahīyanti, anusayā byantīhonti.	By doing so, they give up the fetters and eliminate the underlying tendencies.


Yo hi koci, āvuso, bhikkhu vā bhikkhunī vā mama santike arahattappattiṃ byākaroti, sabbo so imehi catūhi maggehi, etesaṃ vā aññatarenā”ti.	All of the monks and nuns who declare in my presence that they have attained perfection, did so by one or other of these four paths.”

https://buddhism.stackexchange.com/questions/22505/the-fourth-path-to-attaining-arahantship

From the Yuganaddha Sutta, there are four paths described:

Development of insight preceded by development of tranquility

Development of tranquility preceded by development of insight

Tranquility developed in tandem with insight

Mind's restlessness concerning the Dhamma well under control

From my understanding, the first three are about vipassana (insight) and samatha (tranquility).

Questions:

How does the fourth path work? Please elaborate on the fourth path.

How does one practise that? Please provide details.

Why is vipassana and samatha not required on the fourth path?

Frankk response:

There are 4 permutations of samatha and vipassana, the order in which they're developed.

The 3rd permutations is the ideal scenario, where both are developed in tandem.

The 4th permutation, neither are developed, at first. But then:

Puna caparaṃ, āvuso, bhikkhuno dhammuddhaccaviggahitaṃ mānasaṃ hoti.	Another monk’s mind is seized by restlessness to realize the teaching.
Hoti so, āvuso, samayo yaṃ taṃ cittaṃ ajjhattameva santiṭṭhati sannisīdati ekodi hoti samādhiyati.	But there comes a time when their mind is stilled internally; it settles, unifies, and becomes undistractify-&-lucidifyd in samādhi.
Tassa maggo sañjāyati.	The path is born in them.
So taṃ maggaṃ āsevati bhāveti bahulīkaroti.	They cultivate, develop, and make much of it.

To answer the OP questions, two key points need to be understood.

ekodi and samadhi is code phrase indicating second jhana or higher (see MN 20, MN 119, and several others).

dhamma-uddhacca-viggahitaṃ

Uddhacca is referring to the 5 hindrance's restlessnness.

vigggahitam means trying to the ascertain the meaning.

So the meditator, in trying to ascertain the meaning of Dhamma when under the influence of the hindrance of restlessness (4th of 5 hindrances), fails. In other words, he's not doing vipassana successfully because he has insufficient samatha.

But later on, he develops second jhana or higher as indicated by the ekodi and samadhi markers, fullfilling requisite amount of samatha (see SN 46.2, samatha is nutriment for samadhi). Therefore the uddhacca restlessness is no longer a problem, and his Dharma investigation succeeds, his vipassana succeeds because his samatha is good enough.

Since the 4 permutations already covered the samatha preceding vipassanna, then this 4th case is unlikely to be that, and more likely to be having initially no samatha and vipassana, and later samatha and vipassana in tandem.

Conclusion on AN 4.170

So when one considers all 4 permutations, and all 4 attain a path that leads to arahantship, it's clear that samatha and vipassana both need to be present to sufficient quality to succeed, it's just that through inherent weakness in individual meditators, some may mature in samatha first or vipassana first, and not the case that the Buddha recommended one develop samatha before vipassana, or vice versa.

Also, if you study the formula for the four jhanas carefully, and the satipatthana formula carefully, you'll see that 3rd jhana (any of 4 jhanas would fulfill samatha) includes both sati and sampajano (equivalent to pañña and vipassana). And within the satipatthana formula, which also includes sati and sampajano factors, the "vineyya loke abhijja domanassa" is a reference to one having subdued the 5 hindrances, the tough part of samatha training, and putting you right in the doorway of first jhana (which would satisfy samatha). In other words, satipatthana is embedded within jhana, and jhana is embedded within satipatthana. That's the Buddha saying samatha and vipassana is ideally developed in tanndem ( 4 Jhānas🌕 ≈ 4 Satipaṭṭhāna🐘 ).

for a-subha (non-beauty, foul perception), Ajahn Mun points out the skin (from 31asb body parts) is especially important

Ajahn Mun: "When we get infatuated with the human body, the skin is what we are infatuated with. When we conceive of the body as being beautiful (subha) and attractive, and develop love, desire, and longing for it, it’s because of what we conceive of the skin. When we see a body, we suppose it to have a complexion—fair, ruddy, dark, etc.—because of what we conceive the color of the skin to be. If the body didn’t have skin, who would conceive it to be beautiful or attractive? Who would love it, like it, or desire it? We’d regard it with nothing but hatred, loathing, and disgust. If it weren’t wrapped in skin, the flesh, tendons, and other parts of the body wouldn’t hold together and couldn’t be used to accomplish anything at all—which is why we say the skin is especially important. The fact that we can keep on living is because of the skin. The fact that we get deluded into seeing the body as beautiful and attractive is because it has skin. This is why preceptors teach only as far as the skin.

If we set our minds on considering the skin until we see it as disgusting and gain a vision of its unloveliness appearing unmistakably to the heart, we are bound to see the inherent truths of inconstancy, stress, and not-selfness. This will cure our delusions of beauty and attractiveness that are fixated on the skin. We will no longer focus any conceivings on it or find it appealing or desirable, for we have seen it for what it is." - From https://www.dhammatalks.org/books/HeartReleased/Section0005.html