What exactly does the statement “Saṅgha […] consists of the four pairs, the eight individuals” mean? I assume it has a deeper meaning than just a purely mathematical one. Thank you in advance.
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Hi Pulec, as far as I understand it it’s about 4 stages and those 4 stages are divided in a person who walks the path and then attains “the fruit” (4+4=8).
1.) Stream Enterer
2.)Once Returner (Sakadagami)
3.) Non-returner (Anagami)
4.) Arahant
I hope this makes sense 
or maybe someone else can explain it better than I
With much metta 
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This is indeed the commentarial explanation, and IIRC the grouping is not made explicit in Nikāyas proper.
I do however wonder and speculate, whether it could’ve meant “male and female fruit winners” (2x4), since “Sangha” is often also a shorthand for “Ariyasangha” monastics.
That to me makes more sense as “8 individuals”, as for example, someone who’s a sotapanna, would naturally be on the path to other three paths, etc. ? Anyway, that’s just something to consider.
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SN 25.1 describes a category of people who haven’t yet achieved sotāpanna yet who are at “the level of a true person.”
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Though it is in the Abhidhamma, e.g., the Dakkhiṇāvisuddhikathā of the Kathāvatthu. Also, it seems to have been the common understanding of Indian scholastic Buddhism in general, with, afaik, no dissenting opinions.
Attainers of a particular ariyan fruit are not reckoned as being strivers for the next ariyan stage until they actually start exerting themselves for it. This too seems to have been an uncontested view in Indian scholastic Buddhism.
At the moment of the acquisition of a fruit, the practitioners do not acquire the path of a higher fruit; consequently, the abiders in a fruit who do not endeavor with a higher progress in mind are not approachers of a fruit.
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Upon the acquisition of a fruit, one does not acquire a path higher than this fruit; for example, upon the acquisition of the fruit of the stream-enterer, one does not obtain the path of the once-returner: that is the principle. Consequently, the abiders in the fruit, as long as they do not devote themselves (prayujyate) to a higher progress in mind, for the acquisition of a new fruit, i.e., as long as they do not cultivate the preparatory effort (prayoga) which has for its result the abandonment of defilements not yet abandoned and which is the cause for the acquisition of a new fruit, so long are these abiders not approachers of this new fruit.
(Vasubandhu, Abhidharmakośabhāṣya, ch. 6, Mārgapudgalanirdeśa)
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Hi Dogen, in a addition, a really nice and all including suggestion would be: male and female lay person, monk and nun That also makes 4 
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