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Having noticed here that excidere, "to fall out", lacks a perfect participle, a reasonable deficiency given that it's intransitive and has no corresponding passive meaning, I checked Wiktionary's page for its root cadere and found passive forms, including the perfect participle casus. The passive verb forms exist only in the third-person singular, though.

The participle casus would seem to provide the missing link to the fourth-declension noun casus. But what does it mean, exactly?

Is casus like English "fallen", a past participle with active meaning? In English, intransitive verbs all get a past participle to form periphrastic tenses, e.g. "has existed" or, archaically, "is come", but do not function as adjectives. "Fallen" is exceptional. Unlike other English active past participles like "soft-spoken" and "experienced", "fallen" doesn't seem exceptional (at least to this native speaker), because the idea of falling itself seems passive: it's something that happens to you, not something that you actively do.

Wiktionary reports that cadere is impersonal in the passive. That sounds like a clue, but I'm not sure what to make of it. PHI reports no instances of caditur. A brief look at Google Books didn't turn up any legitimate instances of angelus casus. But angelus lapsus is well attested, of course, since lābī is a deponent verb.

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The only example of impersonal passive in L&S's entry of cado comes from Late Latin: Augustinus (De dono perseverantiae), which is beyond the scope of PHI.

Finem autem dico, quo vita ista finitur, in qua tantummodo periculum est ne cadatur (Aug. Don. Persev. 1,1).

The fact that you didn't find any example of impersonal passive caditur in PHI is probably related to an important point you make in your post: "the idea of falling itself seems passive: it's something that happens to you, not something that you actively do". As I pointed out to cmw, a typical semantic restriction associated with impersonal passives in different languages (e.g., German, Dutch, etc.) is that the verb must be (interpreted as) agentive. Agentivity or volitionality are those semantic notions that (typically) allow a verb to enter into an impersonal passive construction. As far as I know, the same semantic restriction (typically) holds for Latin as well: cf. ok Pugnatum est ab utrisque acriter (Caes. BG 4,26,1) vs. *Mortuum est beate. This restriction is often expressed in syntactic terms as well: so-called "unergative" verbs (let's say "agentive" intransitive verbs) enter into the impersonal passive construction, whereas so-called "unaccusative" verbs (let's say "non-agentive", i.e., patient-oriented intransitive verbs) do not. See this link for a brief summary.

As you point out, the verb cadere is typically used as a patient-oriented verb, whereby one would not expect this predicate to enter into an impersonal passive construction. Interestingly, as pointed out by Perlmutter (1978: 172-173), the verb "fall" in Dutch is only allowed to enter into the impersonal passive construction iff it is provided with agentivity/control, which involves a bit marked context. Otherwise, in its more frequent patient-oriented sense, the impersonal construction is ruled out (as expected). Cf. the well-formed agentive impersonal construction in (b) with the ill-formed non-agentive one in (d) (these Dutch examples come from Perlmutter (1978: 172)).

a. De nieuwe acteur is in bet tweede bedrijf op bet juiste ogenblik gevallen. 'The new actor fell at the right moment in the second act. '

b. In bet tweede bedrijf werd er door de nieuwe acteur op bet juiste ogenblik gevallen.

c. Twee mensen zijn uit de venster van de tweede verdieping gevallen. 'Two people fell out of the second-storey window.'

d. *Er werd door twee mensen uit de venster van de tweede verdieping gevallen.

Perhaps something similar to the Dutch scenario could be expected to apply to Classical Latin (cf. the abovementioned example from Augustinus/Late Latin).

As for the explanation of the other constrast you mention (cf. ??puer casus vs. ok puer lapsus), it seems natural to appeal to the fact that the subject of deponent verbs like labi acts as a direct internal argument (i.e., not as a true subject/external argument), whereby only these verbs allow this kind of participial modification (NB: something similar happens with the Ablative Absolute construction). Note then the important parallelism between the subject of deponent verbs like labi and the direct object of transitive verbs like laudare (to repeat it again, both act as direct internal arguments): e.g., cf. ok puer lapsus/natus/mortuus/... and ok puer laudatus/amatus/occisus/.... This parallelism is often accounted for in the light of the so-called "unaccusative hypothesis" (e.g. see this link), but, truth be told, the topic is a bit more complex than what I've sketched out here.

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