How did willa compound with cuma, to signify 'it's well you have come' and 'one who arrives at the pleasure of another'?

The question supposes following agreement in four sources that the univerbated phrasal verb started out as compound of some kind (not adequately specified) in some predecesor of modern English (left open, OE ante quem) whence it derived a noun, "wilcuma", in evidence from Old English, and an opaque, ambivalent word in modern English welcome,

This is fairly inconclusive and raises several questions. Questions are good, but we cannot have too many in one thread, so I will have to focus on the statements emphasised above:

... wilcuma, a noun literally meaning "one who arrives at the pleasure of another"

You can see the "traveling" semantic connection.

So, welcome etymologically means "wish step", because it bids people to come forward.

1. The primary question seems to be only, how does the quoted gloss up hold up to the rest of the given material?

• First of all, you will notice that the quotation-marked bits are not attributed to any reference. You decide if that hold up to scientific rigour, but we may note that it is common practice to mark-up made-up glosses like I were quoting myself, simply to maintain the use-mention distinction. _{Your own cascaded quotation style is rather questionable, too: «"“it is good you have come,” in the sense of a welcome guest"».}

• Second of all, we can ignore the copious space given to the vexilation of will, well or, finally, wel-.

• Third of all, it needs to be noted how there is barely any attention given to the German comparand, willkommen, because the comparative method connects pre-history where the philology of historical evidence ends, but we may ignore this too for sake of the argument, and any chance of cross polination between the two or three West-Germanic branches with this idiom, absent further evidence.

• By the way, a reconstructed root gloss is not necessarily what the word "meant". It is for example very unlikely that there were dozens of words that simply meant "to cut".

2. The second statement requires little clarification. It is an emphatic appeal to the common understanding about semantic fields that join between travel and welcoming in the context of arrivals.

3. We might not have to ignore the third one, if the gloss "wish step" was more than a joke, but I find this subjective. To me it's gibberish, because the gloss is devoid of meaning unless looking for a new interpretation in English, in which case we should simply interpret welcome for the same effect.

The remaining question would be why you have prefaced these with a random internet comment, integrated in one breath with the elaborate source that says something quite different. It sticks out like a sore thumb, because it remains unexplained while quora is hiding the content behind a request for registration. I would have to ignore this, too. For what it's worth:

• A gloss is supposed to be substituted in place of a questionable lexeme, which in this case would only work for the modern interjection, "*Welcome!", equivalent with "Hello!" and "Yabbadabbadoo!!!" as a matter of synchronic analysis, that is largely irrelevant for the setup of this question. We may refer to modern dictionaries for this purpose or, you know, to common sense.

I. "guest" or "one who arrives at the pleasure of another"

With "one who arrives at the pleasure of another" [The Etymology Nerd] loosely resembles "one whose coming suits another's will or wish," [etymonline], who tend to follow the [OED 2nd ed.]. Both variants are basically equivalent in that they try to translate literally and grammatically, that is with fidelity but little felicity, which contrasts sharply with the much simpler gloss "guest", "a wished-for guest", [wiktionary], "welcome guest" [etymonline].

As a mater of native English these are all terrible, unheared of, synthetic constructions, except for "welcome guest", which is problematic (difficult to reason about without circular arguments) and "guest", which is a bit too general (we can have an unwanted guest in hyperbolic speech). If a better gloss cannot be envisioned, these still work as mnemonic. First order in line is a nominative, "one", "guest". Second in line is an attribute akin to the sought sense of will > wel-. Ironically, the most organic interpretation "good" comes closest to that (cf. [Sid Kemp], "It is good you have come", NB: ] good you [, not It is good [that ...], well done) while the other phrasal glosses try to get at a verbal phrase to translate the verbal root, but none pays attention to voice and modus.

I would add the invited guest, albeit imperfect nonetheless, as an ad-hoc translation of German "Willkommener Gast" for evidence from native speaker intuition which,granted,ddoes not account for much. I mean, I would avoid "welcome guest" as ESL in translation because I have come to prefer a different interpretation resting on the exclusively English discourse semantics ("thanks"- "you are welcome"); whether correct or not, this requires an alternative."invited"is satisfying on account of the possibility that" uninvited welcome guest" would be an oxymoron for all intents and purposes, and that an unexpected guest has to be rejected or invited immediately, or they will invite themselves.

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II. How did willa compound with cuma, to signify 'it's well you have come' and 'one who arrives at the pleasure of another'?

The titel question is secondary. Arguably it would never arise while ignorant about the primary issue of the Old English idiom. We have to wonder, in essence, how ought the cited etymologies explain the development towards a modern understanding?

They do not explain it. We are left to fill in the gap with native speaker understanding, like @SipKamp, assuming that the phrase were still transparently analyzable after the reanalysis (viz. "folk-etymolog").

Accepting this as premisses, I can take a stab at it.

a) tl;dr "welcome" > "welcome"?

Ad-hoc I'd say it's very simple. We can draw an analogy to greetings like "Friend!", "Hey, Buddy!", assuming that a similar construction became opaque and was reinterpreted variously, persisting in compounds whence semantic drift from head to attribute confered new meaning, while writen eviden is naturally not to be expected.

b) Welcome vs German Wilkommen

I will ignore the adduced evidence from Romance, since this was said to be ultimately derived from Germanic, anyhow.

I want to remark upon substantivation. It requires a determiner in English, the welcome one, he is welcome, although we won't usually find * he is a welcome one, not the least because it would be ambiguous without context,aand ssimply uunnecessary in any common sense. Although no less unusual in German, substantivation Der Willkommene from the adverb willkommen (sein) "(to be) welcome" is regularly grammatical, usually parsed as plain adjective in a phrase like "der willkommene Gast". The derivation n. cumo ~ v. cuman can be understood similarly, except that the ending is attached directly to the stem (PS: except that it's not entirely clear, to me, what came first, the compounding or the nominalization.)

Therefore I would reason that the gloss "welcome guest" is closest to the truth, with "guest" determining the appropriate semantics of the adjective, while it would be left undetermined by the underspecified pronoun "one", although the field of hospitality remains tangential in most abstract connotations, e.g. a welcome opportunity, in which it might actually be active, i.e. as though a welcom-ing opportunity

I find it notable that PGem *-an would be nearly indistinguishable from the English pronoun. Whereas German conflated infinitive and substantive endings (creating similar morphosyntactic problems like English gerunds versus participles, e.g. swimming), and dative or genetive inflection of certain number+gender combinations, feminine plural noun emdings, and plural verb conjugation endings--most of which is mirrored by Old English in various form (cf. cuman) and reconstructable to Proto-Germanic in different forms.

The early evidence shows no n-stem, although it is derived probably from a participle, cf. willkommen adj. "erwünscht, angenehm", "Part.adj. ‘gern gesehen’, mhd. wille-, wilkomen. Im zweiten Wortteil wohl gebildet zu der Partizipialform von kommen. Vgl. ahd. willikomo m. (10. Jh.), mhd. willekome Adj"; Willkommen n. "freundliche Begrüßung, freundlicher Empfang", synonym Willkomm, in compouns Willkommbecher, Willkommpokal, Will-kom-mens-gruß. [W. Pfeifer, DWDS.de].

This says nothing about a common heritage with English beyond the individual word roots. It still serves well for analogy. (i) The gloss "gern" ("gladly, willingly, freely" [dict.cc]) parallels well and will (cp. to yearn 'to desire' [wiktionary], maybe Ger. begehren, Begierde, gieren [IMHO]). (ii) the variation, does not look like coincidence. Old High German ("ahd") shows a masculine noun. Middle High German, which is marked by Low German influence, that is closer to English and very close to Dutch, shows an adjective with variants. (iii) The bare stem Willkomm would be homonym with English. Alas I don't see it dated.

For comparison of the participle consider up-and-comer, home-coming queen, second coming; for a bare-stem substantive cp. maybe cum (PS: compare by the way Ger. Sam, Samen about vexilations of the endings)

IV. Further evidence

a) Germanic

Wiktionary adduces Old Norse descendents (interjections) and reconstructs a Proto-Germanic noun. Your quotation is incomplete. The formation is yet not well explained, glossed «... Literally "desired comer" (or "desired come!").» or simply «Welcome!». There is added detail for the derivation of the verb from the athematic root-aorist *gʷémt of *gʷem- ("to step").

b) Latin

Compare vale! "Goodbye, farewell", usually thought to be from valeō "I am well, healthy", specificly the second-person singular present active imperative, albeit reconstructing *h₂wl̥h₁éh₁yeti, from *h₂welh₁- (“to rule, be strong”), which is refered to *welh1 however in an unsourced comparison, perhaps because Hittite is the only evidence the laryngeal, where it could be prefixed or else.

Short Discussion

• The Proto-Germanic entry is not sourced, but wiktionary leans inclusion when it comes to reconstruction. However, there is no Gothic comparand to secure old age. It's difficult to exclude borrowing between the given languages.

• Norse and Dutch present reflexes with (regular?) e-vocalism, both known to have contributed significantly to English stock at various times.

• The formation looks on the whole relatively opaque (to me) and possibly archaic. Early PIE is known for noun-noun compounds, but it is not quite clear when and how other compounds emerged. The deverbal noun would be an innovation in Germanic, but compare the 1stP.sg subjunctive *gʷémoh₂, 2ndP.sg Optative *gʷm̥yéh₁s, etc. Also note root aorists in words like cre-d-o (-d- < *dhH-, *dheH- "to do, set, place"). It might be impossible to decide if a noun was compounded or a compound nominalized. Presumably there must be comparable formations to allow the kind of internal reconstruction nneded to secure the roots that are adduced.

• comparing *welh1- and *h2welh1- is as reasonable on semantic grounds as is deriving win and wish from the same root--which isn't much to say about its reality, because a semantic tangent can always be conjured, while morphology and phonology are primary concerns

Conclusion

The question is based on a shaky premisses, which couldn't be confirmed, that the greeting had developed from a noun. A theoretic explanation is easy to imagine, when folk-etymology may be involved. Further considerations suggest the possibilities of either an archaic lexeme or an abberrant development. This should invite serious study of the historical linguistic theory, and might give way to speculative discussion, which has to be omitted for brevities sake, because it is a-priori unlikely. This potential is a notable result, anyway.

Investigation of first hand evidence from literature could be omitted. While lack of evidence has to be expected, it wouldn't constitute evidence of absence at any rate. _{I solely rely on en.wiktionary unless otherwise stated.}