This hits the nail on the head precisely.
My diagram is incorrect, but not because of the ~がある (in brackets to show that it isn’t actually in the poem).
The modifying stem beginning with で should be attached to the object on the left, not the implied verb on the right. I’ll fix it when I’m on a computer. [EDIT: See corrected diagram below]
Note that in either language the poem just provides an adjectival description of an object, not a complete sentence that can be diagrammed. Senryu don’t have to be complete sentences, of course, but the point of diagramming is to understand the underlying grammar with complete sentences.
“The piercing” is not a complete sentence. But something like “The piercing [exists]” is. The brackets denote something implied that isn’t actually in the original.
Discussing with my daughter last night, my favorite translation was:
At my hometown station: a removed nose-piercing.
Like your version above, this is a sentence fragment, not a complete sentence.
Of course, the missing/implied verb could be anything. It makes sense to supply something simple like “exists” / ある in brackets to show that it’s implied and not actually present.
Fragments like “The piercing [I] removed at my home town station” are accurate translations. But “I took off my nose-piercing at the hometown station” wouldn’t be…
It’s just like middle school: my diagramming mistakes (in public) are leading to interesting and valuable discussions. I’m understanding things more clearly.
Here’s my updated diagram. I think this explains best what I’m trying to point out:
There are two sentences:
An adjectival verb-phrase (if that’s the right lingo) that something was removed at a home-town station. This verb-phrase acts like an adjective. It describes something about the nose-ring (that it was removed at a train station).
The core sentence fragment is “nose ring”.
A noun (subject) without some sort of verb can’t be a sentence (a complete thought). Something is implied in order to communicate a complete thought. The most basic interpretation is that they mean “There is a nose-ring” (a complete sentence) but the first two words are only implied, all they actually say is “nose-ring”.
In other words, there has to be a verb for 鼻ピアス in order for it to communicate a complete thought. That verb could be anything since it’s unstated in the poem, but the simplest interpretation is ある. The verb はずした is part of an adjectival phrase, not part of the core sentence (in my opinion).
This was an admittedly very long-winded explanation and doubtless some or even most of you feel it unnecessary or overkill if not outright wrong. But I think the diagrams (including inaccuracies) aided the discussion.
It helped me to understand and make the point that a complete sentence like “I took off my nose-ring at the home-town station” would be a loosely correct interpretation, but less accurate than a sentence fragment like “At the home-town station: a nose ring”.
EXACTLY! This is what I’m trying to point out with the diagram. Sorry, I missed this reply.