I mentioned in the link below that I found about 330 instances of the preposition 'at' in the text, and I’ve shared some of them along with GPT's analysis on that page. Here, I am providing more of these cases, this time with Gemini's analysis:

https://myenglishclub.com/groups/tom-sawyer-reading-discussion-group/forum/topics/day-9-at-1

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In an instant both boys were rolling and tumbling in the dirt, gripped together like cats; and for the space of a minute they tugged and tore at each other’s hair and clothes, punched and scratched each other’s nose, and covered themselves with dust and glory.

Gemini:
The Grammar of "Tearing AT" vs. "Tearing"

In the phrase "tore at each other’s hair," the choice of the preposition at significantly changes the meaning of the verb.

The Logic of "AT" with Aggressive Verbs: When we use at after verbs of physical struggle or aggression (such as pull, tear, kick, punch, or snap), it shifts the focus from the result to the attempt and the process.

Tore hair: This implies a completed result—the hair was actually removed or completely ripped.

Tore AT hair: This indicates an action that is repetitive, chaotic, and forceful.

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and when she saw the state his clothes were in her resolution to turn his Saturday holiday into captivity at hard labor became adamantine in its firmness.

Gemini:

At hard labor

The Logic: Why do we use At? It is because "hard labor" (forced labor or intense work) is conceptualized here as an activity or a functional role, rather than a physical location.

Just as we say at work or at school to describe someone's current engagement in a task, at hard labor places the person in the position of performing a specific duty or function. In this context, the person is not just in a place; they are "at" a point of obligation and activity.
Quick Comparison: Activity vs. Location

In labor: Usually refers to the physical state of giving birth (surrounded by the process).

At labor: Refers to being occupied with a task or a job (functional engagement).

Modern Shortcut: Whenever a noun represents a job, a task, or a functional state, AT is the preposition used to show that the person is currently "occupied" by that specific point of activity.

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There was a song in every heart; and if the heart was young the music issued at the lips.

Gemini:
Issued at the lips

Mental Model: Point of Exit.

Why At? Because the lips are conceptualized as the specific point of interaction where the internal music meets the outside world.

The lips are the "gate" or the "exit node." As we saw with "climbing in at the window" or standing "at the gate," At is used for the exact location where a transition happens.

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He remembered that there was company at the pump.

Gemini:

The choice of preposition depends on how the speaker conceptualizes the relationship between the people and the object.

AT the pump (Point of Function)

Mental Model: Functional Node / Destination.

Logic: This is the most natural choice here because the pump is a "destination." It implies the people are there for a reason (to get water, to socialize, or as a meeting point).

Context: "I'll meet you at the pump" (The pump is a specific node on a map).

BY the pump (Proximity)

Mental Model: Physical distance.

Logic: This focuses purely on being near the object. It does not imply any interaction with it. You could be "by the pump" accidentally while looking for something else.

Context: "I left my shoes by the pump" (Near it, but not using it).

BEHIND the pump (Relative Position)

Mental Model: Directional / Obstruction.

Logic: This focuses on the spatial arrangement. It often implies being hidden from view or being on the far side of the object relative to the observer.

Context: "The boy hid behind the pump so his aunt wouldn't see him."