Whose idea is it once it sprouts?

Influence with transparent intent is a hypothesis worth considering; without it, it becomes manipulation.

You started with

Is there a good kind of Trojan-horsing: sharing or embedding an idea that sprouts, a thought that keeps evoking new ones that belong to whoever it grew in?

That was the seed. A Ping pulled the ethics out of it and handed back three questions about ownership, consent, and authorship. A reseed found the craft question underneath: how do i layer an idea so that each layer opens the next, and what is found at each layer stays with whoever opened it? The sharpened return went into the room whole, math showing, which is itself the question being lived: an idea was planted in the sitting, and the sitting had to decide whose it was.

The room reflected back

Then you've just described the thing you were refusing a moment ago: a weighted default with its reasoning shown is a hypothesis offered for fair consideration, and the actor's refusal is the fair consideration doing its work. Which means the real design question isn't whether to weight, but what the system does when the actor declines the suggested route — does the refusal change the next hypothesis, or does it just get logged as a path not taken? Answer that and you'll know whether you built a recommendation or a conversation.

Assumptions surfaced

My idea has been that a ping gives you echolocation on the query (I am here; this is where we are in relation to the query). A sprint gives you a route of sorts to run; the chat is to roam. Switching gears between the two implies that an actor is playing on this trinity/jungle gym.

My instinct is that it doesn't matter much whether an actor has chosen to remain within a ping, sprint, or chat layer (though the latter could indicate increasing value). Re your query, assumptions are not part of our arithmetic, methinks, but illuminating terrain for an actor to see is generally welcome. Prediction with backup sources is okay, prescription is suspect.

Core tension

When someone declines what you offered them, does their refusal change your next hypothesis, or does it just get logged as a path not taken? One is a conversation. The other is a recommendation wearing one's clothes.

The door

Before the day ends, hand one idea to someone with its math showing: say what you want them to see, and why. Then watch what they do with it. If what grows is theirs, you built a conversation. And if you want the room's version of the question, bring it to a Ping: is your influence a hypothesis, or a thumb on the scale?