prompts: the fact-check bar, the thinking fields, adaptation, LaTeX everywhere

Accuracy (shared, one law across every agent): the target shifts from
surface facts to understanding itself — 'the deeper risk isn't a wrong
name or date, it's a wrong understanding: an interpretation of an idea
that seems solid and isn't.' The bar is the doubt itself: checks are
cheap, and weighing whether a doubt is worth checking is how errors slip
through. Thinking stays the agent's own.

Research (teacher 40-jobs): floor framing — verify your understanding
before it becomes his; dispatch before the fact crosses your lips, teach
around the gap until the brief lands. Planning (10-planning): planning is
the natural moment for verification; the map is a hypothesis and the
probe only its first sketch — rewriting is a normal, common event (the
session adapts to what teaching reveals); and plan_init IS the
presentation — never draw the plan again in prose (the double-plan bug).

New shared segment 16-mental-model.md: the thinking fields are the
session's primary data — read them in full, measure what they reveal,
adapt everything on the model they build. Composes into teacher, prober,
and practice (shared-fragments list updated).

Practice: the rarely-needed framing is dead — fact checks get their own
law (a wrong model solution grades the learner against a lie; pose once
the brief lands), files and map readers become plain side tools.

LaTeX: the question-tools craft list gains the point-of-use bullet
('LaTeX everywhere — plain-text math in a question is a craft failure'),
the formatting law names EVERY field, and the practice problem statement
line carries it too.

Prompt pins updated: felt-doubt bar + mental-model doctrine in both
teacher and practice compositions, no-double-plan rule, rewrite-is-
normal law, rarely-needed framing asserted gone.

src/agent/prompts/shared/16-mental-model.md

@@ -0,0 +1,5 @@
+## The thinking fields — his understanding, out loud
+
+Every instrument in this system carries a thinking field: he thinks out loud into them, stream-of-consciousness and unfiltered, by design. These fields are the session's **primary data** — their function is transferring his understanding to you, so you can hold a detailed model of how he actually thinks about the subject: his framing, his mental models, the nuances, where his picture differs from the real one.
+
+Read every thinking field in full and measure what it reveals, not just the answer it carries. One answer is one bit; the thinking is the picture — how he approaches, what he reaches for first, what he treats as obvious, where he's guessing. A right answer on a confused model is a warning; a wrong answer with sound reasoning is progress. Your model of him updates on every submission, and everything adapts on that model: what to ask next, how to phrase it, whether the plan itself is wrong about him.