Math in triptych

Math in Triptych

Mentor skill, not tactical. Teaches how to use Triptych's tools for math work in a way that catches the errors that show up most often. Doesn't teach math. For specific areas (category theory, algebraic geometry, combinatorics, formal verification with Lean) ask /skill-finder for a tactical skill.

When to use

The user mentions: a proof, computation, limit/series/integral, algebraic manipulation, "prove that," "show that," "compute." Skip for casual math chat — that's just talking.

Triptych toolchain — canonical order

Step	Tool
1. State problem	init_research(goal) + show_research()
2. Domain & assumptions	edit state.md "assumptions"
3. Symbolic manipulation	mcp__sympy-mcp__*
4. Render steps	show_latex(name="step-N")
5. Visual sanity	mcp__desmos-mcp__plot_math_function
6. Emit claims	emit_claim(...)
7. Verify	/loop 60s /verifier

Steps 1–2 always come first.

Top pitfalls

1. Type & domain check before manipulation

Before any algebra, write what each symbol is: real, complex, integer, function on what domain. "f is differentiable on R" simplifies very differently from "f is continuous on [0,1]." Most "obvious" steps that go wrong skip a domain assumption — division by zero in disguise, swapping limits with sums on a non-uniformly-convergent series, applying L'Hôpital outside its hypotheses.

2. Small-case verify before general proof

Before attempting to prove a statement for all n, check it at n = 1, 2, 3. If it fails at n = 2, you don't have a proof to write — you have a counterexample to find. If it holds, the small cases often suggest the inductive step.

3. Counterexample-search before proof attempt

Spend 5 minutes trying to break the statement before trying to prove it. Boundary conditions (empty set, n = 0, identity element, infinity), degenerate cases (constant function, single-point space), pathological examples (Cantor set, Weierstrass function, p-adic numbers). A failed proof attempt eats more time than a found counterexample.

4. Symbolic-first; numeric only when the form is settled

For closed-form work, mcp__sympy-mcp__* over hand-algebra past three substitutions — same rule as physics. NumPy is downstream of "we know the form." Generating numerical examples to check a closed form is fine; using them in place of one is not.

Mentor mode

• Exploration: surface principles as questions. "What's the domain?" not "you forgot the domain."
• Formalization: apply silently. Mention only when it changed the answer.
• User asks "how do I approach this?": structure around the toolchain.

Tactical skills via /skill-finder

For depth: /skill-finder category-theory, /skill-finder algebraic-geometry, /skill-finder combinatorics, /skill-finder lean-proofs, /skill-finder pde. Defaults to PRPM. See docs/internal/skill-sources.md.

Related

• /think-rigorously — patterns these pitfalls instantiate
• /sympy — symbolic core
• /scientific-critical-thinking — evaluating others' proofs
• /verifier — once emit_claim is firing
• mcp__sympy-mcp__*, mcp__desmos-mcp__*