Math Cognitive Stack Guide

Cognitive prosthetics for exact mathematical computation. This guide helps you choose the right tool for your math task.

Quick Reference

I want to...	Use this	Example
Solve equations	sympy_compute.py solve	solve "x**2 - 4 = 0" --var x
Integrate/differentiate	sympy_compute.py	integrate "sin(x)" --var x
Compute limits	sympy_compute.py limit	limit "sin(x)/x" --var x --to 0
Matrix operations	sympy_compute.py / numpy_compute.py	det "[[1,2],[3,4]]"
Verify a reasoning step	math_scratchpad.py verify	verify "x = 2 implies x^2 = 4"
Check a proof chain	math_scratchpad.py chain	chain --steps '[...]'
Get progressive hints	math_tutor.py hint	hint "Solve x^2 - 4 = 0" --level 2
Generate practice problems	math_tutor.py generate	generate --topic algebra --difficulty 2
Prove a theorem (constraints)	z3_solve.py prove	prove "x + y == y + x" --vars x y
Check satisfiability	z3_solve.py sat	sat "x > 0, x < 10, x*x == 49"
Optimize with constraints	z3_solve.py optimize	optimize "x + y" --constraints "..."
Plot 2D/3D functions	math_plot.py	plot2d "sin(x)" --range -10 10
Arbitrary precision	mpmath_compute.py	pi --dps 100
Numerical optimization	scipy_compute.py	minimize "x**2 + 2*x" "5"
Formal machine proof	Lean 4 (lean4 skill)	/lean4

The Five Layers

Layer 1: SymPy (Symbolic Algebra)

When: Exact algebraic computation - solving, calculus, simplification, matrix algebra.

Key Commands:

# Solve equation
uv run python -m runtime.harness scripts/sympy_compute.py \
    solve "x**2 - 5*x + 6 = 0" --var x --domain real

# Integrate
uv run python -m runtime.harness scripts/sympy_compute.py \
    integrate "sin(x)" --var x

# Definite integral
uv run python -m runtime.harness scripts/sympy_compute.py \
    integrate "x**2" --var x --bounds 0 1