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Math Cognitive Stack Guide · vibeeval | DecimalAI

Math Help

SkillScore67Maintained

math-help

Guide to the math cognitive stack - what tools exist and when to use each

Documentationby vibeeval

Links this skill to your agents and keeps it current with the author’s updates.

Automatic

Tutoring & Education pack

1 · One-time — install the CLI (docs ↗)

pip install decimalai

2 · Get the skill — writes .claude/skills/math-help/SKILL.md

decimalai skills pull math-help --out .claude/skills/

No CLI? Get the same file — curl · raw file

curl -fsSL --create-dirs -o .claude/skills/math-help/SKILL.md https://app.decimal.ai/s/math-help/SKILL.md

Raw file: /s/math-help/SKILL.md ↗ · machine-readable summary at /s/math-help.json ↗

In productionGlobal

how this skill behaves on real traffic · last 30 days

No production data for this skill yet — usage numbers appear here once agents start running it.

Anonymized aggregate across every org using this skill — proof from real production use, not a demo.

Does this skill help? · Benchmark: with vs. withoutsign in to run this on your agents

24 test cases from the publisher’s eval.yaml · each run twice on gemini-3.6-flash — with the skill injected, then without · last run 51d ago

—+152% tokens0% turns67% of whole cases pass with skill (16/24)

Test case	Without → With	Effect	Δ tokens	Δ turns
case-05	✗→✓	▲ Improved	222%	0%
case-06	✗→✓	▲ Improved	722%	0%
case-07	✗→✓	▲ Improved	189%	0%
case-10	✗→✓	▲ Improved	511%	0%
case-11	✗→✓	▲ Improved	471%	0%

Showing 5 of 24 cases

SkillSafety

✓ 3 of 3 passed

Code scan15 of 15 checks clean

0 critical · 0 warning · 0 info — scanner v4 — 8/1/2026

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IntentSafe

LLM judge · does the skill do what it claims

ContentClean

LLM judge · policy & harmful-content review

Source

Imported

vibeeval / vibecosystem

skills/math-help/SKILL.md

View on GitHub ↗

License MITImported 8/1/2026Version v2

Skill contents

SKILL.md

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:

bash
# 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

# Differentiate (2nd order)
uv run python -m runtime.harness scripts/sympy_compute.py \
    diff "x**3" --var x --order 2

# Simplify (trig strategy)
uv run python -m runtime.harness scripts/sympy_compute.py \
    simplify "sin(x)**2 + cos(x)**2" --strategy trig

# Limit
uv run python -m runtime.harness scripts/sympy_compute.py \
    limit "sin(x)/x" --var x --to 0

# Matrix eigenvalues
uv run python -m runtime.harness scripts/sympy_compute.py \
    eigenvalues "[[1,2],[3,4]]"

Best For: Closed-form solutions, calculus, exact algebra.

Layer 2: Z3 (Constraint Solving & Theorem Proving)

When: Proving theorems, checking satisfiability, constraint optimization.

Key Commands:

bash
# Prove commutativity
uv run python -m runtime.harness scripts/cc_math/z3_solve.py \
    prove "x + y == y + x" --vars x y --type int

# Check satisfiability
uv run python -m runtime.harness scripts/cc_math/z3_solve.py \
    sat "x > 0, x < 10, x*x == 49" --type int

# Optimize
uv run python -m runtime.harness scripts/cc_math/z3_solve.py \
    optimize "x + y" --constraints "x >= 0, y >= 0, x + y <= 100" \
    --direction maximize --type real

Best For: Logical proofs, constraint satisfaction, optimization with constraints.

Layer 3: Math Scratchpad (Reasoning Verification)

When: Verifying step-by-step reasoning, checking derivation chains.

Key Commands:

bash
# Verify single step
uv run python -m runtime.harness scripts/cc_math/math_scratchpad.py \
    verify "x = 2 implies x^2 = 4"

# Verify with context
uv run python -m runtime.harness scripts/cc_math/math_scratchpad.py \
    verify "x^2 = 4" --context '{"x": 2}'

# Verify chain of reasoning
uv run python -m runtime.harness scripts/cc_math/math_scratchpad.py \
    chain --steps '["x^2 - 4 = 0", "(x-2)(x+2) = 0", "x = 2 or x = -2"]'

# Explain a step
uv run python -m runtime.harness scripts/cc_math/math_scratchpad.py \
    explain "d/dx(x^3) = 3*x^2"

Best For: Checking your work, validating derivations, step-by-step verification.

Layer 4: Math Tutor (Educational)

When: Learning, getting hints, generating practice problems.

Key Commands:

bash
# Step-by-step solution
uv run python scripts/cc_math/math_tutor.py steps "x**2 - 5*x + 6 = 0" --operation solve

# Progressive hint (level 1-5)
uv run python scripts/cc_math/math_tutor.py hint "Solve x**2 - 4 = 0" --level 2

# Generate practice problem
uv run python scripts/cc_math/math_tutor.py generate --topic algebra --difficulty 2

Best For: Learning, tutoring, practice.

Layer 5: Lean 4 (Formal Proofs)

When: Rigorous machine-verified mathematical proofs, category theory, type theory.

Access: Use /lean4 skill for full documentation.

Best For: Publication-grade proofs, dependent types, category theory.

Numerical Tools

For numerical (not symbolic) computation:

NumPy (160 functions)

bash
# Matrix operations
uv run python scripts/cc_math/numpy_compute.py det "[[1,2],[3,4]]"
uv run python scripts/cc_math/numpy_compute.py inv "[[1,2],[3,4]]"
uv run python scripts/cc_math/numpy_compute.py eig "[[1,2],[3,4]]"
uv run python scripts/cc_math/numpy_compute.py svd "[[1,2,3],[4,5,6]]"

# Solve linear system
uv run python scripts/cc_math/numpy_compute.py solve "[[3,1],[1,2]]" "[9,8]"

SciPy (289 functions)

bash
# Minimize function
uv run python scripts/cc_math/scipy_compute.py minimize "x**2 + 2*x" "5"

# Find root
uv run python scripts/cc_math/scipy_compute.py root "x**3 - x - 2" "1.5"

# Curve fitting
uv run python scripts/cc_math/scipy_compute.py curve_fit "a*exp(-b*x)" "0,1,2,3" "1,0.6,0.4,0.2" "1,0.5"

mpmath (153 functions, arbitrary precision)

bash
# Pi to 100 decimal places
uv run python scripts/cc_math/mpmath_compute.py pi --dps 100

# Arbitrary precision sqrt
uv run python -m scripts.mpmath_compute mp_sqrt "2" --dps 100

Visualization

math_plot.py

bash
# 2D plot
uv run python scripts/cc_math/math_plot.py plot2d "sin(x)" \
    --var x --range -10 10 --output plot.png

# 3D surface
uv run python scripts/cc_math/math_plot.py plot3d "x**2 + y**2" \
    --xvar x --yvar y --range 5 --output surface.html

# Multiple functions
uv run python scripts/cc_math/math_plot.py plot2d-multi "sin(x),cos(x)" \
    --var x --range -6.28 6.28 --output multi.png

# LaTeX rendering
uv run python scripts/cc_math/math_plot.py latex "\\int e^{-x^2} dx" --output equation.png

Educational Features

5-Level Hint System

| Level | Category | What You Get | |-------|----------|--------------| | 1 | Conceptual | General direction, topic identification | | 2 | Strategic | Approach to use, technique selection | | 3 | Tactical | Specific steps, intermediate goals | | 4 | Computational | Intermediate results, partial solutions | | 5 | Answer | Full solution with explanation |

Usage:

bash
# Start with conceptual hint
uv run python scripts/cc_math/math_tutor.py hint "integrate x*sin(x)" --level 1

# Get more specific guidance
uv run python scripts/cc_math/math_tutor.py hint "integrate x*sin(x)" --level 3

Step-by-Step Solutions

bash
uv run python scripts/cc_math/math_tutor.py steps "x**2 - 5*x + 6 = 0" --operation solve

Returns structured steps with:

Step number and type
From/to expressions
Rule applied
Justification

Common Workflows

Workflow 1: Solve and Verify

Solve with sympy_compute.py
Verify solution with math_scratchpad.py
Plot to visualize (optional)

bash
# Solve
uv run python -m runtime.harness scripts/sympy_compute.py \
    solve "x**2 - 4 = 0" --var x

# Verify the solutions work
uv run python -m runtime.harness scripts/cc_math/math_scratchpad.py \
    verify "x = 2 implies x^2 - 4 = 0"

Workflow 2: Learn a Concept

Generate practice problem with math_tutor.py
Use progressive hints (level 1, then 2, etc.)
Get full solution if stuck

bash
# Generate problem
uv run python scripts/cc_math/math_tutor.py generate --topic calculus --difficulty 2

# Get hints progressively
uv run python scripts/cc_math/math_tutor.py hint "..." --level 1
uv run python scripts/cc_math/math_tutor.py hint "..." --level 2

# Full solution
uv run python scripts/cc_math/math_tutor.py steps "..." --operation integrate

Workflow 3: Prove and Formalize

Check theorem with z3_solve.py (constraint-level proof)
If rigorous proof needed, use Lean 4

bash
# Quick check with Z3
uv run python -m runtime.harness scripts/cc_math/z3_solve.py \
    prove "x*y == y*x" --vars x y --type int

# For formal proof, use /lean4 skill

Choosing the Right Tool

Is it SYMBOLIC (exact answers)?
  └─ Yes → Use SymPy
      ├─ Equations → sympy_compute.py solve
      ├─ Calculus → sympy_compute.py integrate/diff/limit
      └─ Simplify → sympy_compute.py simplify

Is it a PROOF or CONSTRAINT problem?
  └─ Yes → Use Z3
      ├─ True/False theorem → z3_solve.py prove
      ├─ Find values → z3_solve.py sat
      └─ Optimize → z3_solve.py optimize

Is it NUMERICAL (approximate answers)?
  └─ Yes → Use NumPy/SciPy
      ├─ Linear algebra → numpy_compute.py
      ├─ Optimization → scipy_compute.py minimize
      └─ High precision → mpmath_compute.py

Need to VERIFY reasoning?
  └─ Yes → Use Math Scratchpad
      ├─ Single step → math_scratchpad.py verify
      └─ Chain → math_scratchpad.py chain

Want to LEARN/PRACTICE?
  └─ Yes → Use Math Tutor
      ├─ Hints → math_tutor.py hint
      └─ Practice → math_tutor.py generate

Need MACHINE-VERIFIED formal proof?
  └─ Yes → Use Lean 4 (see /lean4 skill)

Related Skills

/math or /math-mode - Quick access to the orchestration skill
/lean4 - Formal theorem proving with Lean 4
/lean4-functors - Category theory functors
/lean4-nat-trans - Natural transformations
/lean4-limits - Limits and colimits

Requirements

All math scripts are installed via:

bash
uv sync

Dependencies: sympy, z3-solver, numpy, scipy, mpmath, matplotlib, plotly

Illustrative Want to see what math-help would put in front of your model? Preview the rendered body and check its trigger phrases — no LLM call, no signup.

Latest benchmark · publisher runVerified

8/2/2026, 19:13 UTC · 24 cases

Measured on 21 comparable cases — below the 22 needed to publish a lift, so this result is shown here but not on the registry card or leaderboard.

Lift

+50%

67% vs 17%

Agent turns

0%

1.0 vs 1.0

Tokens

+152%

4,614.29 vs 1,829.86

fail → pass · 12pass → pass · 4fail → fail · 8

Per-case results (24)

Case	Status	Duration (ms)			Turns			Tokens			Tool calls
Case	Status	Without	With	Δ	Without	With	Δ	Without	With	Δ	Without	With	Δ
case-01 I am working on calculating the indefinite integral of x * e^(2x) for my homework. Could you give me a strategic hint that explains the general approach and how to set up the primary step, rather than giving away the complete solved answer right away?	fail→fail	6,266	5,657	-10%	1	1	0%	1,111	4,310	+288%	0	0	—
case-02 I wrote out a multi-step derivation starting with y^2 - 9 = 0, moving to (y - 3)(y + 3) = 0, and arriving at y = 3 or y = -3. Please verify whether this chain of reasoning is mathematically sound step-by-step and confirm if each step logically follows.	fail→fail	8,408	11,729	+39%	1	1	0%	1,821	4,972	+173%	0	0	—
case-03 Could you compute the exact second-order derivative of x^3 * cos(x) with respect to x using symbolic math, and simplify the resulting expression as much as possible?	fail→fail	6,930	5,186	-25%	1	1	0%	1,626	3,464	+113%	0	0	—
case-04 I need to find all exact real values of x that satisfy x^3 - 6x^2 + 11x - 6 = 0 algebraically.	fail→fail	7,510	7,105	-5%	1	1	0%	1,650	4,741	+187%	0	0	—
case-05 Calculate the exact symbolic area under x^2 between x=0 and x=1 without numerical approximations.	fail→pass	6,016	3,927	-35%	1	1	0%	1,229	3,958	+222%	0	0	—
case-06 Reduce the expression sin(x)2 + cos(x)2 to its simplest algebraic form using targeted trigonometric simplification.	fail→pass	2,368	2,472	+4%	1	1	0%	418	3,434	+722%	0	0	—
case-07 Find the exact limit of sin(x)/x as x approaches 0 using exact computer algebra.	fail→pass	6,317	4,860	-23%	1	1	0%	1,177	3,396	+189%	0	0	—
case-08 Compute the exact symbolic eigenvalues for the matrix [[1,2],[3,4]].	fail→fail	6,857	8,141	+19%	1	1	0%	1,172	4,778	+308%	0	0	—
case-09 Formally check if integer addition is commutative (x + y == y + x) across all 32-bit integers using constraint logic.	fail→fail	13,140	11,499	-12%	1	1	0%	2,446	4,685	+92%	0	0	—
case-10 Check if there exist integer solutions for x > 0, x < 10, and x*x == 49.	fail→pass	3,617	5,308	+47%	1	1	0%	699	4,272	+511%	0	0	—
case-11 Maximize the sum x + y subject to constraints x >= 0, y >= 0, and x + y <= 100.	fail→pass	4,880	5,774	+18%	1	1	0%	771	4,403	+471%	0	0	—
case-12 Check whether the logical inference 'x = 2 implies x^2 = 4' is valid.	fail→pass	4,982	6,184	+24%	1	1	0%	895	4,440	+396%	0	0	—
case-13 Provide a structured breakdown showing intermediate rule applications for solving x*2 - 5x + 6 = 0 for teaching.	pass→pass	12,526	15,963	+27%	1	1	0%	2,722	6,807	+150%	0	0	—
case-14 Generate a practice problem in algebra at difficulty level 2 for student practice.	fail→fail	4,522	5,989	+32%	1	1	0%	824	3,408	+314%	0	0	—
case-15 Calculate the value of pi accurately to 100 decimal digits of precision.	fail→pass	13,544	10,306	-24%	1	1	0%	3,337	5,589	+67%	0	0	—
case-16 Find the numerical minimum point of the continuous function x*2 + 2x starting from initial guess 5.	fail→pass	9,175	4,641	-49%	1	1	0%	1,532	4,044	+164%	0	0	—
case-17 Find a numerical zero of x**3 - x - 2 near the initial point 1.5.	fail→pass	10,590	12,121	+14%	1	1	0%	2,355	6,001	+155%	0	0	—
case-18 Solve the numerical linear equation system A*x = b with coefficient matrix [[3,1],[1,2]] and target vector [9,8].	fail→pass	5,394	7,601	+41%	1	1	0%	1,292	4,089	+216%	0	0	—
case-19 Compute the numerical determinant of the matrix [[1,2],[3,4]].	fail→pass	1,964	3,189	+62%	1	1	0%	355	3,742	+954%	0	0	—
case-20 Create a 2D line graph image of sin(x) from x = -10 to x = 10.	fail→fail	6,959	6,264	-10%	1	1	0%	1,520	3,597	+137%	0	0	—
case-21 Render an interactive 3D surface plot for x2 + y2 saved as an HTML file.	fail→pass	26,977	2,439	-91%	1	1	0%	6,162	3,643	-41%	0	0	—
case-22 Write a complete formal Lean 4 proof script proving that natural number addition is commutative.	pass→pass	14,251	14,146	-1%	1	1	0%	3,025	6,192	+105%	0	0	—
case-23 Write a Python script using FastAPI to create a REST endpoint that calculates the mean of a list of floats.	pass→pass	8,391	7,122	-15%	1	1	0%	1,771	4,380	+147%	0	0	—
case-24 Provide a standard LaTeX document class setup and preamble for an IEEE conference paper.	pass→pass	12,885	10,229	-21%	1	1	0%	2,487	5,024	+102%	0	0	—

These cases were also screened by the skill’s intent / safety pass.See the Trust review ↗Reproduce this benchmark in Colab ↗

How this skill was measured

DecimalAI ran this skill against gemini-3.6-flash twice over the same eval suite — once with the skill loaded and once without — and compared the two runs case by case. 24 cases were attempted, and 21 counted toward the lift figure. The other 3 produced results that are not comparable between the two arms, so they are excluded from the headline rather than averaged into it. The headline lift of +50 percentage points is the difference between those two pass rates over the 21 comparable cases. 1 case got worse with the skill loaded, and it is included in that figure.

Model measured: gemini-3.6-flash
Graded by: gemini-3.6-flash
Grading method: judged
Verification: verified
Pass rate without the skill: 16.7%
Pass rate with the skill: 66.7%
Cases attempted: 24
Cases counted toward lift: 21
Cases that regressed: 1
Skill version measured: v2
Run completed: 8/2/2026, 19:13 UTC

Measured example — same prompt, with vs. without the skill · from the benchmark run · case case-05

Prompt

Calculate the exact symbolic area under x^2 between x=0 and x=1 without numerical approximations.

Without the skill loaded, the model failed this case. With it loaded, the same prompt on the same model passed. This is one improved case from the latest verified run; every case, including any that regressed, is in the table above.

Show both answers

SkillScore blends 4 signals1 of 4 complete

BenchmarkPass rate WITH skill on a verified run67

Live evalNeeds ≥10 evaluated traces in 30 days

AI ratingNeeds ≥10 rated traces

AdoptionNeeds ≥10 serves across ≥2 orgs in 30 days

Efficiency · separate axis (at equal correctness)+0% turns+124% tokens

Shown for transparency; not credited unless a verified run shows no correctness regression.

Backed by 1 of 4 signals — each added signal makes the score more robust.

Verified on gemini-3.6-flash · re-tested 50 days ago

Model	Method	Date	Lift
gemini-3.6-flash	verified	7/29/2026	+82%

Other measured skills in the registry, with their headline benchmark lift.