Math

Verify exact numerical answers with typed output and deterministic code. Single products saturate strong models, so these tasks grow through digit transforms and modular recurrences until attempts genuinely disagree.

Files

• basic.py — contrast a saturated multiplication with an eight-step recurrence in the learning zone.
• chained_arithmetic.py — compose multiplication, digit sums, scaling, and a modulus instead of grading an easy one-step product.
• recurrence_boundary.py — grow the same nonlinear recurrence from eight to ten rounds and read the changing pass-rate band.

When to use

Use exact math when a programmatic verifier can decide correctness without judgement. The calibration method comes from _07_difficulty_calibration/; the same principle is applied to executable queries in _22_sql_generation/.

Run

.venvs/demo/bin/python cookbook/environments/_21_math/basic.py
.venvs/demo/bin/python cookbook/environments/_21_math/chained_arithmetic.py
.venvs/demo/bin/python cookbook/environments/_21_math/recurrence_boundary.py

Requires OPENAI_API_KEY.