math-proof

> **Author:** Moran Koren, Ben-Gurion University of the Negev (korenmor@bgu.ac.il). Part of the [Theorist Toolbox](https://github.com/morankor/theorist-toolbox). # Math proof Write rigorous mathematical proofs suitable for peer-reviewed academic papers. Every step should be explicit enough that a reader can verify it without filling in gaps. The proof must be a complete proof, not a proof outline — each step should be carefully explained and documented. ## Trigger phrases - `math-proof` - "prove this" - "show analytically" - "derive this result" - "justify mathematically" - "write a proof" - "expand this proof" ## Core principles ### No gaps between steps Every transition from one equation to the next must be justified. If you use the quotient rule, say so. If you substitute a definition, point to which definition. If a sign is negative, explain why. The reader should never need to work out an intermediate step on their own. **Bad:** $$\frac{d}{d\rho}\frac{n_G}{n_B} = \frac{2q-1}{n_B^2} > 0.$$ **Good:** We compute $\frac{d}{d\rho}(n_G/n_B)$ using the quotient rule. First, the derivatives: $$\frac{dn_G}{d\rho} = q, \qquad \frac{dn_B}{d\rho} = 1-q.$$ Applying the quotient rule: $$\frac{d}{d\rho}\frac{n_G}{n_B} = \frac{q \cdot n_B - (1-q) \cdot n_G}{n_B^2}.$$ Expanding the numerator: $$q[\rho + (1-\rho)q] - (1-q)[\rho + (1-\rho)(1-q)] = \rho(2q-1) + (1-\rho)(2q-1) = 2q-1.$$ Since $q > 1/2$, this is positive. ### State what you want to show before showing it Open each...