proof-architect

Proof Architect

Structured methodology for constructing and verifying mathematical proofs in statistical research

Use this skill when working on: mathematical proofs, theorem development, derivations, consistency proofs, asymptotic arguments, identification proofs, or verifying proof correctness.

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Proof Structure Framework

Standard Proof Components

Every rigorous statistical proof should contain:

1. Claim Statement - Precise mathematical statement of what is being proved
2. Assumptions - All conditions required (clearly enumerated A1, A2, ...)
3. Notation - Define all symbols before use
4. Proof Body - Logical sequence of justified steps
5. Conclusion - Explicit statement that claim is established

Proof Skeleton Template

\begin{theorem}[Name]
\label{thm:name}
Under Assumptions \ref{A1}--\ref{An}, [precise claim].
\end{theorem}

\begin{proof}
The proof proceeds in [n] steps.

\textbf{Step 1: [Description]}
[Content with justification for each transition]

\textbf{Step 2: [Description]}
[Content]

\vdots

\textbf{Step n: Conclusion}
Combining Steps 1--[n-1], we obtain [result], completing the proof.
\end{proof}

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Proof Types in Statistical Methodology

1. Identification Proofs

Goal: Show that a causal/statistical quantity is uniquely determined from observed data distribution.

Standard Structure:

1. Define target estimand (e.g., $\psi = E[Y(a)]$)
2. State identifying assumptions (consistency, positivity, exchangeability)
3. Apply identification formula derivation
4. Show formula depends only on observable quantities

Template:

\begin{theorem}[Identification of $\psi$]
Under Assumptions \ref{A:consistency}--\ref{A:positivity}, the causal effect
$\psi = E[Y(a)]$ is identified by
\[
\psi = \int E[Y \mid A=a, X=x] \, dP(x).
\]
\end{theorem}

\begin{proof}
\begin{align}
E[Y(a)] &= E[E[Y(a) \mid X]] && \text{(law of iterated expectations)} \\
        &= E[E[Y(a) \mid A=a, X]] && \text{(A\ref{A:exchangeability}: $Y(a) \indep A \mid X$)} \\
        &= E[E[Y \mid A=a, X]] && \text{(A\ref{A:consistency}: $Y = Y(A)$)} \\
        &= \int E[Y \mid A=a, X=x] \, dP(x) && \text{(definition)}
\end{align}
which depends only on the observed data distribution.
\end{proof}