E.2 Dependency structure
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The main proof dependencies are:
\[
\begin{gathered}
\text{restored oracle convention}\ \Longrightarrow\ \text{cut comparison},\\
\text{cut comparison}+\text{degree per multiplication}\
\Longrightarrow\ \text{complete-cover lower bound},\\
\text{safe ideal}+\text{finite linear duality}\
\Longrightarrow\ \text{evaluation-span certificates},\\
\text{monotone ideal}+\text{affine input/output changes}\
\Longrightarrow\ \text{calibration formulas},\\
\text{affine restriction}+\text{quadratic polar rank}\
\Longrightarrow\ \text{projection-cost lower bound},\\
\text{orthogonal idempotents}\ \Longrightarrow\ \text{gluing dimension},\\
\text{Boolean coefficient identity}+\text{root bound}\
\Longrightarrow\ \text{conditional fingerprint test}.
\end{gathered}
\]
The gluing-dimension theorem and fingerprint construction are separate from the circuit-cut theorem. No proof step identifies algebra dimension with unrestricted circuit size.