Structure before circuit tricks
Are coefficient preparation and controlled operator access both available?
Prepare coefficients sqrt(w_a/alpha), select the corresponding unitary, and unprepare. Phases for complex coefficients must be assigned explicitly.
This is an authored reusable derivation guide, not a claim that the full family has been source-assimilated.
No local transport theorem is bound to this record. Do not infer formal truth from its position in the atlas.
SP can be a BE ingredient, but one isolated state does not determine a general matrix. This explanatory bridge has no newly certified transport root.
SP + SELECT gives an LCU block — proposal
primary-text-checked Results: circuit complexity lower bound; construction of LCU-based block-encoding; Methods: state preparation
Explicit access construction is not a free oracle. PREPARE together with SELECT and uncomputation can supply a block encoding; a single prepared state alone does not determine an arbitrary operator.
% Authored mechanism lesson; not a new theorem certificate.
\section*{PREPARE–SELECT–unprepare}
Are coefficient preparation and controlled operator access both available?
\[
(\langle0|\otimes I)(P^\dagger\otimes I)\operatorname{SELECT}(U)(P\otimes I)(|0\rangle\otimes I)=\sum_a w_aU_a/\alpha
\]
Prepare coefficients sqrt(w\_a/alpha), select the corresponding unitary, and unprepare. Phases for complex coefficients must be assigned explicitly.
\paragraph{Hypotheses and contracts.}
\begin{enumerate}
\item Nonnegative weights w\_a and alpha=sum w\_a>0, or an explicit complex-phase adapter
\item Implementable SELECT and adjoint PREPARE, not merely copies of a state
\item Compatible data/ancilla layouts and normalization
\end{enumerate}
\paragraph{Mathematical proof mechanism.}
This is a reusable derivation guide; exact certified scope is given by the linked Lean signatures.
\begin{enumerate}
\item Expand PREPARE on the clean ancilla.
\item Apply SELECT branch by branch.
\item Contract with the same coefficient preparation to obtain weighted matrix entries.
\end{enumerate}
\paragraph{Boundary.} SP can be a BE ingredient, but one isolated state does not determine a general matrix. This explanatory bridge has no newly certified transport root.