Block-encoding normalization, error, and cost accounting

For one LCU construction, the normalization is $\mathrm{\Lambda }$ and each additive cost coordinate for the sequential operations is the sum of the declared costs of PREPARE, SELECT, and PREPARE${}^{†}$; a SELECT implementation assembled from controlled component unitaries includes those calls and phases in its declared cost. For a two-factor product construction, the normalization is ${\alpha }_A{\alpha }_B$ and the query and gate counts are the sums of the two supplied encoding costs in the chosen order. Any explicit wire permutation or control wrapper is included in a supplied operation's cost or charged as a separate operation. The normalized error is bounded by Product block-encoding error bound.

Matrix-data access, coefficient-table access and storage, state preparation and its inverse, controlled access, angle or entry precision, classical preprocessing, and compilation remain separate cost coordinates. This result counts declared primitives; it does not synthesize a circuit from a bare matrix or assign an absent primitive zero cost.