Scaled complex Householder reflector step with explicit range status

For a stored complex column $x\in {\mathbb{C}}^m$, $m\ge 1$, and block $B\in {\mathbb{C}}^{m\times q}$, $q\ge 0$, call the Householder branch of Scaled real and complex Householder and Givens update arithmetic on $[xB]$. This is the canonical scaled real-component norm and separately scaled phase-modulus realization, with fixed sequential dots, real-coefficient multiplication, complex product and subtraction order. It includes the exact target-column overwrite. Return its $(v,\widehat{\tau },\widehat{\alpha },\widehat{x},\widehat{B})$ on success; a zero column returns identity, zero column and copied $B$. Report an invalid/nonpositive stored-vector norm as breakdown; every other invalid/out-of-range operation returns range-failure with its operation index and partial values.

The exact representative is $P(v)=I-2v{v}^{\ast }\mathrm{/}(v^{\ast }v)$ for nonzero stored $v$. Store this product representation rather than identifying the rounded rank-one action with an exactly unitary dense matrix. For polynomial-vector initialization, form the reflector from that external vector and use its separate-block application on the entire active row panel; do not overwrite an unrelated matrix column. The proof-backed target and separate-block coefficients are supplied by Scaled complex Householder and Givens steps have explicit local backward errors.