Orthogonal-Projector Reflection and Access Contract

Let $H$ be a finite-dimensional complex-inner-product space and let $\mathrm{\Pi }$ be an orthogonal projection on $H$. Fix the two signs

$R_{\mathrm{\Pi }}:=2\mathrm{\Pi }-I,S_{\mathrm{\Pi }}:=I-2\mathrm{\Pi }=-R_{\mathrm{\Pi }}\mathrm{.}$

Thus $R_{\mathrm{\Pi }}$ acts as $+I$ on $\mathrm{ran}\mathrm{\Pi }$ and $-I$ on $\ker \mathrm{\Pi }$; $S_{\mathrm{\Pi }}$ reverses those signs. They are not interchangeable inside a controlled block. For a unit vector $\psi$, the rank-one reflection is $R_{\psi }=2\mathrm{\mid }\psi ⟩⟨\psi \mathrm{\mid }-I$ with the Dirac convention of Dirac Notation Conversion.

An access model counts a direct reflection as one primitive call only when that call is explicitly declared. Controlled reflection access is a separate declaration. If $\mathrm{\mid }0⟩$ is a specified unit input in the same space $H$, the state is prepared as $\psi =U\mathrm{\mid }0⟩$ for a declared unitary $U$, and a reflection is implemented by conjugating $R_0=2\mathrm{\mid }0⟩⟨0\mathrm{\mid }-I$, the construction is $U{R}_0{U}^{†}$ and its cost must include the calls to $U$, $R_0$, and $U^{†}$. A state vector or matrix description alone implies neither an efficient reflection nor inverse access.

The root theorem An endomorphism is an orthogonal projection exactly when it is idempotent and self-adjoint identifies an orthogonal projection as a self-adjoint idempotent. Projector Reflections Are Hermitian Unitary Involutions ↗ proves the algebraic identities for both signs; Rank-One Reflection Is Ray-Invariant and Its Signs Differ Under Control ↗ proves ray invariance and the controlled sign relation. Their actual proof inputs remain subject to cross-pair supplier reconciliation.