Quantum convex membership and weak-separation model

Let $B(x,t)=\{z\in {\mathbb{R}}^n:\mathrm{\parallel }z-x{\mathrm{\parallel }}_2\le t\}$ and let $B(K,t)={\bigcup }_{x\in K}B(x,t)$ for $t\ge 0$; $B(K,-t)$ denotes the inward erosion consisting of points whose radius-$t$ ball is contained in $K$. A convex body is a closed convex set with nonempty interior. Assume rationally encoded $0<r\le R$ and $B(0,r)\subseteq K\subseteq B(0,R)$. The deterministic weak-membership oracle ${\mathrm{M}\mathrm{E}\mathrm{M}}_{\epsilon ,0}(K)$ is queried at a finite-precision point $y$ and must correctly assert either $y\in B(K,\epsilon )$ or $y\mathrm{\notin }B(K,-\epsilon )$. In the overlap it may return either valid assertion. The coherent interface is a supplied reversible Boolean unitary with a fixed deterministic completion on every encoded input, together with its inverse and any controlled forms used.

A weak separation oracle ${\mathrm{S}\mathrm{E}\mathrm{P}}_{\eta ,\rho }(K)$, queried at $y$, with probability at least $1-\rho$, either asserts $y\in B(K,\eta )$ or asserts $y\mathrm{\notin }B(K,-\eta )$ and returns a unit vector $g$ satisfying $⟨g,z⟩\le ⟨g,y⟩+\eta$ for every $z\in B(K,-\eta )$. The membership-to-separation algorithm is promised only on the query domain in its theorem; no unique answer is required in the boundary bands. A finite unit-normal encoding may be a nonzero rational vector v with the normalization g=v/||v|| interpreted exactly; rational bisection supplies an upper norm bound N between ||v|| and2||v|| when a classical reduction requires rational halfspaces. Radius, point, unit-normal and offset encodings and all query/inverse/gate/bit costs are part of the contract.

This specifies oracle behavior; a membership-to-separation theorem and any separation-to-optimization composition are separate claims with their own hypotheses.