Ordered-Angle Coding for Exact Multiuser Unanimity Testing

We introduce ordered-angle coding for binary-unanimity testing among $n$ transformation-only users in a serial quantum architecture inherited from Loop-Back communication. User $B_i$ fixes one private sign across a logical word and applies $R(s_i{\alpha }_j)$ in trial $j$. If $w$ users choose the negative sign, serial composition gives $R[(n-2w){\alpha }_j]$. We show that every fixed same-axis pulse that is deterministic for both unanimous inputs under the binary Bell readout has ${\alpha }_j=q_j\pi \mathrm{/}(2n)$ with integer $q_j$. For a nonadaptive word $\mathbf{q}=(q_1,\dots ,q_m)$, a mixed Hamming weight imitates the unanimous signature with probability $$M_{\mathbf{q}}(w)=\prod _{j=1}^m{\cos }^2\text{\hspace{0.17em}⁣}\left(\frac{\pi q_{j}w}{n}\right)\mathrm{.}$$ Within this complete deterministic-unanimity pulse family, a finite perfect word exists if and only if $n$ is a power of two. For $n=2^r$, the dyadic word $(1,2,4,\dots ,2^{r-1})$ is exact and pulse-minimal with $m_{\min }=r={\log }_{2}n$ trials. It replaces the $O(n^2\log (1\mathrm{/}\epsilon ))$ worst-case burden of repeated smallest-angle tests by exact $O(\log n)$ verification. Odd primes instead admit balanced statistical words with uniform mixed-weight imitation $2^{-(p-1)}$. For actual operations $R(s_i{\alpha }_j+{\delta }_{ij})$, an ideal rejecting position is lifted to ${\sin }^2{\mathrm{\Delta }}_j$, where ${\mathrm{\Delta }}_j={\sum }_i{\delta }_{ij}$. This yields explicit robustness bounds with angular and binary-readout errors. Bell entanglement is not required for the additive algebra, but it keeps the traveler locally maximally mixed in every honest trial. We therefore present the result as a coded multiuser relation primitive with optional raw conference-key-candidate use, not as a composably secure conference-key protocol.