\section{LILY / compiling complete certificates} \label{sec:compiler} \subsection{Why the original single-sensor greedy rule does not transfer} AELHYEUM's sensor objective is a multiplicative reduction of pairwise overlap. The exact probability of obtaining a semantic certificate can have increasing returns. If $C=\{1,2\}$ is a minimal forbidden support and both input coordinates are active, selecting only sensor 1 or only sensor 2 produces no certificate, while selecting both produces one with probability $q_1q_2>0$. The second sensor's marginal benefit is larger after selecting the first. Therefore the semantic objective over individual sensor positions need not be submodular. A greedy guarantee for the source objective cannot simply be copied into the new language. The compiler instead treats a \emph{complete sound certificate} as the atomic selectable object. \subsection{Packet semantics} For each minimal forbidden support $C$, make a packet that prepares the needed coordinate symbols and tests all of them by their conclusive local projectors. In a binary alphabet with fixed nonzero activation probabilities, its refusal probability is \begin{equation} r_C(x)=\gamma_C\1\{C\subseteq\supp(x)\},\qquad \gamma_C=\prod_{i\in C}q_i. \label{eq:packet} \end{equation} Every successful packet emits \texttt{REFUSE(C)}. Its certificate is independently checkable from the grammar because $C\notin\K$, and it is sound in the calibrated ideal model. A local implementation on the minimal forbidden support is optimal for that packet's one-sided task by Theorem~\ref{thm:hereditary}. Choose packets before observing the unknown input. When selected packets use separate preparations with conditionally independent outcomes given the same fixed $x$, and $\pi(x)$ is a declared design prior, their expected success is \begin{equation} J(\mathcal S)=\sum_x\pi(x)\left[1-\prod_{C\in\mathcal S}(1-r_C(x))\right]. \label{eq:outerobjective} \end{equation} Repeated preparation is a charged resource. The formula does not assume that overlapping tests on a single quantum system are independent, and it does not clone an unknown supplied word. \begin{theorem}[Certificate selection and an exact fast marginal] \label{thm:compiler} The function $J$ is normalized, monotone, and submodular. Under a cardinality budget of $b$ packets, ordinary greedy selection achieves at least $(1-1/e)$ times the optimal value. In the binary support model \eqref{eq:packet}, define \[ v_{\mathcal S}(x)=\pi(x)\prod_{C\in\mathcal S}(1-r_C(x)). \] The exact marginal of a candidate $C$ is \begin{equation} \Delta_C J(\mathcal S) =\gamma_C\sum_{x:\supp(x)\supseteq C}v_{\mathcal S}(x). \label{eq:zetagain} \end{equation} All candidate marginals can be computed from one upper subset-zeta transform in $O(n2^n)$ arithmetic operations and $O(2^n)$ memory. \end{theorem} \begin{proof} For any candidate packet $C$, \[ \Delta_CJ(\mathcal S)=\sum_x\pi(x)r_C(x) \prod_{B\in\mathcal S}(1-r_B(x)). \] Every term is nonnegative and decreases when $\mathcal S$ grows, proving monotonicity and diminishing returns. The classical cardinality-constrained greedy theorem\ \cite{nemhauser} gives the approximation factor. Substituting \eqref{eq:packet} proves \eqref{eq:zetagain}. The upper subset transform stores $z(C)=\sum_{X\supseteq C}v(X)$ and performs one addition for each missing coordinate in each subset, giving the stated cost. \end{proof} The approximation theorem and subset transform are established algorithms. The proposed coupling is the replacement of raw sensor choices by globally optimal one-sided certificate packets, with an exact probability model and source-aligned transform oracle. The $O(n2^n)$ bound is exponential in word length. It is useful for a finite dictionary and does not establish a polynomial-time method for arbitrary succinctly represented languages. \subsection{A fully enumerated eight-coordinate instance} The supplied program uses the at-most-two grammar on eight binary coordinates, hence 56 possible three-coordinate packets. It fixes the prior and local activation probabilities in the source code. All 27,720 three-packet choices are evaluated under the same objective. \begin{center} \begin{tabular}{@{}lll@{}} \toprule \textbf{Budget} & \textbf{Greedy expected refusal} & \textbf{Complete optimum if enumerated}\\ \midrule 1 & 0.2268502804 & --\\ 2 & 0.3271777764 & --\\ 3 & 0.4062552360 & 0.4062552360\\ 4 & 0.4618133396 & --\\ \bottomrule \end{tabular} \end{center} At budget three, greedy chooses $\{1,2,3\}$, $\{1,2,6\}$, and $\{4,6,7\}$ and attains the complete finite optimum. The transform and direct marginals agree within $8.33\times10^{-17}$. This one instance verifies the implementation; it does not improve the general approximation factor. \begin{figure}[ht] \centering\includegraphics[width=.85\textwidth]{figures/certificate_compiler.pdf} \caption{Synthetic packet selection with all preparation uses charged as separate packets. The diamond marks the exhaustive three-packet optimum.} \end{figure} \subsection{An executable semantic interface} The minimal compiler record consists of: \begin{enumerate} \item \textbf{Alphabet:} the character labels and amplitudes, or calibrated local density operators and background supports. \item \textbf{Meaning:} the Boolean function, sparse-repair dictionary and budget, or hereditary grammar. \item \textbf{Contract:} minimum error, exact reading, or a specified false-refusal tolerance. \item \textbf{Receiver:} a theorem-backed measurement and a deterministic classical decision rule. \item \textbf{Certificate:} a forbidden support, a finite-field dual witness, or a recorded undecided outcome. \item \textbf{Resource record:} preparations, active dimension, interactions, readout round, classical work, and transmitted metadata. \end{enumerate} The source manuscripts' decoder-synchronized coding interface can serialize this record. A fixed certificate-informed prior may drive a causal mixture whose predictions are reproduced at the decoder. The present work contributes the semantic probability and certificate; it does not add a new NNCP or cmix implementation. Shared calibration or grammar tables must genuinely be shared or charged in the packet. Quantum readout does not remove the source manuscripts' information-accounting constraints. \subsection{A practical decision map} \begin{tabularx}{\textwidth}{@{}p{.30\textwidth}X@{}} \toprule \textbf{Required meaning} & \textbf{Proved design consequence}\\ \midrule Uniform-prior Boolean guess from binary pure product signals, with $0