AI 中文总结
提出可微算子LatentGamma作为Horn程序Tarski算子的平滑替代，证明构造性表达性定理，在有限窗口内精确复现推导结果，并验证噪声鲁棒性与参数下界。
AI 中文摘要
我们引入\textsc{LatentGamma}，一种定义在单位立方体$[0,1]^M$上的可微算子，设计为与$M$个原子上的确定Horn程序$P$相关的Tarski直接后果算子$\TP$的平滑替代。该算子由五阶段组合构建，结合了sigmoid门控、softmax路由和通过构造强制单调性的残差更新。我们建立了四个理论结果。首先，迭代序列逐坐标非递减且有界于$\ind$，因此收敛到\textsc{LatentGamma}的不动点；该算子本身在$[0,1]^M$上是格单调的。其次，我们的\emph{构造性表达性定理}表明，对于每个确定Horn程序$P$，存在一个闭式参数赋值$\theta^*(P)$和一个显式时间界限$T_{\max}(P)$，使得迭代序列在窗口$[D(P, F_0), T_{\max}(P)]$内精确复现$\TPinf(F_0)$，其中$D(P, F_0) \leq M$是推导深度。窗口$T_{\max}$在实践中很大（对于稀疏程序$\geq 10^5$），反映了平滑sigmoid门计算的有限时间性质。第三，该oracle对主体和头部logits上的高斯噪声具有鲁棒性，具有显式的非渐近界限。第四，我们证明了参数数量的匹配信息论下界。我们在从$M = 33$到$M = 504$个原子的程序上提供了完整的数值验证：oracle在$2000$个测试用例上达到$1.0000$的准确率，假阳性和假阴性率为零，经验噪声容限$\sigma_{\max}$精确地按照联合界$RM \cdot \Phi(-10/\sigma_\eta)$的预测进行缩放。
英文摘要
We introduce \textsc{LatentGamma}, a differentiable operator on the unit cube $[0,1]^M$ designed as a smooth surrogate of Tarski's immediate consequence operator $\TP$ associated with a definite Horn program $P$ on $M$ atoms. The operator is built as a five-stage composition combining sigmoidal gating, softmax routing, and a residual update that enforces monotonicity by construction. We establish four theoretical results. First, the iterated sequence is coordinate-wise non-decreasing and bounded by $\ind$, hence converges to a fixed point of \textsc{LatentGamma}; the operator itself is lattice-monotone on $[0,1]^M$. Second, our \emph{constructive expressivity theorem} shows that for every definite Horn program $P$ there exists a closed-form parameter assignment $θ^*(P)$ and an explicit time bound $T_{\max}(P)$ such that the iterated sequence \emph{exactly} reproduces $\TPinf(F_0)$ for every initial fact set $F_0$ throughout the window $[D(P, F_0), T_{\max}(P)]$, where $D(P, F_0) \leq M$ is the derivation depth. The window $T_{\max}$ is large in practice ($\geq 10^5$ for sparse programs) and reflects the finite-time nature of computation by smooth sigmoidal gates. Third, the oracle is robust to Gaussian noise on its body and head logits, with explicit non-asymptotic bounds. Fourth, we prove a matching information-theoretic lower bound on the parameter count. We provide a complete numerical validation on programs ranging from $M = 33$ to $M = 504$ atoms: oracle accuracy reaches $1.0000$ on $2000$ test cases with zero false-positive and false-negative rates, and the empirical noise tolerance $σ_{\max}$ scales precisely as the union bound $RM \cdot Φ(-10/σ_η)$ predicts.
