We give structured proofs for five mathematical propositions governing synchronous peer-to-peer computation on a finite grid graph embedded in $\mathbb{Z}^2$. Proposition 1 gives three lower bounds: a transport-work bound $\sum_i a_i \ell_i \geq W_1(μ,ν)$ attained by every shortest-path schedule; a completion-depth bound $D_{\min} \geq r_μ$ attained by non-congesting parallel routing; and a compressive-reduction edge bound $|E'| \geq \mathrm{St}_G(\mathrm{supp}(μ)\cup\{x_\star\})$. A negative result refutes naive $O(f_{\text{act}}P^{3/2})$ concentration for sink-trunk loads under corner-sink dimension-order routing, showing variance $Θ(f_{\text{act}}(1-f_{\text{act}})P^2)$. Proposition 2 establishes, under the $α$-$β$-$γ$ collective-communication and a Mixture-of-Experts sparse-activation model, that the grid-to-cluster latency ratio improves monotonically as $f_{\text{act}}$ shrinks whenever cluster fixed overhead dominates the grid geometric constant. Proposition 3 identifies a sufficient algebraic criterion for schedule-independent reduction: update rules decomposing into a local map and an abelian-monoid merge, expressed as a product-preserving functor from the Lawvere theory of commutative monoids into the hardware-state category. Proposition 4 bounds the conditional expected route length under i.i.d. site failure in the subcritical regime $δ< p_c^{\text{site}}(\mathbb{Z}^2)$ by an additive detour, using Aizenman-Barsky exponential cluster-size decay. Proposition 5 augments the grid with $k$ uniform long-range shortcuts per node, collapsing the typical shortest-path length from $Θ(\sqrt{P})$ to $O(\log P)$ under a mean-field (Erdős-Rényi) universality argument -- rigorous for the 1-D-ring base (Newman-Watts-Strogatz), conjectural for the 2-D-grid base.

翻译：我们给出了关于嵌入$\mathbb{Z}^2$的有限网格图上同步对等计算的五个数学命题的结构化证明。命题1给出三个下界：一个由每条最短路调度达到的运输功下界$\sum_i a_i \ell_i \geq W_1(μ,ν)$；一个由非拥塞并行路由达到的完成深度下界$D_{\min} \geq r_μ$；以及一个压缩约简边数下界$|E'| \geq \mathrm{St}_G(\mathrm{supp}(μ)\cup\{x_\star\})$。一个否定性结果反驳了在角落-沉没维度序路由下关于沉没-主干负载的朴素$O(f_{\text{act}}P^{3/2})$集中性，其方差为$Θ(f_{\text{act}}(1-f_{\text{act}})P^2)$。命题2在$α$-$β$-$γ$集体通信与混合专家稀疏激活模型下证明：当簇固定开销主导网格几何常数时，网格到簇的延迟比随$f_{\text{act}}$减小而单调提升。命题3给出一个调度无关约简的充分代数判据：更新规则可分解为局部映射与阿贝尔幺半群合并，表示为从交换幺半群的Lawvere理论到硬件态范畴的积保持函子。命题4在次临界状态$δ< p_c^{\text{site}}(\mathbb{Z}^2)$下，利用Aizenman-Barsky指数级簇尺寸衰减，通过附加迂回对独立同分布站点故障下的条件期望路径长度进行界定量化。命题5通过每节点添加$k$条均匀长程捷径，在平均场（Erdős-Rényi）普适性论证下将典型最短路径长度从$Θ(\sqrt{P})$压缩至$O(\log P)$——对一维环基（Newman-Watts-Strogatz）为严格证明，对二维网格基仍为猜想。