Danil Gorinevski
Five propositions on synchronous peer-to-peer computation on a grid graph in $\mathbb{Z}^2$. \textbf{Proposition~1} gives three lower bounds: a transport-work bound $\sum_i a_i \ell_i \ge W_1(μ,ν)$, a completion-depth bound $D_{\min} \ge r_μ$ on the support radius, and a compressive-reduction edge bound $|E'| \ge \operatorname{St}_G(\operatorname{supp}(μ)\cup\{x_\star\})$. A negative result: for corner-sink dimension-order routing, the sink-trunk route-load functional has variance $Θ(f(1-f)P^2)$ under i.i.d.\ Bernoulli activation, refuting $O(fP^{3/2})$ concentration. \textbf{Proposition~2}: under an $αβγ$ collective-communication cost model and a sparse-activation model, the grid-to-cluster latency ratio improves monotonically as $f_{\mathrm{act}}$ shrinks whenever the cluster's fixed overhead dominates the grid's geometric constant. \textbf{Proposition~3}: a sufficient algebraic criterion for schedule-independent reduction semantics is that update rules decompose into a local map and an abelian-monoid merge, as a product-preserving functor from the Lawvere theory of commutative monoids into a hardware-state category. Under an idealised non-interfering scheduler this yields a wall-clock bound; the criterion is PCC-certifiable. \textbf{Proposition~4} bounds the conditional expected route length under i.i.d.\ site failure in the subcritical regime $δ< p_c^{\mathrm{site}}(\mathbb{Z}^2)$ by an additive detour term, via Aizenman--Barsky cluster-size decay and a boundary-length bound. \textbf{Proposition~5} augments the grid with $k$ long-range Watts--Strogatz shortcuts per node, collapsing typical shortest-path length from $Θ(\sqrt{P})$ to $O(\log P)$; percolation is placed in the mean-field universality class (rigorous for the $1$-D-ring base, conjectural for the $2$-D-grid base).