Mathematical Foundations for Peer-to-Peer Lattice Computation
This work provides theoretical foundations for peer-to-peer computation on grids, offering rigorous bounds and criteria that are relevant for distributed computing and network design, though the results are largely theoretical and incremental.
The paper presents five propositions on synchronous peer-to-peer lattice computation, establishing lower bounds for transport work, completion depth, and edge count, analyzing latency ratios under sparse activation, providing algebraic criteria for schedule-independent reduction, bounding route lengths under site failure, and showing that adding long-range shortcuts collapses typical shortest-path length from Θ(√P) to O(log P).
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).