Mathematical Foundations for Peer-to-Peer Lattice Computation

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 $\sum_i a_i \ell_i \geq W_1(\mu,\nu)$ attained by every shortest-path schedule; a completion-depth bound $D_{\min} \geq r_\mu$ attained by non-congesting parallel routing; and a compressive-reduction edge bound $|E'| \geq \mathrm{St}_G(\mathrm{supp}(\mu)\cup\{x_\star\})$. 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 $\Theta(f_{\text{act}}(1-f_{\text{act}})P^2)$. Proposition 2 establishes, under the $\alpha$-$\beta$-$\gamma$ 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 $\delta < p_c^{\text{site}}(\mathbb{Z}^2)$ 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 $\Theta(\sqrt{P})$ to $O(\log P)$ under a mean-field (Erd\H{o}s-R\'enyi) universality argument -- rigorous for the 1-D-ring base (Newman-Watts-Strogatz), conjectural for the 2-D-grid base.