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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.22832 |
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| _version_ | 1866913161074966528 |
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| author | Gorinevski, Danil |
| author_facet | Gorinevski, Danil |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_22832 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Mathematical Foundations for Peer-to-Peer Lattice Computation Gorinevski, Danil Distributed, Parallel, and Cluster Computing Data Structures and Algorithms Combinatorics Probability 68M14 (Primary) 68Q25, 60K35, 49Q22, 05C80 (Secondary) D.2.4; C.2.4; F.2.2; F.1.2; G.2.2 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. |
| title | Mathematical Foundations for Peer-to-Peer Lattice Computation |
| topic | Distributed, Parallel, and Cluster Computing Data Structures and Algorithms Combinatorics Probability 68M14 (Primary) 68Q25, 60K35, 49Q22, 05C80 (Secondary) D.2.4; C.2.4; F.2.2; F.1.2; G.2.2 |
| url | https://arxiv.org/abs/2605.22832 |