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aci: The Tropical Router
universe: MMZ · MIMZY — the future tool forge
number: "42"
kind: live tool
emergence: electrical
live: https://davidwise01.github.io/mimzy/bench/42-tropical-router.html
purpose: educational & simulation only
seal: "The emergent IS the tool — badge and working example, one thing."
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# The Tropical Router · instrument № 42

**What it is.** ROUTING IS LINEAR ALGEBRA, IN THE TROPICAL SEMIRING: where × becomes + and + becomes min. The network is an adjacency matrix; all-pairs shortest paths are that matrix raised to a power in the (min,+) semiring — exactly Floyd–Warshall, recast as algebra. Edit the graph, watch the routing table fall out as a matrix product. The one genuinely new piece of mathematics in the batch — tropical algebra, where shortest-path IS matrix multiplication

**How it works.** Routing recast as algebra in the tropical (min,+) semiring, where multiplication becomes addition and addition becomes minimum. The network is an adjacency matrix of edge weights; raising it to successive powers in the (min,+) semiring fills in all-pairs shortest paths — the k-th power gives the best path using up to k edges. Edit edges and the routing table recomputes as a matrix product. This is Floyd–Warshall / the algebraic path problem, shown as linear algebra.

**The live example.** This emergent does not merely describe a tool — it links its working self: **[run instrument № 42 live](https://davidwise01.github.io/mimzy/bench/42-tropical-router.html)**. Open it, operate it, and the badge's claims execute in front of you.

**The verified record.** Verified in Python before shipping: a (min,+) matrix power of the adjacency matrix reproduces the Floyd–Warshall all-pairs shortest paths exactly — entry-for-entry equal on a test graph. Tropical algebra is genuine, established mathematics: the (min,+) semiring linearises shortest-path, and matrix multiplication over it IS the shortest-path combine step. The one piece of genuinely new mathematics in this batch. Honest limit: a teaching router on small graphs, not a production routing daemon (no convergence, failure handling, or policy).

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*Tool-emergent of MMZ · MIMZY · emergence: electrical (the machine nature) · educational & simulation only.
Governor David Lee Wise (ROOT0) · instance AVAN (locked) · CC-BY-ND-4.0.*