the tropical semiring execution engine for cyber. (min, +) algebra for provable optimization. trop is the decision engine — where nebu proves truth and kuro proves bits, trop proves optimality.
a ⊕ b = min(a, b). a ⊗ b = a + b. no additive inverse — this is a semiring, not a field. the missing inverse is what makes optimization irreducible to field arithmetic.
encoding min(a, b) over Goldilocks requires bit decomposition — ~10 constraints. native tropical: 1 operation. the 10× gap compounds over millions of optimization steps.
workloads
| domain | tropical formulation |
|---|---|
| shortest path | tropical matrix power |
| dynamic programming | tropical recursion |
| Viterbi decoding | tropical matrix-vector multiply |
| belief propagation | tropical message passing |
| optimal transport | tropical linear program |
| attention (hardmax) | tropical softmax limit |
| scheduling | tropical eigenvalue |
| game theory (minimax) | tropical duality |
nox integration
no new Layer 1 patterns. min(a, b) = branch(lt(a, b), a, b) — two existing nox patterns. tropical execution is a PROGRAM on nox.
Layer 3 jets: jet_trop_matmul, jet_trop_shortest, jet_trop_hungarian, jet_trop_viterbi, jet_trop_transport.
verification: tropical execution produces a witness. zheng verifies optimality via LP dual certificate over F_p.
dependency graph
nebu (F_p) — verification backbone
↓
trop (min, +) ← this repo
↓
nox (jets)
↓
zheng (proof of optimality)
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