**Theorem R (reciprocity).** For `n > 0` and every factorization type `σ`: the universal monic density is the root-law coefficient with `q ↦ 1/q` substituted, `A_{n,σ}(q) = B_{n,σ}(1/q)` (where `B_{n,

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**Theorem R (reciprocity).** For `n > 0` and every factorization type `σ`: the universal monic density is the root-law coefficient with `q ↦ 1/q` substituted, `A_{n,σ}(q) = B_{n,σ}(1/q)` (where `B_{n, · Prove2Me