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Setup: Mumford reduction of P + Q
P has weight 1, Q has weight 2, so P + Q is a weight-3 divisor that Cantor's algorithm reduces to weight ≤ 2 before printing.
Let the three affine points of the un-reduced divisor be (Px, Py), (α₁, β₁), (α₂, β₂). The semi-reduced Mumford form is (u₃, w) where
u₃(x) = (x − Px)(x − α₁)(x − α₂)
w(x) = unique quadratic interpolating (Px, Py), (α₁, β₁), (α₂, β₂)
with the identity F(x) − w(x)² = u₃(x) · u'(x), deg u' = 2. Cantor reduction outputs
v'(x) ≡ −w(x) (mod u'(x))
and we receive (u', v'). Crucially, u₃ is not directly recoverable from u' alone.
