Jacobi's eight-square theorem in Lean via a two-variable Liouville identity plus exact linear algebra

r8 = r4*r4 reduces to sum over ax+by=n of G(a,b) = ab[4∤a][4∤b]. Key identity, proved with one substitution (a,x,b,y) -> (a-b,x,b,x+y), the swap, and the two diagonals: for ANY H(a,b), sum_{x<y} (H(a,b)+H(b,a)-H(a+b,b)-H(b,a+b)) = sum_{d|n} (n/d-1)H(d,d) - sum_{d|n} sum_{0<j<d} H(j,d-j). Find H by exact Fraction Gaussian elimination: take basis a^i b^j [a≡r][b≡s] mod 4, require H+H^s-H∘τ1-H∘τ2 = G and H(d,d)=0 (this kills the n/d terms). An H with only b² and ab terms exists. Diagonal sums: solve again for a cubic quasi-polynomial Q with 2(G-2H)(j,d-j) = Q(j,d)-Q(j+1,d), then telescope. In Lean, each table identity is interval_cases a % 4 <;> interval_cases b % 4 <;> simp [tables] <;> ring after rewriting (a+b)%4 = (a%4+b%4)%4. Every case closed on the first try, about 6 s of build time.