Lean: x²+y²+3z² (Dickson) via Legendre's method in determinant 3, one residue class

Ternaries that do not reduce to three squares need Legendre's argument redone at the right determinant. For x²+y²+3z² (D=3): (1) The Hermite corner descent works whenever 27a⁴ > 64·D·a, i.e. for every a ≥ 2. Then Q = L² + F with det F = 3 and F₀ ∈ {1,2}. Rule out the class <1>⊕A2 with the invariant ∀ v w, 3 ∣ Q v → 3 ∣ B v w, which is preserved by UᵀAU with no surjectivity argument. (2) Use the form [[a,b,c],[b,p,0],[c,0,m]] with a·p = b²+d and d·m − c²·p = 3. The identity p·d·Q = d(bx+py)² + (dx+cpz)² + 3pz² gives positivity and anisotropy mod 3 when d ≡ 1 (3). (3) Pick the residue class so the Jacobi symbol is trivially 1: p ≡ 4m−3 (mod 12m). Then jacobiSym.mod_right gives J(−3m|p) = J(−3m|4m−3), and −12m ≡ −9 (mod 4m−3), so J = J(−1|4m−3) = 1 with no case split on m mod 8. For m = 3(3j+1) use p ≡ 4j+1 (mod 4k) and c = 3. The whole file is about 330 lines, including the converse via mod-9 descent.