The A₈ Snowflake Target Star of the Residual Erdős–Straus Obstruction: A support-idempotent and Fricke-sheet refinement of the six-class modulo-840 datum

For hard prime denominators in the Erdős–Straus equation, the fixed-shell divisor calculus reduces each residual shell to two normalized targets, −1 and −p⁻¹, in a finite signed divisor-ratio box. After the unconditional R = 3, 7 sieve, the remaining prime residue classes modulo 840 form the six-element cyclic square fiber S₈₄₀ = {1, 121, 169, 289, 361, 529} = ⟨289⟩ ≅ C₆. This note proves that the two-target problem is equivalently a positivity problem on a six-edge target star. The six vertices are realized as the six nonzero isotropic endpoints {±3e₀, ±3e₁, ±3e₂} in the discriminant form of A₈³. The normalized target −1 marks a distinguished hub, and a prime class p ≡ 289ᵏ (mod 840) determines the target edge {v₃, v₃₋ₖ}. Thus the remaining arithmetic assertion is exactly a support-positive hitting problem on this snowflake star.

The same construction explains the 744 ↦ 750 shift. The shift by 6 is invisible modulo the residual Coxeter fiber C₆ but acts as the sign reversal −3 ↦ +3 on the unique nonzero isotropic 3-torsion line in the A₈ discriminant group. We compute the two relevant Coxeter eta sheets: T₆(τ) = η(τ)⁵ η(3τ) / [η(2τ) η(6τ)⁵], T₉(τ) = η(τ)³ / η(9τ)³, together with their Fricke-even, Fricke-odd, and Eisenstein logarithmic derivative channels. Jacobi's eta-cube identity proves that the raw A₈³ sheet satisfies [q⁷⁴⁴] T₉ = [q⁷⁵⁰] T₉ = 0, while the Fricke and Eisenstein completions recover the champion source and certificate residues modulo 107. In particular, [q⁷⁴⁴] J₉ = N∗, [q⁷⁴⁴] Y₉ = −N∗, [q⁷⁴⁴] f₉ = a∗², and [q⁷⁵⁰] T₉ = 0, ([q⁷⁵⁰] J₉)² = [q⁷⁵⁰] f₉, ([q⁷⁵⁰] f₉)⁻¹ = N∗. The note includes explicit diagrams and exact finite verification recurrences.