Split-Zero Residual Moonshine and the Snowflake Obstruction: Support-Exact Divisor Shells, q-Series Transforms, and the A₅⁴D₄ Umbral Completion

We consolidate the split-zero, residual-divisor, finite conservative-matrix-field, snowflake-symmetry, and lambency-six moonshine layers attached to the modulo-840 hard residue datum for the Erdős–Straus equation. The output is a support-complete preprint rather than a compressed note. The central convention is that the split-zero semiring G(R)=R⊔{τ} has two zero-like states: the unsupported semiring zero τ and the supported arithmetic zero e=0_R∈R. This distinction changes every statement about zero, primehood, support, linear algebra, and residual coefficients.

For a fixed shell R=4a-p, N=pa, we prove the divisor-pair certificate, its projective normalization, the fully reduced shell criterion, and the two-target form R works ⟺ λ_R(B(a))∩{-1,-p⁻¹}≠∅. The finite support dynamics is recast as a flat conservative matrix field over the finite group K_R(a) and diagonalized by finite Fourier kernels. The resulting shell coefficient is then lifted to G(ℤ), where the three states τ, 0_ℤ, and ℤ_{>0} are topologically separated by the primes P_τ and P_0. This gives an exact counterfactual theorem: any residual failure is either pure absence, generic supported-zero failure, or a finite closed congruence-prime obstruction.

The modulo-840 fibration is shown to be a 24→4×6 support fibration with principal fiber S_840=⟨289⟩≅C_6. Its four augmentation fibers give A₅⁴, the discriminant base gives D₄, and the explicit isotropic glue subgroup of order 72 completes the frame to the Niemeier lattice of root type A₅⁴D₄. We spell out the group-theoretic snowflake model: the raw signed three-arm symmetry C₂³⋊S₃ has even projective subgroup C₂²⋊S₃≅S₄, while the actual glue lift is the non-split double cover GL₂(3)→S₄. This separates the D₄ triality S₃ from the terminal C₂ support signs and from the central lift.

Finally, the residual q-series coordinate is identified with the eta quotient T₆(τ) = (η(τ)⁵η(3τ))/(η(2τ)η(6τ)⁵), with Fricke relation T₆(W₆τ)=72/T₆(τ), Monster 6B trace coordinate T₆+5+72/T₆, and umbral weight-two shadow -2η(6τ)⁴T₆. We also include the hidden-scale q-rational transport theorem that packages Euler–Mellin periods, Hurwitz-zeta jets, and finite beta shadows through the single span scale h_x=log(V_x-U_x). The paper ends with the support-prime formulation of the remaining obstruction: the Niemeier and moonshine structures rigidify the coordinate system, but the arithmetic problem remains the positivity of the split-zero shell coefficient.