Inductive Cayley–Dickson Shell Algebras and a Finite Sedenion Ghost-Operator Invariant of the 107 Residual Erdős–Straus Shell

For every reduced fixed-denominator shell in the Erdős–Straus equation, the completion problem is equivalent to a divisor congruence in a finite exponent box. From the prime support of the shell we construct inductively a Cayley–Dickson coefficient algebra together with tagged quotient refinements. This yields a finite shell invariant package consisting of the support rank, the blade-count vector, the tagged target coefficient, the origin-sector decomposition, and, for shell-realized zero-divisor pairs, the spectral data of the finite ghost operator Γ_{a,b} = [L_a, L_b] − L_{[a,b]}. We evaluate the package completely at the residual shell p∗ = 8803369, R∗ = 107, a∗ = (p∗ + 107)/4 = 3² · 11² · 43 · 47, for which the projective coefficient algebra is the sedenion stage A₄. Exact logarithmic enumeration modulo 107 yields Θ^{A₄,C₂}₁₀₇(p∗) = 2o₃₄[1̅], equivalently [53](([34] + [72])([19] + [87])) = 2. The shell contains the zero-divisor quartet X_ε = o₁[0̅] + ε o₁₂₃₄[1̅], Y_η = o₂[0̅] + η o₃₄[1̅], with X_ε Y_−ε = Y_−ε X_ε = 0. For the associated operator Γ₁₀₇ := Γ_{X_−, Y_+} we prove ker(Γ₁₀₇) = ℍ₁₂,₄ ⊗ ℝ[C₂], χ_{Γ₁₀₇}(x) = x⁸(x² + 4)⁸(x² + 16)⁴, μ_{Γ₁₀₇}(x) = x(x² + 4)(x² + 16), Γ₁₀₇(o₁[0̅]) = 2o₃₄[1̅]. We further integrate the carrier symmetry package generated by the commuting involutions κ_{o₃} and κ_{o₄}: on V = Span{1, o₃, o₄, o₃₄} they define a Klein-four action, its normalizer is S₄, the full blade basis splits into Klein-four sectors of dimensions 1, 1, 1, 13, and the neutral atom Z₁ = 1[0̅] and the target branch o₃₄[1̅] lie in distinct joint eigenspaces. A final comparison remark records the exact shell counts for R = 11, 43, 107, displaying a positive octonionic shell, a vanishing quaternionic shell, and the positive sedenionic shell treated in detail.