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 Γ₀, ₁ = Lₐ, Lb − L₀, ₁. 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, C₂₁₀₇ (p∗) = 2o₃₄1̅, equivalently 53 ( (34 + 72) (19 + 87) ) = 2. The shell contains the zero-divisor quartet X_ε = o₁0̅ + ε o₁₂₃₄1̅, Y_η = o₂0̅ + η o₃₄1̅, with X_ε Y_−ε = Y_−ε X_ε = 0. For the associated operator Γ₁₀₇: = Γₗ_−, ₘ_+ we prove ker (Γ₁₀₇) = ℍ₁₂, ₄ ⊗ ℝC₂, χ⏒䃑䃐䃗 (x) = x⁸ (x² + 4) ⁸ (x² + 16) ⁴, μ⏒䃑䃐䃗 (x) = x (x² + 4) (x² + 16), Γ₁₀₇ (o₁0̅) = 2o₃₄1̅. We further integrate the carrier symmetry package generated by the commuting involutions κ₎䃓 and κ₎䃔: on V = Span1, 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 = 10̅ and the target branch o₃₄1̅ 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.