Inductive Cayley–Dickson Shell Algebras and the Sedenion Ghost Invariant of the 107 Residual Erdős–Straus Shell
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2026
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| author | The Clankers |
| author_facet | The Clankers |
| contents | <p>For every reduced fixed denominator shell in the Erdős–Straus equation, an exact divisor congruence is converted into a finite exponent-box coefficient problem. This paper constructs, by induction over the prime support of the shell, the corresponding Cayley–Dickson coefficient algebra and its quotient-tag refinements. The construction gives explicit closed invariants for every shell: the support rank, the blade-count vector, the tagged target coefficient, the origin-sector coefficient, the Wedderburn sector projections, the zero-divisor shadow set, and the nonassociative ghost spectrum attached to any shell-realized zero-divisor pair.</p> <p>The general construction is then evaluated at the residual shell</p> <p>p∗ = 8803369, R∗ = 107, a∗ = (p∗ + 107)/4 = 3² · 11² · 43 · 47.</p> <p>This is the first champion shell in the displayed computation for which the projective divisor coefficient algebra is sedenionic. The exact logarithmic enumeration modulo 107 gives precisely two target hits, both on the o₃₄-blade. In the quadratic-character quotient the shell contains the primary sedenion zero-divisor quartet</p> <p>X_ε = o₁[0̅] + ε o₁₂₃₄[1̅], Y_η = o₂[0̅] + η o₃₄[1̅],</p> <p>with cross-polarized products X_ε Y_−ε = Y_−ε X_ε = 0. The induced finite ghost operator</p> <p>Γ₁₀₇ = [L_{X_−}, L_{Y_+}] has kernel</p> <p>ℍ₁₂,₄ ⊗ ℝ[C₂], ℍ₁₂,₄ = ℝ⟨1, o₁₂, o₄, o₁₂₄⟩ ≃ ℍ,</p> <p>characteristic polynomial x⁸(x² + 4)⁸(x² + 16)⁴, and minimal polynomial x(x² + 4)(x² + 16). The target coefficient is localized by the blade-one identity</p> <p>Θ^{A₄,C₂}₁₀₇(p∗) = 2o₃₄[1̅] = Γ₁₀₇(o₁[0̅]).</p> <p>The six-sector algebra used in the origin refinement is treated as the ordinary Wedderburn decomposition of ℂ[C₂ × C₃]. The completed tessarine square appears only as the explicit C₂ × C₂ transport algebra generated by the free blades o₃, o₄ and the diagonal target blade o₃₄.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20343718 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | Inductive Cayley–Dickson Shell Algebras and the Sedenion Ghost Invariant of the 107 Residual Erdős–Straus Shell The Clankers Erdős–Straus equation residual shells Cayley–Dickson algebras sedenions ghost invariant zero divisors completed tesserine algebra finite-dimensional real algebras <p>For every reduced fixed denominator shell in the Erdős–Straus equation, an exact divisor congruence is converted into a finite exponent-box coefficient problem. This paper constructs, by induction over the prime support of the shell, the corresponding Cayley–Dickson coefficient algebra and its quotient-tag refinements. The construction gives explicit closed invariants for every shell: the support rank, the blade-count vector, the tagged target coefficient, the origin-sector coefficient, the Wedderburn sector projections, the zero-divisor shadow set, and the nonassociative ghost spectrum attached to any shell-realized zero-divisor pair.</p> <p>The general construction is then evaluated at the residual shell</p> <p>p∗ = 8803369, R∗ = 107, a∗ = (p∗ + 107)/4 = 3² · 11² · 43 · 47.</p> <p>This is the first champion shell in the displayed computation for which the projective divisor coefficient algebra is sedenionic. The exact logarithmic enumeration modulo 107 gives precisely two target hits, both on the o₃₄-blade. In the quadratic-character quotient the shell contains the primary sedenion zero-divisor quartet</p> <p>X_ε = o₁[0̅] + ε o₁₂₃₄[1̅], Y_η = o₂[0̅] + η o₃₄[1̅],</p> <p>with cross-polarized products X_ε Y_−ε = Y_−ε X_ε = 0. The induced finite ghost operator</p> <p>Γ₁₀₇ = [L_{X_−}, L_{Y_+}] has kernel</p> <p>ℍ₁₂,₄ ⊗ ℝ[C₂], ℍ₁₂,₄ = ℝ⟨1, o₁₂, o₄, o₁₂₄⟩ ≃ ℍ,</p> <p>characteristic polynomial x⁸(x² + 4)⁸(x² + 16)⁴, and minimal polynomial x(x² + 4)(x² + 16). The target coefficient is localized by the blade-one identity</p> <p>Θ^{A₄,C₂}₁₀₇(p∗) = 2o₃₄[1̅] = Γ₁₀₇(o₁[0̅]).</p> <p>The six-sector algebra used in the origin refinement is treated as the ordinary Wedderburn decomposition of ℂ[C₂ × C₃]. The completed tessarine square appears only as the explicit C₂ × C₂ transport algebra generated by the free blades o₃, o₄ and the diagonal target blade o₃₄.</p> |
| title | Inductive Cayley–Dickson Shell Algebras and the Sedenion Ghost Invariant of the 107 Residual Erdős–Straus Shell |
| topic | Erdős–Straus equation residual shells Cayley–Dickson algebras sedenions ghost invariant zero divisors completed tesserine algebra finite-dimensional real algebras |
| url | https://doi.org/10.5281/zenodo.20343718 |