Algebraic Hiddenness in the Sedenion Algebra: Characterization, Spectral Structure, and Bundle Topology

The 16-dimensional sedenion algebra 𝕊, obtained as the fourth Cayley–Dickson doubling of the reals, is the lowest level of the CD tower at which the composition law fails and zero divisors appear. We develop a complete algebraic and geometric theory of its zero-divisor locus and the rank-4 vector bundle naturally associated with it.

The principal results are geometric. (i) The Bundle Nontriviality Theorem — the rank-4 kernel bundle over G₂/SU(2) carries a nonzero ℤ/2 secondary characteristic class (the aphanic twist), and parallel transport around certain geodesic loops is −id. (ii) The Holonomy Decomposition Theorem — given the previously established holonomy U(2) = (U(1) × SU(2))/ℤ₂ of the canonical connection, the SU(2) factor is the restriction to the kernel of the lifted G₂-stabilizer action, and the U(1) factor is generated, at the basepoint, by a commutator of right-multiplications R_e₁, R_e₂. (iii) The R_ℓ Centrality Theorem — that U(1) generator is the restriction to the kernel of a single global ambient operator R_ℓ on 𝕊, with [R_ℓ, Lₚᵀ Lₚ] = 0 on the entire imaginary locus, of which the aphanic locus is a strict subset. (iv) The CD-Tower Recursion Theorem — that centrality identity is level-uniform, holding at every Cayley–Dickson level by an argument using only flexibility, inner-product duality, and the conjugation anti-homomorphism property.

These rest on an algebraic foundation which we develop by a self-contained route, and whose structural conclusions are in substantial part already known. (v) A Pfaffian route to the annihilator structure of 𝕊: the identity Pf(Lₚ) = −(‖p‖²)² T, with T = 4‖a‖² Re(b)² + 4⟨a, b⟩² + (‖a‖² − ‖b‖²)², certified by exact finite computation, from which follow in one derivation the zero-divisor criterion of Moreno and Khalil–Yiu, the {0, 4} nullity dichotomy of Moreno and Biss–Dugger–Isaksen, and the determinant factorization det Lₚ = (‖p‖²)⁴ T² obtained independently by Koebisu. (vi) A derivation of the 4-8-4 spectral structure of the Gram operator Gₚ = Lₚᵀ Lₚ — Spec(Gₚ) = {(‖p‖² − σ)⁴, (‖p‖²)⁸, (‖p‖² + σ)⁴}, σ = 2‖x ∧ y‖ — by an associator-norm argument; this is equivalent to the eigenvalue computation of Biss–Christensen–Dugger–Isaksen for their operator Mₐ = (1/‖a‖²) L_ā Lₐ, and is used here as the structural input to the bundle and holonomy results above. (vii) A unified eleven-step proof of the Aphanic Characterization Theorem, the doubly-imaginary case of the criterion in (v). All proofs are constructive and certified by self-contained C++17 probes.