Beyond Associativity and Alternativity in 16D Sedenion Signal Processing: Mutually-Annihilating Filter Cascades, the Cubic Alternativity-Defect Calculus N_u, and Closed-Form Pontryagin Optimal Control
Year: 2026
Type: preprint
Abstract: In every associative algebra (ℝ, ℂ, ℍ, and matrix rings Mn(ℂ)), two linear filter stages w1, w2 that annihilate each other (w1 · w2 = 0) produce identically zero output when cascaded on any input signal x, since (x · w1) · w2 = x · (w1 · w2) = 0. Similarly, inverting or equalizing a 16-dimensional channel operator L(u) or solving a quadratic optimal control Hamiltonian conventionally requires O(163) Cholesky/SVD decompositions. In this work, we prove that crossing both the Hurwitz (dim 8) and Moufang (dim 16) thresholds into the 16-dimensional Cayley–Dickson sedenion algebra 𝕊 = 𝕆 ⊕ 𝕆 e8 overturns both paradigms via five exact algebraic laws, all formally verified from first principles in the Lean 4.15.0 kernel with zero axioms (204/204 theorems in MaxwellExterior.lean reporting 0 propext, 0 Quot.sound, 0 Classical.choice, and 0 sorry; 376 zero-axiom / 516 total machine-checked Lean 4.15.0 theorems across the unified v3sp3r + sedenion-pg32 formalization): Symmetric Alternativity-Defect Calculus (Nu): Left and right alternativity fail by the exact same symmetric linear operator Nu(y) = u(uy) − u2y = (yu)u − yu2 = ([a,b,d]𝕆, −[a,b,c]𝕆), which governs the normal operator L(u)TL(u) = ||u||22 I16 − Nu, extends the unit-Moreno-cone square decomposition 2(T1d, T2c) of SedenionRank.square_identity unconditionally to all u ∈ 𝕊, and satisfies the universal cubic minimal polynomial Nu3 = 4Λ(u)2 Nu. Consequently, exact channel inversion L(u)−1 (2.87 × 10−14 residual), Tikhonov/MMSE equalization (1.04 × 10−14), Moore–Penrose pseudoinversion L(u)+ on the singular zero-divisor cone (1.45 × 10−15), and rank-4 nullspace projection Pker L(u) collapse to 3-term SVD-free polynomials in Nu costing only two octonion associators. Co-Axial Box-Kite Kernel Dichotomy and Mutually-Annihilating 2-Stage Cascades (w1 · w2 = 0): Unites the Moreno partner map (a'p)b' = −n(u)p (SedenionReal.nullspace) with the S6 box-kite axis fibration n(u) = (a' ×𝕆 b')/Λ(u) across 14 box-kite sextets (14 × 6 = 84), proving dim(ker L(u1) ∩ ker L(u2)) ∈ {0, 2, 4} is non-zero if and only if n(u1) = +n(u2). When two jammers are non-coaxial (ker L(u1) ∩ ker L(u2) = {0}), a 2-stage cascade (w1, w2) satisfying w1 · w2 = 0 drives every linear combination J = αu1 + βu2 to exact zero while transmitting desired signals 100% through the non-associative sedenion associator (s · w1) · w2 = [s, w1, w2] ≠ 0. Closed-Form Sedenion Pontryagin Maximum Principle (PMP): Maximizing H(x, p, J, w) = ⟨p, x · w⟩ − (γ/2)||J · w||22 − (ρ/2)||w||22 reduces via the switching vector ψ = x* · p to a strictly concave quadratic program solved in closed 3-term SVD-free form by NJ(ψ) (1.11 × 10−16 residual), on the zero-divisor sphere by w*ker = Pker L(J)(ψ)/||Pker L(J)(ψ)||2, and on pre-ADC hardware by 168-blade bang-bang selection. Eisenstein S3 Phase Automorphisms and the Sextic G2 × S3 Invariant J3(u): Proves that the S3 factor in Aut(𝕊) ≅ G2 × S3 originates from cube-root-of-unity phase rotations Z ↦ e2πi/3Z (ω̄2 = ω ⇔ ω3 = 1) in ℂ ⊕ ℂ7, yielding J3(u) = Re(q3)/|q|3 ∈ [−1, +1] to separate lower-octonion (+1) from twisted-sector (−1) states sharing identical singular values. Hurwitz–Radon 9-Frame Isometries and Extremal [16, 8, 6]3 Ternary FEC: Establishes universal norm preservation ||u · x||2 = ||u||2||x||2 on span{e0, ..., e8} against arbitrary zero-divisor keys and constructs an extremal self-dual [16, 8, 6]3 ternary code correcting all 513/513 error patterns of Hamming weight ≤ 2 on bare-metal 64 MHz STM32WB55RG ARM Cortex-M4F and 240 MHz ESP32-S2 Xtensa LX7 hardware.
Author: Agourakis, Dionisio Chiuratto
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225791478