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Quaternionic Algebra as Sole Mathematical Structure in Quantum Superposition Search Results — E8 Intelligence Research

Andrew Stewart Caldin

Published Sep 23, 2026• 0 citations
Publisher

Abstract

FINDING: The search results are dominated by generic educational content on quantum superposition, with only one substantive mathematical item: a 1996 arXiv paper on quaternionic quantum interferometry. No direct T^3 scaling law or Folman experiment derivation was retrieved. | MATH: The sole mathematical structure present is the quaternionic algebra (non-commutative, 4-dimensional division algebra) applied to scattering amplitudes. The paper posits a universal algebraic identity among six coherent cross-sections (σ_ij for i,j ∈ {1,2,3}, i≠j) if amplitudes are complex: σ_12 σ_13 σ_23 = σ_12 σ_13 σ_23 (cyclic permutations) — a specific relation that would be violated if amplitudes were quaternionic. No explicit T^3 scaling, no constants, no ratios extracted from the retrieved text. | CONNECTION: Quaternions are intimately linked to SU(2) symmetry and the 3-sphere S^3 — the unit quaternions form a double cover of SO(3), the rotation group. This connects to crystallographic point groups (e Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI
10.5281/zenodo.22909826

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FINDING: The search results are dominated by generic educational content on quantum superposition, with only one substantive mathematical item: a 1996 arXiv paper on quaternionic quantum interferometry. No direct T^3 scaling law or Folman experiment derivation was retrieved. | MATH: The sole mathematical structure present is the quaternionic algebra (non-commutative, 4-dimensional division algebra) applied to scattering amplitudes. The paper posits a universal algebraic identity among six coherent cross-sections (σ_ij for i,j ∈ {1,2,3}, i≠j) if amplitudes are complex: σ_12 σ_13 σ_23 = σ_12 σ_13 σ_23 (cyclic permutations) — a specific relation that would be violated if amplitudes were quaternionic. No explicit T^3 scaling, no constants, no ratios extracted from the retrieved text. | CONNECTION: Quaternions are intimately linked to SU(2) symmetry and the 3-sphere S^3 — the unit quaternions form a double cover of SO(3), the rotation group. This connects to crystallographic point groups (e Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Search results are dominated by generic quantum superposition explainers and one specific paper on quaternionic quantum interferometry; no direct T^3 scaling law or Folman massive-superposition derivation surfaced. The only mathematically substantive item is the quaternionic interferometry paper (arXiv:quant-ph/9605024v1). MATH: The paper establishes a universal algebraic identity for six coherent cross-sections (σ_ij) of three scatterers with complex amplitudes: σ_12 σ_13 σ_23 = (1/4)(σ_1+σ_2−σ_12)(σ_1+σ_3−σ_13)(σ_2+σ_3−σ_23) — but with a specific phase constraint. For complex amplitudes, the product of the three pairwise cross-sections satisfies a linear relation involving the three single-scatterer cross-sections. If amplitudes were quaternionic, this relation breaks. The key invariant is the *triple product phase*: Φ = arg( A_12 A_23 A_31 ) — for complex numbers, Φ ≡ 0 mod π (due to associativity and commutativity of C). For quaternions, Φ can be non-trivial, revealin Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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