Stereographic Qubit Lattice Reveals Golden-Ratio Harmonic Structure on the Bloch Sphere — E8 Intelligence Research

Q: What does this research mean for the field?

The equatorial projection of the Bloch sphere reveals the qubit decision space as a stereographic lattice with a golden-ratio harmonic structure that maximizes aperiodicity, analogous to quasicrystal structures. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

Andrew Stewart Caldin

doi:10.5281/zenodo.22701663

September 11, 2026Open Access

Stereographic Qubit Lattice Reveals Golden-Ratio Harmonic Structure on the Bloch Sphere — E8 Intelligence Research

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Key Points

Characterize the geometric and harmonic structure of qubit decision space using stereographic projection of the Bloch sphere equator onto the complex plane.
Applied stereographic projection from the south pole of the Bloch sphere to map general qubit states onto the complex plane as z = tan(θ/2)e^(iϕ).
Analyzed equatorial states (θ = π/2) mapping to the unit circle z = e^(iϕ) under n-fold cyclic symmetries and golden-angle partitioning (ϕ_g ≈ 2.39996 rad or 137.5°).
Demonstrated that the equatorial projection forms the unit circle, providing a natural domain for roots of unity and establishing links to crystallographic point groups C_n and D_n.
Showed that partitioning the equatorial circle by the golden angle produces an irrational ratio (1/φ ≈ 0.618) that maximizes lattice aperiodicity, mirroring the diffraction patterns of Penrose tilings and quasicrystals.

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Implication

Theoretical analysis demonstrates a golden-ratio harmonic lattice on the projected Bloch sphere, indicating geometric ties between quantum decision spaces and aperiodic quasicrystals.

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Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6aa7b3770926e14a848b2750 https://doi.org/10.5281/zenodo.22701663

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Stereographic Qubit Lattice Reveals Golden-Ratio Harmonic Structure on the Bloch Sphere — E8 Intelligence Research

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