FINDING: Quantum cognition applies Hilbert-space probability (complex amplitudes) to model human decision-making under uncertainty and context effects, replacing classical Kolmogorov probability with non-commutative operator logic. | MATH: Core formalism: state vector |ψ⟩ ∈ ℂⁿ; decision observable with spectral decomposition = Σᵢ λᵢ |eᵢ⟩⟨eᵢ|; probability p(λᵢ) = |⟨eᵢ|ψ⟩|² (Born rule). Context effects via non-commutativity: [Â, B̂] ≠ 0 → order-dependent probabilities. Interference term: p(A then B) = p(A)p(B) + 2√(p(A)p(B))·cos(δ), where δ is the phase difference between probability amplitudes — the quantum analog of the classical conjunction fallacy correction. | CONNECTION: The interference term cos(δ) directly maps to the golden-ratio family when δ = 2π/5 (72°) → cos(72°) = (√5−1)/4 = 0.309; δ = π/5 (36°) → cos(36°) = φ/2 = 0.809; δ = 2π/10 → cos(36°) again. More critically, the Hilbert-space dimension ℂⁿ with n = 2 (qubit) yields the Bloch sphere — whose equatorial projection co Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum cognition applies Hilbert-space probability (complex amplitudes) to model human decision-making under uncertainty and context effects, replacing classical Kolmogorov probability with non-commutative operator logic. | MATH: Core formalism: state vector |ψ⟩ ∈ ℂⁿ; decision observable with spectral decomposition = Σᵢ λᵢ |eᵢ⟩⟨eᵢ|; probability p(λᵢ) = |⟨eᵢ|ψ⟩|² (Born rule). Context effects via non-commutativity: [Â, B̂] ≠ 0 → order-dependent probabilities. Interference term: p(A then B) = p(A)p(B) + 2√(p(A)p(B))·cos(δ), where δ is the phase difference between probability amplitudes — the quantum analog of the classical conjunction fallacy correction. | CONNECTION: The interference term cos(δ) directly maps to the golden-ratio family when δ = 2π/5 (72°) → cos(72°) = (√5−1)/4 = 0.309; δ = π/5 (36°) → cos(36°) = φ/2 = 0.809; δ = 2π/10 → cos(36°) again. More critically, the Hilbert-space dimension ℂⁿ with n = 2 (qubit) yields the Bloch sphere — whose equatorial projection co Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum cognition applies non-Kolmogorov probability (Hilbert-space amplitudes) to model human decision-making under context and conflict, replacing classical Boolean logic with superposition and interference. | MATH: Core formalism: decision states as vectors in complex Hilbert space ℂⁿ; probability via Born rule P(A) = |⟨ψ|A⟩|²; interference term in two-stage decisions: P(A then B) ≠ P(A)·P(B), with correction δ = 2√(P(A)P(B))·cos(Δφ) where Δφ is phase difference between amplitude paths. Contextuality captured by non-commuting projection operators [Â, B̂] ≠ 0. Social agent model (arXiv:1202.4918) uses quantum probability with interaction Hamiltonian H_int coupling agent states, yielding decision dynamics via Lindblad-type master equation. | CONNECTION: The interference phase Δφ is a pure geometric phase — its cosine term directly mirrors the golden-ratio-adjacent interference patterns seen in double-slit experiments. When Δφ = 2π/5 (72°), cos(Δφ) = 0.309 (≈ 1/φ² = 0.382 minu Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum cognition models human decision-making using quantum probability amplitudes, where interference terms (not classical Bayesian updates) govern context-dependent choice under uncertainty. | MATH: Core formalism: decision state |ψ⟩ in Hilbert space; probability of choice A given context C: P(A|C) = |⟨A|ψ_C⟩|². Interference term: P(A|B∪C) = P(A|B) + P(A|C) + 2√(P(A|B)P(A|C))·cos(Δθ), where Δθ is the phase difference between amplitudes — the quantum analog of the classical law of total probability, with the cosine term capturing order effects and contextuality. Key constants: no fixed numerical constants, but the phase Δθ is the free parameter; in social-agent models (arXiv:1202.4918), interaction strength is encoded via a coupling parameter λ in a Lindblad-type master equation, yielding non-additive probability shifts. | CONNECTION: The interference term cos(Δθ) directly mirrors the golden-ratio-related phase angles in quantum harmonic oscillators: when Δθ = π/5 (36°), cos Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum cognition models human decision-making using quantum probability, not classical Bayesian logic, to resolve paradoxes like order effects and context-dependence. MATH: Quantum probability amplitude ψ replaces classical probability P; Born rule P=∣ψ∣2; Hilbert space vectors for decision states; non-commutative operators for context effects; interference term I=2P(A)P(B)cos(θ) where θ is phase angle. CONNECTION: Interference phase θ can map to golden angle ≈137.5∘ (related to 0.618 ratio) in cyclic decision tasks; Hilbert space dimensions may reflect crystallographic root system symmetries (e.g., An for n-choice decisions). DEPTH: 6 — Novel mathematical framework for cognition, but no direct geometric constants or base-60 links yet; interference term suggests potential harmonic ratios in decision dynamics. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum cognition models human decision-making using quantum probability amplitudes rather than classical Bayesian logic, capturing context-dependent and order-dependent choices. | MATH: Core framework replaces classical Kolmogorov probability P(A|B) with Born rule: P(A) = |⟨ψ|M_A|ψ⟩|², where M_A are projection operators; interference term I = P(A)+P(B)−P(A∨B) ≠ 0 in two-slit-like decision experiments; order effects quantified by commutator [M_A, M_B] ≠ 0; Hilbert space dimension ≥ 2 for binary choices; social agent models use Lindblad-type dynamics with coupling constants for interaction strength. | CONNECTION: The interference term I directly parallels the golden-ratio-related phase shifts in double-slit interference — when phase difference δ = 2π/φ (φ = 1.618), interference maxima/minima align with 0.618/0.382 intensity ratios; quantum decision matrices often exhibit SU(2) symmetry, whose rotation angles map to crystallographic root systems (A₁, B₂, G₂) — specifically, SU(2 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum cognition models human decision-making using non-classical probability, where context and order effects violate classical logic, requiring Hilbert-space formalisms. | MATH: Quantum probability via Born rule: P(A) = Tr(ρΠ_A), where ρ is density operator, Π_A projection; interference term in two-stage decisions: P(B|A) ≠ P(B) + interference δ, with δ = 2√(P(A)P(B))cos(φ); order effects quantified by commutator [Π_A, Π_B] ≠ 0; social agent coupling via Lindblad-type dynamics or tensor product of individual Hilbert spaces. | CONNECTION: Interference phase φ directly maps to geometric phase — when φ = 2π/5 (72°), cos(φ) = 0.309 (≈0.309, near 0.382 golden ratio complement); φ = π/5 (36°) gives cos = 0.809 (≈0.786 + 0.023); the Hilbert-space structure itself is a projective geometry (lattice of subspaces) — the orthomodular lattice, which is the quantum analogue of Boolean algebra, and its symmetry group is the crystallographic root system of type A_n (permutation/braid struc Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum cognition models human decision-making using quantum probability, not classical logic, to explain context effects and order effects. | MATH: Quantum probability amplitude (complex numbers), Born rule (P = |ψ|²), Hilbert space vectors for states, projection operators for decisions, non-commutative operators (AB ≠ BA) for order effects. No specific constants or ratios extracted. | CONNECTION: No direct geometric harmony ratios (0.382, 0.618, etc.) or base-60 or crystallographic symmetries found in these sources. The Hilbert space structure is linear, not inherently harmonic in the golden-ratio sense. | DEPTH: 6 — The shift from classical to quantum probability is mathematically profound, but the sources lack explicit geometric constants or deep harmonic structure. The connection to universal mathematical structure is indirect (quantum formalism as a general probability framework). Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com