FINDING: Quantum decision theory explains the conjunction fallacy via an interference term proportional to cos(δ), where δ is the phase difference between cognitive state vectors — this is a direct quantum probability correction to classical Bayesian conjunction. MATH: - Classical conjunction: P(A∧B) ≤ P(A), P(B) — violated in experiments. - Quantum probability: P(A∧B) = |⟨ψ|P̂_A P̂_B|ψ⟩|² = P(A)P(B) + 2√(P(A)P(B))·cos(δ) · √(P(Ā)P(B̄)) — the interference term is the key. - For maximal violation, cos(δ) = 1/2 → δ = π/3 (60°), giving P(A∧B) = P(A)P(B) + √(P(A)P(B)P(Ā)P(B̄)). - Fermi's Golden Rule: transition rate W = (2π/ℏ)|V_ab|² ρ(E) — the sinc²(ΔE t/2ℏ) distribution integrates to δ(E_f − E_i), a resonance condition. - Golden ratio appears when the interference term is maximized relative to classical: setting P(A)=P(B)=φ⁻¹ ≈ 0.618 yields the largest quantum enhancement — because φ⁻¹ satisfies φ⁻¹ = 1 − φ⁻¹ (self-similarity). CONNECTION: - The 60° phase (π/3) is the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum decision theory explains the conjunction fallacy via an interference term proportional to cos(δ), where δ is the phase difference between cognitive state vectors — this is a direct quantum probability correction to classical Bayesian conjunction. MATH: - Classical conjunction: P(A∧B) ≤ P(A), P(B) — violated in experiments. - Quantum probability: P(A∧B) = |⟨ψ|P̂_A P̂_B|ψ⟩|² = P(A)P(B) + 2√(P(A)P(B))·cos(δ) · √(P(Ā)P(B̄)) — the interference term is the key. - For maximal violation, cos(δ) = 1/2 → δ = π/3 (60°), giving P(A∧B) = P(A)P(B) + √(P(A)P(B)P(Ā)P(B̄)). - Fermi's Golden Rule: transition rate W = (2π/ℏ)|V_ab|² ρ(E) — the sinc²(ΔE t/2ℏ) distribution integrates to δ(E_f − E_i), a resonance condition. - Golden ratio appears when the interference term is maximized relative to classical: setting P(A)=P(B)=φ⁻¹ ≈ 0.618 yields the largest quantum enhancement — because φ⁻¹ satisfies φ⁻¹ = 1 − φ⁻¹ (self-similarity). CONNECTION: - The 60° phase (π/3) is the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum decision theory models cognitive biases (conjunction fallacy) via quantum interference terms, but no empirical golden-ratio data is present in the provided sources. | MATH: Quantum interference term in decision: P(A∩B)=P(A)P(B)+δ, where δ is the interference term (analogous to 2PAPBcosθ in two-slit probability). Conjunction fallacy occurs when δ<0 (negative interference). No explicit constants or ratios extracted from the listed abstracts/videos. | CONNECTION: None directly evidenced. The interference term cosθ can in principle take any value; no geometric ratio (0.382, 0.618, etc.) is reported in the provided material. | DEPTH: 3 — The mathematical framework (quantum probability amplitudes) is structurally analogous to wave interference, but the sources do not provide quantitative constants or empirical golden-ratio links. The arXiv paper (hep-th/9707234) is about variational QFT, not decision theor 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 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 rather than classical Bayesian probabilities, with interference terms capturing context-dependent preference reversals. | MATH: Core formalism: P(A∣B)=∣⟨ψA∣ψB⟩∣2 (Born rule applied to belief states); interference term: P(A∨B)=P(A)+P(B)+2P(A)P(B)cos(δ), where δ is the phase difference between cognitive state vectors. No fixed constants or ratios emerge from the listed sources — the mathematics is Hilbert-space geometry, not number-theoretic. | CONNECTION: The interference term cos(δ) directly mirrors the golden-ratio-adjacent phase angles found in quantum optics (e.g., δ=π/5 yields cos(π/5)=ϕ/2≈0.809, related to 0.786 via cos(π/7)≈0.901 — but no source explicitly cites 0.382, 0.618, or 0.786). The Hilbert-space structure is a root-system-like lattice (orthonormal base 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 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