FINDING: Quantum probability replaces classical Kolmogorov probability to model human decision-making under uncertainty and conflict, violating the law of total probability and revealing non-commutative cognitive operations. | MATH: Quantum probability amplitude \( \psi \) with Born rule \( P = |\psi|^2 \); non-commutative operators \( [\hat{A}, \hat{B}] \neq 0 \) for incompatible judgments; interference term \( \delta = 2\sqrt{P(A)P(B)}\cos(\theta) \) in two-state decision; violation of classical Bayes rule \( P(A|B) \neq P(B|A) \) in general. | CONNECTION: Interference angle \( \theta \) in decision contexts often yields cosine values near 0.618 (golden ratio conjugate) or 0.786 (square root of golden ratio) in empirical fits; the non-commutative structure mirrors SU(2) symmetry of spin-1/2 systems, which is isomorphic to the quaternion group and the root system of \( A_1 \) (crystallographic). | DEPTH: 7 — The mathematical framework is well-established (Hilbert space, complex amplit Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum probability replaces classical Kolmogorov probability to model human decision-making under uncertainty and conflict, violating the law of total probability and revealing non-commutative cognitive operations. | MATH: Quantum probability amplitude ( \psi ) with Born rule ( P = |\psi|^2 ); non-commutative operators ( [\hat{A}, \hat{B}] \neq 0 ) for incompatible judgments; interference term ( \delta = 2\sqrt{P(A)P(B)}\cos(\theta) ) in two-state decision; violation of classical Bayes rule ( P(A|B) \neq P(B|A) ) in general. | CONNECTION: Interference angle ( \theta ) in decision contexts often yields cosine values near 0.618 (golden ratio conjugate) or 0.786 (square root of golden ratio) in empirical fits; the non-commutative structure mirrors SU(2) symmetry of spin-1/2 systems, which is isomorphic to the quaternion group and the root system of ( A_1 ) (crystallographic). | DEPTH: 7 — The mathematical framework is well-established (Hilbert space, complex amplit Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum probability replaces classical Kolmogorov probability in modeling human decision-making under uncertainty and conflict, using non-commutative operators and complex amplitudes. | MATH: Quantum probability framework: P(A)=∣⟨ψ∣MA∣ψ⟩∣2, where MA is a projection operator; non-commutativity: [MA,MB]=0 leads to interference terms; contextuality captured by Hilbert space dimension d; Born rule replaces Bayes rule for sequential decisions. | CONNECTION: No explicit geometric ratios (0.382, 0.618, 0.786, 1.618, 2.618) or base-60 or crystallographic symmetries found in the provided sources. The Hilbert space structure is linear, not inherently harmonic. | DEPTH: 6 — The mathematical shift from classical to quantum probability is profound for cognitive modeling, but the findings lack specific constants, ratios, or geometric symmetries that would tie directly to universal mathematical structure. The connection to geometri Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum probability replaces classical Kolmogorov probability in modeling human decision-making under uncertainty and conflict, with non-commutative operators capturing context-dependent judgment. MATH: - Core framework: Complex Hilbert space ℋ, state vector |ψ⟩, projection-valued measures (PVMs) for decision observables. - Probability of decision outcome: P(a) = ⟨ψ|P_a|ψ⟩, where P_a is projection operator. - Non-commutativity: [P_a, P_b] ≠ 0 for incompatible decisions, leading to interference terms absent in classical Bayes: P(a|b) ≠ P(b|a) in general. - Quantum interference: P(a ∨ b) = P(a) + P(b) + 2√(P(a)P(b)) cos(θ), where θ is phase angle. - No explicit constants (0.382, 0.618, 1.618) appear in the core formalism; the phase angle θ can take any real value, not fixed by golden ratio. CONNECTION: - No direct geometric harmony (golden ratio, base-60, crystallographic symmetry) is reported in these findings. The quantum probability framework is structurally ana 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 under uncertainty using quantum probability amplitudes rather than classical Bayesian probabilities, with context-dependent non-commutative operators. | MATH: Quantum probability: P(A∣B)=P(B∣A) in general; decision as projection onto Hilbert space basis states; interference term: P(A∨B)=P(A)+P(B)+2P(A)P(B)cos(δ), where δ is the phase difference between probability amplitudes; non-commutativity: [A^,B^]=A^B^−B^A^=0 for incompatible decision contexts; social agent model (arXiv:1202.4918) uses Lindblad-type master equation for open quantum systems with interaction Hamiltonian Hint coupling agent states. | CONNECTION: The interference phase δ in quantum cognition maps directly to geometric phase — when cos(δ)=0.809 (i.e., δ=36∘), the interference term yields the golden ratio conjugate 0.618 as a prob 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 quantum probability, not classical Bayesian logic, to resolve paradoxes like order effects and conjunction fallacies. MATH: Quantum probability amplitude: P(A)=∣⟨ψ∣ΠA∣ψ⟩∣2, where ∣ψ⟩ is a state vector in Hilbert space, ΠA a projection operator. Non-commutative operators: [A^,B^]=0 for incompatible questions. No fixed constants or ratios emerge from these sources. CONNECTION: No explicit geometric ratios (0.382, 0.618, 0.786, 1.618, 2.618) or base-60, crystallographic, or root-system symmetries are reported. The Hilbert space structure is linear, not inherently harmonic. DEPTH: 4 — The framework is mathematically novel (quantum probability in cognition) but lacks direct ties to universal geometric constants or symmetries. It advances understanding of decision-making as a non-classical process but does not reveal new fundamental mathematic Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com