Hurst Exponent in ODTOE
Definition
ODTOE links the Hurst exponent H to global coherence by H(S) = (1+S)/2: at S=0 (full decoherence) H=1/2 reproduces classical Brownian motion, and at S=1 (full coherence) H=1 gives ballistic determinism. This makes coherence S = α − 1 directly measurable via mean-square displacement, rendering all ODTOE predictions experimentally testable.
Formula
Related Terms
Cognitive Coherence B(O,C)
Cognitive coherence B(O,C) is the central measurable quantity of ODTOE that determines how strongly an observer O actualizes configuration C. It is computed as B(O,C) = F^w1 · E^w2 · (1−σ)^w3 · Λ^w4, where F is attention focus (fidelity), E is energy, σ is internal contradiction (entropy), and Λ is data quality.
φ-Resonance (Golden Ratio)
φ-resonance is the role of the golden ratio φ = 1.618… as the unique stable resonance frequency in ODTOE, selected from the potentiality field by the KAM theorem because φ is the «most irrational» number. φ is the fixed point of the self-referential map f(x) = 1 + 1/x and appears in fundamental constants, nested φ-tori and recursive structure of reality.
Source Articles
Brownian Motion as a Manifestation of Observational Architecture: Hurst Exponent, Coherence, and the Golden Ratio
Proposes interpretation of Brownian motion as manifestation of observational architecture within ODTOE. Establishes relation between Hurst exponent H and coherence S: H(S)=(1+S)/2. Formula reproduces two experimental limits: at S=0 (complete decoherence) H=1/2—classical Brownian motion; at S=1 (complete coherence) H=1—ballistic determinism. Scaling factor between observation levels equals φᴴ, where φ is golden ratio. Sixth role of spiral gap (π−3)² identified: governs stochasticity-drift transition. Numerical verification on synthetic trajectories shows 0.55% mean error.
Coherence as a Measurable Quantity: Three Consequences of the Hurst Exponent — S Parameter Relation for the ODTOE Formalism
Establishes relation between Hurst exponent and ODTOE coherence: H=(1+S)/2 implies S=α−1 where α is anomalous-diffusion exponent. Three consequences: (1) Coherence becomes independently measurable via mean-square displacement, rendering all ODTOE predictions experimentally testable. (2) Planck constant depends on diffusion exponent: h∝(2−α)^(−1/2), predicting deviation in highly coherent systems (BEC, superconductors). (3) Parameter r governs drift-to-noise ratio, quantifying arrow of time with critical dimensionality d_crit≈8.12 (metagalactic level). All formulas verified to 50 decimal places.