ODTOE Formulas
Complete mathematical apparatus of Observer theory
20 formulas
ODTOE Axiom
Reality R is the result of applying observation operator Ô to potentiality field Ψ. The Observer is not a passive spectator but an active participant in reality formation.
Contextual Belief
Belief B depends on attention focus F, energy E, internal coherence (1-σ) and resonance Λ. Weights wᵢ determine each factor's contribution.
Belief Dynamics
Rate of belief change follows logistic dynamics with saturation. Parameter γ is learning rate, β is sensitivity to contradictions.
Infinity of Realities (P1)
Cardinality of reality set M grows exponentially with observer count N(t). K is reality branching coefficient.
Reconfiguration Speed (P2)
Reality reconfiguration speed is inversely proportional to configuration inertia I(C). ε is minimum threshold, α is coupling constant.
Configuration Lifetime (P3)
Configuration existence time grows as power law when coherence S approaches unity. T₀ is base time, n is stability exponent.
Belief and Probability (P4)
Event E probability given belief B follows power law. Exponent k characterizes nonlinearity of belief-realization connection.
Collective Observation (P5)
Collective event realization probability is determined by product of individual observer contributions.
Number of Theories (P6)
Number of valid theories decreases with coherence growth. At S→1 only one theory remains — fully coherent reality.
Reconfiguration Dynamics
Configuration evolution is determined by potential gradient U, inertia I and stochastic noise η(t) with variance D₀(1-S).
Cluster Coherence
Observer group coherence is measured through pairwise belief differences. Fewer divergences mean higher S.
Reality Intersection
Effective reality count is limited by coherence. At S→1 all observers converge to one shared reality.
Fixed Point
Self-consistent configuration Ψ* is a fixed point of mapping Φ. This is a stable equilibrium state of reality.
Self-Observation Mapping
Composition of observation operator Ô and inclusion ι gives mapping Φ, generating strange loop of self-reference.
Observation Grain per Loop Turn
Reduced Planck constant ℏ defines minimum action quantum per one strange observation loop turn.
Configuration Inertia
Configuration inertia is defined by density integral over configuration gradient. Higher gradient means more resistance to changes.
Observer-Environment Resonance
Resonance Λ is measured by scalar product of observer and environment wave functions.
Observer Energy
Observer energy is quantized like harmonic oscillator. n is attention state quantum number.
Coherence Continuity Equation
Coherence obeys conservation law with source σ_S. Coherence can be created and destroyed locally.
Reality Manifestation
Manifested reality is the limit of repeated observation operator application to initial state.