ODTOE
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ODTOE Formulas

Complete mathematical apparatus of Observer theory

20 formulas

AxiomR = Ô(Ψ)
1

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.

BeliefB(O;C) = F^w₁ · E^w₂ · (1-σ)^w₃ · Λ^w₄
2

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.

DynamicsdB/dt = γ · tanh(β·d̄') · d̄ · B · (1-B)
3

Belief Dynamics

Rate of belief change follows logistic dynamics with saturation. Parameter γ is learning rate, β is sensitivity to contradictions.

Postulate|M| = K^N(t)
4

Infinity of Realities (P1)

Cardinality of reality set M grows exponentially with observer count N(t). K is reality branching coefficient.

Postulatev = α / (I(C) + ε)
5

Reconfiguration Speed (P2)

Reality reconfiguration speed is inversely proportional to configuration inertia I(C). ε is minimum threshold, α is coupling constant.

PostulateT(C) = T₀ / (1-S)^n
6

Configuration Lifetime (P3)

Configuration existence time grows as power law when coherence S approaches unity. T₀ is base time, n is stability exponent.

PostulateP(E|B) = B^k
7

Belief and Probability (P4)

Event E probability given belief B follows power law. Exponent k characterizes nonlinearity of belief-realization connection.

PostulateP_coll = 1 - ∏ᵢ(1 - Bᵢᵏ)
8

Collective Observation (P5)

Collective event realization probability is determined by product of individual observer contributions.

PostulateN_theories = N₀(1-S)^m + 1
9

Number of Theories (P6)

Number of valid theories decreases with coherence growth. At S→1 only one theory remains — fully coherent reality.

DynamicsdC/dt = -α·∇U(C)/(I(C)+ε) + η(t)
10

Reconfiguration Dynamics

Configuration evolution is determined by potential gradient U, inertia I and stochastic noise η(t) with variance D₀(1-S).

CollectiveS = 1 - (1/n²) Σᵢ<ⱼ |Bᵢ - Bⱼ|
11

Cluster Coherence

Observer group coherence is measured through pairwise belief differences. Fewer divergences mean higher S.

Collective|M_eff| ≤ K^(N·(1-S))
12

Reality Intersection

Effective reality count is limited by coherence. At S→1 all observers converge to one shared reality.

StructureΨ* = Φ(Ψ*)
13

Fixed Point

Self-consistent configuration Ψ* is a fixed point of mapping Φ. This is a stable equilibrium state of reality.

StructureΦ = ι ∘ Ô
14

Self-Observation Mapping

Composition of observation operator Ô and inclusion ι gives mapping Φ, generating strange loop of self-reference.

Quantumℏ = h/(2π)
15

Observation Grain per Loop Turn

Reduced Planck constant ℏ defines minimum action quantum per one strange observation loop turn.

DynamicsI(C) = ∫ ρ(x)·|∇C|² dx
16

Configuration Inertia

Configuration inertia is defined by density integral over configuration gradient. Higher gradient means more resistance to changes.

BeliefΛ = ⟨Ψ_obs|Ψ_env⟩
17

Observer-Environment Resonance

Resonance Λ is measured by scalar product of observer and environment wave functions.

QuantumE_obs = ℏω·(n + 1/2)
18

Observer Energy

Observer energy is quantized like harmonic oscillator. n is attention state quantum number.

Dynamics∂S/∂t + ∇·(S·v) = σ_S
19

Coherence Continuity Equation

Coherence obeys conservation law with source σ_S. Coherence can be created and destroyed locally.

StructureR_manifest = lim(t→∞) Ô^t(Ψ₀)
20

Reality Manifestation

Manifested reality is the limit of repeated observation operator application to initial state.