Appendix 31 · The Rigorous Derivation of the World‑Migration Equation dp/dt

世界迁移方程 dp/dt 的严格推导（英文版）

1. Introduction: Worlds Do Not “Change”—Probabilities Migrate

From previous appendices, we already have:

Φ = Σ_i A_i e^{iθ_i} Ω_i

World(t) = O(t) ∘ T(Φ)

But a crucial question remains: How does the experienced world shift over time among different possible worlds? This requires a dynamical equation for the probability distribution over worlds:

dp/dt

2. The World Probability Distribution p_i(t)

p_i(t) = Pr( experience lies in world Ω_i )

Symbol	Meaning
p_i(t)	Probability weight of world Ω_i at time t
Ω_i	Structural core of world i

Normalization:

p_i(t) ≥ 0,   Σ_i p_i(t) = 1

World‑migration = the evolution of p_i(t) in “world‑space.”

3. The Master Equation for dp_i/dt

dp_i/dt = Σ_j K_ij p_j(t)

Symbol	Meaning
dp_i/dt	Rate of change of probability for world Ω_i
K_ij	Transition rate from world Ω_j to Ω_i

Probability conservation requires:

Σ_i dp_i/dt = 0  ⇒  Σ_i K_ij = 0  (for all j)

4. Structure of the Transition Kernel K_ij

K_ij = Γ · C_ij · D_ij

Symbol	Meaning
Γ	Global migration scale (variability strength)
C_ij	Compatibility factor compatibility(O, Ω_i, Ω_j)
D_ij	Distance factor f(d(Ω_i, Ω_j))

Example distance factor:

D_ij = exp( -α · d(Ω_i, Ω_j) )

Symbol	Meaning
d(Ω_i, Ω_j)	Structural distance between worlds
α	Sensitivity to distance

5. Continuous Form: A Fokker–Planck‑Type Equation

If worlds are labeled by a continuous parameter λ:

∫ p(λ, t) dμ(λ) = 1

The migration equation becomes:

∂p(λ, t)/∂t
= -∇_λ · ( v(λ) p(λ, t) )
  + ∇_λ² ( D(λ) p(λ, t) )

Symbol	Meaning
p(λ, t)	Probability density at world‑parameter λ
v(λ)	Drift field (world‑attractor direction)
D(λ)	Diffusion coefficient (random wandering strength)
∇_λ, ∇_λ²	Gradient and Laplacian in world‑parameter space

Discrete form = jumps; continuous form = flow.

6. Coupling to the Observer Operator O

O(t) = O(ν_O(t), θ_O(t), A_O(t), 𝕀(t))

The transition kernel depends on O(t):

K_ij(t) = K_ij( O(t), Ω_i, Ω_j )

The master equation becomes:

dp_i/dt = Σ_j K_ij( O(t), Ω_i, Ω_j ) p_j(t)

Symbol	Meaning
ν_O(t)	Observer frequency
θ_O(t)	Observer phase
A_O(t)	Awareness amplitude
𝕀(t)	Interpenetration tensor (collective structure)

As O evolves, the set of “resonant worlds” and “reachable paths” changes.

7. World Attractors and Steady‑State Distributions p_i*

If O is approximately stable over some time scale:

dp_i/dt = 0  ⇒  Σ_j K_ij p_j* = 0

Symbol	Meaning
p_i*	Steady‑state world distribution under fixed O

This means: For a relatively stable observer structure, there exists a “world‑attractor distribution” around which the experienced world tends to hover.

8. Unified View with the Universe Equation

0 = 1 + T(Φ)

Φ = Σ_i A_i e^{iθ_i} Ω_i

W = T(Φ)

World(t) = O(t) ∘ T(Φ)

p_i(t) = Pr( experience in world Ω_i )

dp_i/dt = Σ_j K_ij( O(t), Ω_i, Ω_j ) p_j(t)

The Universe Equation provides Φ and T; the observer operator O provides appearance; the world‑migration equation provides the temporal dynamics of appearance.

9. Final Unified Summary

The world‑migration equation dp/dt does not describe “the universe changing.” It describes how, given Φ and T, the evolving observer operator O(t) causes the probability distribution p_i(t) to flow, drift, and migrate through world‑space, eventually forming steady or quasi‑steady attractor distributions.