RANSOC Proof: Step by Step Visualization
1
Initial Setup
Given a set of planets P = {p₁, p₂, ..., pₙ} with initial masses and distances from query star s.
w(pᵢ, s) = mᵢ/d(pᵢ, s)
0 < α < 1 (learning rate)
2
Mass Conservation
Proof that total mass M remains constant after updates:
M = Σmᵢ (constant)
mₕᵢₜ_new = (1-α)mₕᵢₜ
mⱼ_new = (1 + α/(n-1))mⱼ for j ≠ hit
ΔM = -αmₕᵢₜ + α/(n-1) * (M - mₕᵢₜ) = 0
3
Mass Growth for Unselected Planets
After k iterations without selection, a planet's mass grows exponentially:
mᵢ(k) = mᵢ(0) * (1 + α/(n-1))ᵏ
4
Eventual Selection Guarantee
Any unselected planet will eventually be chosen because:
w(pᵢ, s) = mᵢ(k)/d(pᵢ, s)
As k → ∞, mᵢ(k) → ∞
Therefore w(pᵢ, s) will exceed any finite weight