SOUL CONSERVATION

Formal Conservation Theorem

Theorem (Soul Conservation):

Let ψ_S be a soul-field satisfying:

1. ψ_S is a solution to the soul-field equation (Klein-Gordon or generalization)

2. The soul-field Lagrangian has U(1) symmetry

3. ψ_S has finite energy and normalization

Then: The soul number N_S = ∫ ρ_S d³x is conserved: dN_S/dt = 0.

Proof:

1. By Noether's theorem, U(1) symmetry implies conserved current j^μ_S

2. The conserved charge is N_S = ∫ j⁰_S d³x

3. By the continuity equation ∂_μj^μ = 0:

(by Gauss's theorem, assuming j → 0 at infinity)

4. Therefore N_S is constant in time ∎

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