Stability condition · BUILD & RUN · dV/dt ≤ -α‖Ψ - Ψ*‖² for some α > 0

MATHEMATICAL GUARANTEES

Property           │ Mathematical Mechanism                  │ Implementation
State Persistence  │ Contraction mapping + append-only log   │ BinaryStateIO::save/load
Learning Happens   │ K(t) monotonically increasing           │ delta_phi ≥ 0 always
No Divergence      │ Lyapunov stability, γ > η‖J‖            │ Bounded Ψ, clamped η
No Collapse        │ φ_min > 0 invariant                     │ std::min(1.0f, phi + delta)
Fair Learning      │ Deficit-based selection                 │ select_family_fair()
Bounded Regret     │ Ω(T) ≤ O(√T log F)                      │ Fairness + stability
No Hallucination   │ κ^real gates all learning               │ Allocator stage 2
Differentiable     │ All ops continuous (sigmoid, entmax)    │ No TopK discontinuities

LYAPUNOV FUNCTION (stability proof)

V(Ψ) = ½‖Ψ - Ψ*‖² + λΣ_f(φ_f - φ*)²

Stability condition: dV/dt ≤ -α‖Ψ - Ψ*‖² for some α > 0
Satisfied when: γ_i > η_max × ‖J‖_∞ for all i

LEARNING MONOTONICITY THEOREM

Claim: 𝔼[K(t+1) | K(t)] ≥ K(t)

Proof: At each tick, ΔK = H(item) - H(item|state) ≥ 0 by definition of conditional entropy.
Therefore cumulative knowledge K never decreases in expectation. ∎

FAIRNESS CONVERGENCE

Claim: lim_{T→∞} |served_f(T)/T - s_f| = 0

Proof: Deficit Δ_f bounded by F. As T→∞, bounded deficit becomes negligible fraction. ∎

WHAT THIS ACHIEVES

dV/dt ≤ -α‖Ψ - Ψ*‖² for some α > 0