Read the equation
For discrete random variables, H(item)−H(item|state) equals mutual information and is nonnegative when conditional entropy is averaged over the joint distribution. A single realized item can have negative pointwise information gain, so the phrase at each tick is stronger than that theorem. A reproducible per-update ΔK needs the probability model, averaging convention, and estimator.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| Delta_K | Claimed knowledge increment |
| H | Discrete Shannon entropy |
How it works
Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.
Assumptions & limits
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Original context
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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 ACHIEVESExact original notation
At each tick, ΔK = H(item) - H(item|state) ≥ 0 by definition of conditional entropy.
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Sources & provenance
At each tick, ΔK = H(item) - H(item|state) ≥ 0 by definition of conditional entropy.
At each tick, ΔK = H(item) - H(item|state) ≥ 0 by definition of conditional entropy.
At each tick, ΔK = H(item) - H(item|state) ≥ 0 by definition of conditional entropy.
At each tick, ΔK = H(item) - H(item|state) ≥ 0 by definition of conditional entropy.