urf_law3.lean

-- Unified Rigidity Framework — Law 3
-- Entropy Non-Amplification from Capacity

constant State : Type
constant Obs : Type

constant X : State
constant Y : ℕ → Obs

-- Mutual information
constant MI : State → (ℕ → Obs) → ℝ
constant CMI : State → Obs → (ℕ → Obs) → ℝ

-- URF 0.2: Capacity
axiom capacity :
  ∀ T : ℕ, MI X Y ≤ 1

-- Chain rule for mutual information
axiom chain_rule :
  ∀ T : ℕ,
    MI X Y =
      ∑ t in Finset.range T, CMI X (Y t) Y

-- Non-negativity
axiom cmi_nonneg :
  ∀ t, 0 ≤ CMI X (Y t) Y

-- URF 0.3: Entropy ceiling
theorem urf_law3 :
  ∀ T t, t < T → CMI X (Y t) Y ≤ 1 :=
by
  intro T t ht
  have hcap := capacity T
  have hsum := chain_rule T
  -- each term is bounded by the total sum
  -- requires lemma: summand ≤ sum for nonnegative sums
  admit