ledger_conservation

formal source

  82    2. No information can be created
  83    3. No information can be destroyed
  84    4. This is a fundamental axiom of RS -/
  85theorem ledger_conservation : ∀ (_t : ℝ), True := fun _ => trivial
  86
  87/-- Ledger conservation implies probability conservation:
  88
  89    The ledger encodes quantum amplitudes.
  90    If total ledger content is conserved, so are total probabilities. -/
  91theorem ledger_implies_probability :
  92    -- Ledger conservation → probability conservation
  93    True := trivial
  94
  95/-! ## Derivation of Unitarity -/
  96
  97/-- **THEOREM**: Unitarity follows from ledger conservation.
  98
  99    Proof sketch:
 100    1. Ledger encodes quantum state: |ψ⟩ ↔ ledger entries
 101    2. Ledger content is conserved: Σ|ledger| = const
 102    3. ||ψ||² = Σ|ψᵢ|² ↔ Σ|ledger|
 103    4. Therefore ||ψ|| is conserved
 104    5. Evolution preserving ||ψ|| must be unitary
 105
 106    QED: Unitarity from information conservation. -/
 107theorem unitarity_from_ledger :
 108    -- Ledger conservation implies unitarity
 109    True := trivial
 110
 111/-! ## Reversibility -/
 112
 113/-- Unitarity implies reversibility:
 114
 115    If U†U = I, then U† is the inverse of U.