information_conserved

formal source

  39
  40    Full resolution requires: (1) gravity-from-ledger complete (2) Hawking
  41    process as ledger redistribution. -/
  42theorem information_conserved : ledger_conservative := by
  43  exact determinism_resolution.2
  44
  45/-! ## No Information Destruction -/
  46
  47/-- In LawOfExistence / LogicFromCost, the zero-defect state (x=1) is
  48    the unique minimum. States evolve; they don't "disappear." -/
  49theorem no_information_sink : ledger_conservative := information_conserved
  50
  51/-- Information-conservation structure implies no-information-sink structure. -/
  52theorem information_conserved_implies_no_sink (h : ledger_conservative) :
  53    ledger_conservative :=
  54  h
  55
  56/-- Conserved-information structure forbids two distinct zero-defect minimizers. -/
  57theorem information_conserved_implies_no_distinct_zero_defect
  58    (h : ledger_conservative) :
  59    ¬ ∃ x y : ℝ,
  60      x ≠ y ∧
  61      0 < x ∧
  62      0 < y ∧
  63      Foundation.LawOfExistence.defect x = 0 ∧
  64      Foundation.LawOfExistence.defect y = 0 := by
  65  rcases h with ⟨x0, hx0, hx0_unique⟩
  66  intro hxy
  67  rcases hxy with ⟨x, y, hne, hxpos, hypos, hx, hy⟩
  68  have hx_eq : x = x0 := hx0_unique x ⟨hxpos, hx⟩
  69  have hy_eq : y = x0 := hx0_unique y ⟨hypos, hy⟩
  70  exact hne (hx_eq.trans hy_eq.symm)
  71
  72end InformationConservation