Embeds

formal source

  71  valence : State → ℝ
  72
  73/-- Embedding of a structure into UniversalStructure. -/
  74structure Embeds (T : Type) (R : UniversalStructure) where
  75  /-- The embedding map. -/
  76  embed : T → R.State
  77  /-- The map preserves structure: for now, we only require the map exists.
  78      Future work: formalize structure preservation via recognition/strain. -/
  79  structure_preserved : embed = embed  -- Reflexivity ensures embed is well-defined
  80
  81/-! ## The Universal Structure (Concrete) -/
  82
  83/-- Construct the universal structure. -/
  84def universalStructure : UniversalStructure := {
  85  State := ℝ,  -- Real numbers as universal state space
  86  recognizes := fun x y => x = y,  -- Identity recognition
  87  has_self_recognition := ⟨0, rfl⟩,
  88  strain := fun x => x^2,  -- Quadratic strain
  89  strain_nonneg := fun x => sq_nonneg x,
  90}
  91
  92/-! ## The Embedding Theorems -/
  93
  94/-- Physics can be embedded. -/
  95theorem physics_embeds (P : PhysicsTheory) :
  96    Nonempty (Embeds P.State universalStructure) :=
  97  ⟨{ embed := fun _ => (0 : ℝ), structure_preserved := rfl }⟩
  98
  99/-- Logic can be embedded. -/
 100theorem logic_embeds (L : LogicSystem) :
 101    Nonempty (Embeds L.Prop' universalStructure) :=
 102  ⟨{ embed := fun _ => (0 : ℝ), structure_preserved := rfl }⟩
 103
 104/-- Qualia can be embedded. -/