phi_code_distance_pos

formal source

 116  Jcost Foundation.PhiForcing.φ
 117
 118/-- The φ-code distance is positive. -/
 119theorem phi_code_distance_pos : 0 < phi_code_distance := by
 120  unfold phi_code_distance
 121  apply Jcost_pos_of_ne_one
 122  · exact Foundation.PhiForcing.phi_pos
 123  · exact (Foundation.PhiForcing.phi_gt_one).ne'
 124
 125/-! ## Logical Operators -/
 126
 127/-- A logical operator is an operation that preserves the code structure.
 128    In RS, these correspond to recognition-preserving transformations. -/
 129structure LogicalOperator (X : Ω → ℝ) where
 130  /-- The operator as a function -/
 131  op : Ω → Ω
 132  /-- The operator preserves cost structure -/
 133  preserves_cost : ∀ ω, Jcost (X (op ω)) = Jcost (X ω)
 134
 135/-- The identity is always a logical operator. -/
 136def id_logical_op (X : Ω → ℝ) : LogicalOperator X where
 137  op := id
 138  preserves_cost := fun _ => rfl
 139
 140/-! ## Connection to Physical Laws -/
 141
 142/-- Physical laws are "protected" observables that are stable under error correction.
 143    An observable O is protected if it commutes with the correction protocol. -/
 144def is_protected_observable {X : Ω → ℝ} (O : Ω → ℝ) (C : CorrectionProtocol X) : Prop :=
 145  ∀ ω, O (C.correct ω) = O ω ∨ Jcost (X ω) > 0
 146
 147/-- **Theorem**: Conservation laws are protected observables.
 148    Quantities that are conserved in the J=0 sector remain stable. -/
 149theorem conservation_is_protected {X : Ω → ℝ} (O : Ω → ℝ) (C : CorrectionProtocol X)