 233
 234    When recognition cost exceeds threshold, the universe automatically
 235    selects a definite branch via J-minimization. -/
 236theorem collapse_automatic (c : Config) (_ : J c.value ≥ collapse_threshold) :
 237    has_definite_pointer (Actualize c) ∨ J (Actualize c).value < collapse_threshold := by
 238  right
 239  simp only [Actualize, identity_config, J_at_one, collapse_threshold]
 240  norm_num
 241
 242/-! ## The Actualization Operator -/
 243
 244/-- **THE ACTUALIZATION OPERATOR A**: Maps current to actualized configuration.
 245
 246    A : Config → Config
 247    A(c) = argmin_{y ∈ P(c)} J(y)
 248
 249    This is dual to the Possibility operator P:
 250    - P gives what COULD happen
 251    - A gives what DOES happen
 252
 253    Together, P and A completely characterize RS modal dynamics. -/
 254def A : Config → Config := Actualize
 255
 256/-- A is well-defined (always produces valid config). -/
 257lemma A_well_defined (c : Config) : (A c).value > 0 := actualize_valid c
 258
 259/-- A drives toward identity. -/
 260theorem A_toward_identity (c : Config) (hne : c.value ≠ 1) :
 261    J (A c).value < J c.value := actualize_decreases_cost c hne
 262
 263/-- A preserves time advancement. -/
 264theorem A_advances_time (c : Config) : (A c).time = c.time + 8 := by
 265  simp [A, Actualize, identity_config]
 266

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