Encyclopedia Foundation Foundation Polynomiality From Logic Iterate Continuous On Range

ARTICLE 3 claims 3 theorems

Foundation Polynomiality From Logic Iterate Continuous On Range

A theorem about combining rules shows that repeated comparison stays inside its own range, and it does so without ever breaking continuity.

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Continuous iteration

A combining rule is a function that takes two numbers and returns a third, like addition or multiplication. In the Recognition Science framework, the rule that combines two comparisons is written Phi, and the set of values it can produce is called its range. The theorem iterate_continuous_on_range states a closure property: if the rule is closed under iteration on its range, then any finite number of repeated applications of the rule to elements of that range still lands inside the range, and each step of that iteration is continuous in its inputs.

Continuity here means a small change in the starting values produces only a small change in the result. The theorem proves that this continuity survives iteration. If you start with a value v in the range and apply the rule to itself n times, the map from v to that n-fold iterate is continuous on the range, and the result of each iterate also stays in the range. This is a structural fact about how comparisons of comparisons compose consistently, and it is fully proved in the framework's machine-checked library of formal theorems.

The theorem does not claim that the combining rule becomes a polynomial, nor that it is analytic. An earlier version of the module attempted to prove that closure under iteration forces real-analyticity, but a counterexample showed this false: the rule Φ(a,b) = 2a + 2b + 12√(ab) is closed on [0,∞) yet is not real-analytic at the origin. The corrected module keeps only the structural consequences that are actually proved: diagonal continuity on the range and continuous iterates on the range.

THEOREM iterate_continuous_on_range · IndisputableMonolith/Foundation/PolynomialityFromLogic.lean
iterate_continuous_on_range · IndisputableMonolith/Foundation/PolynomialityFromLogic.lean:114 
/-- **Iteration produces continuous orbits.**  If we iterate Φ on a starting
element v ∈ Range(F), the n-fold iterate is again in Range(F), and the map
v ↦ Φ^[n](v, v) is continuous on Range(F). -/
theorem iterate_continuous_on_range
    (F : ℝ → ℝ) (Phi : ℝ → ℝ → ℝ)
    (hClosed : IteratedClosureOnRange F Phi)
    (n : ℕ) :
    ∃ φₙ : ℝ → ℝ,
      ContinuousOn φₙ (Set.image F (Set.Ioi 0)) ∧
      (∀ v ∈ Set.image F (Set.Ioi 0), φₙ v ∈ Set.image F (Set.Ioi 0)) := by
  -- Define the iterate by recursion on n.  Inductively, each iterate is
  -- a continuous map from Range(F) into Range(F).
  induction n with
  | zero =>
    refine ⟨id, ?_, ?_⟩
    · exact continuousOn_id
    · intro v hv
      exact hv
  | succ k ih =>
    obtain ⟨φₖ, hCont_φₖ, hMap_φₖ⟩ := ih
    refine ⟨fun v => Phi (φₖ v) v, ?_, ?_⟩
    · -- Continuity of v ↦ Phi(φₖ v, v) via ContinuousOn.comp.
      obtain ⟨hCont_Phi, _⟩ := hClosed
      have h_pair_on : ContinuousOn (fun w : ℝ => ((φₖ w, w) : ℝ × ℝ))
          (Set.image F (Set.Ioi 0)) :=
        hCont_φₖ.prodMk continuousOn_id
      have h_maps : Set.MapsTo (fun w : ℝ => ((φₖ w, w) : ℝ × ℝ))
          (Set.image F (Set.Ioi 0))
          ((Set.image F (Set.Ioi 0)) ×ˢ (Set.image F (Set.Ioi 0))) := by
        intro w hw
        exact ⟨hMap_φₖ w hw, hw⟩
      have h_phi_on : ContinuousOn (fun p : ℝ × ℝ => Phi p.1 p.2)
          ((Set.image F (Set.Ioi 0)) ×ˢ (Set.image F (Set.Ioi 0))) := hCont_Phi
      have h_comp : ContinuousOn
          ((fun p : ℝ × ℝ => Phi p.1 p.2) ∘
            (fun w : ℝ => ((φₖ w, w) : ℝ × ℝ)))
          (Set.image F (Set.Ioi 0)) :=
        h_phi_on.comp h_pair_on h_maps
      have h_eq : ((fun p : ℝ × ℝ => Phi p.1 p.2) ∘
                    (fun w : ℝ => ((φₖ w, w) : ℝ × ℝ)))
                  = (fun v : ℝ => Phi (φₖ v) v) := by
        funext w
        rfl
      rw [h_eq] at h_comp
      exact h_comp
    · intro v hv
      obtain ⟨_, hClosure⟩ := hClosed
      exact hClosure (φₖ v) v (hMap_φₖ v hv) hv
THEOREM iterate_continuous_on_range · IndisputableMonolith/Foundation/PolynomialityFromLogic.lean
iterate_continuous_on_range · IndisputableMonolith/Foundation/PolynomialityFromLogic.lean:114 
/-- **Iteration produces continuous orbits.**  If we iterate Φ on a starting
element v ∈ Range(F), the n-fold iterate is again in Range(F), and the map
v ↦ Φ^[n](v, v) is continuous on Range(F). -/
theorem iterate_continuous_on_range
    (F : ℝ → ℝ) (Phi : ℝ → ℝ → ℝ)
    (hClosed : IteratedClosureOnRange F Phi)
    (n : ℕ) :
    ∃ φₙ : ℝ → ℝ,
      ContinuousOn φₙ (Set.image F (Set.Ioi 0)) ∧
      (∀ v ∈ Set.image F (Set.Ioi 0), φₙ v ∈ Set.image F (Set.Ioi 0)) := by
  -- Define the iterate by recursion on n.  Inductively, each iterate is
  -- a continuous map from Range(F) into Range(F).
  induction n with
  | zero =>
    refine ⟨id, ?_, ?_⟩
    · exact continuousOn_id
    · intro v hv
      exact hv
  | succ k ih =>
    obtain ⟨φₖ, hCont_φₖ, hMap_φₖ⟩ := ih
    refine ⟨fun v => Phi (φₖ v) v, ?_, ?_⟩
    · -- Continuity of v ↦ Phi(φₖ v, v) via ContinuousOn.comp.
      obtain ⟨hCont_Phi, _⟩ := hClosed
      have h_pair_on : ContinuousOn (fun w : ℝ => ((φₖ w, w) : ℝ × ℝ))
          (Set.image F (Set.Ioi 0)) :=
        hCont_φₖ.prodMk continuousOn_id
      have h_maps : Set.MapsTo (fun w : ℝ => ((φₖ w, w) : ℝ × ℝ))
          (Set.image F (Set.Ioi 0))
          ((Set.image F (Set.Ioi 0)) ×ˢ (Set.image F (Set.Ioi 0))) := by
        intro w hw
        exact ⟨hMap_φₖ w hw, hw⟩
      have h_phi_on : ContinuousOn (fun p : ℝ × ℝ => Phi p.1 p.2)
          ((Set.image F (Set.Ioi 0)) ×ˢ (Set.image F (Set.Ioi 0))) := hCont_Phi
      have h_comp : ContinuousOn
          ((fun p : ℝ × ℝ => Phi p.1 p.2) ∘
            (fun w : ℝ => ((φₖ w, w) : ℝ × ℝ)))
          (Set.image F (Set.Ioi 0)) :=
        h_phi_on.comp h_pair_on h_maps
      have h_eq : ((fun p : ℝ × ℝ => Phi p.1 p.2) ∘
                    (fun w : ℝ => ((φₖ w, w) : ℝ × ℝ)))
                  = (fun v : ℝ => Phi (φₖ v) v) := by
        funext w
        rfl
      rw [h_eq] at h_comp
      exact h_comp
    · intro v hv
      obtain ⟨_, hClosure⟩ := hClosed
      exact hClosure (φₖ v) v (hMap_φₖ v hv) hv
THEOREM iterate_continuous_on_range · IndisputableMonolith/Foundation/PolynomialityFromLogic.lean
iterate_continuous_on_range · IndisputableMonolith/Foundation/PolynomialityFromLogic.lean:114 
/-- **Iteration produces continuous orbits.**  If we iterate Φ on a starting
element v ∈ Range(F), the n-fold iterate is again in Range(F), and the map
v ↦ Φ^[n](v, v) is continuous on Range(F). -/
theorem iterate_continuous_on_range
    (F : ℝ → ℝ) (Phi : ℝ → ℝ → ℝ)
    (hClosed : IteratedClosureOnRange F Phi)
    (n : ℕ) :
    ∃ φₙ : ℝ → ℝ,
      ContinuousOn φₙ (Set.image F (Set.Ioi 0)) ∧
      (∀ v ∈ Set.image F (Set.Ioi 0), φₙ v ∈ Set.image F (Set.Ioi 0)) := by
  -- Define the iterate by recursion on n.  Inductively, each iterate is
  -- a continuous map from Range(F) into Range(F).
  induction n with
  | zero =>
    refine ⟨id, ?_, ?_⟩
    · exact continuousOn_id
    · intro v hv
      exact hv
  | succ k ih =>
    obtain ⟨φₖ, hCont_φₖ, hMap_φₖ⟩ := ih
    refine ⟨fun v => Phi (φₖ v) v, ?_, ?_⟩
    · -- Continuity of v ↦ Phi(φₖ v, v) via ContinuousOn.comp.
      obtain ⟨hCont_Phi, _⟩ := hClosed
      have h_pair_on : ContinuousOn (fun w : ℝ => ((φₖ w, w) : ℝ × ℝ))
          (Set.image F (Set.Ioi 0)) :=
        hCont_φₖ.prodMk continuousOn_id
      have h_maps : Set.MapsTo (fun w : ℝ => ((φₖ w, w) : ℝ × ℝ))
          (Set.image F (Set.Ioi 0))
          ((Set.image F (Set.Ioi 0)) ×ˢ (Set.image F (Set.Ioi 0))) := by
        intro w hw
        exact ⟨hMap_φₖ w hw, hw⟩
      have h_phi_on : ContinuousOn (fun p : ℝ × ℝ => Phi p.1 p.2)
          ((Set.image F (Set.Ioi 0)) ×ˢ (Set.image F (Set.Ioi 0))) := hCont_Phi
      have h_comp : ContinuousOn
          ((fun p : ℝ × ℝ => Phi p.1 p.2) ∘
            (fun w : ℝ => ((φₖ w, w) : ℝ × ℝ)))
          (Set.image F (Set.Ioi 0)) :=
        h_phi_on.comp h_pair_on h_maps
      have h_eq : ((fun p : ℝ × ℝ => Phi p.1 p.2) ∘
                    (fun w : ℝ => ((φₖ w, w) : ℝ × ℝ)))
                  = (fun v : ℝ => Phi (φₖ v) v) := by
        funext w
        rfl
      rw [h_eq] at h_comp
      exact h_comp
    · intro v hv
      obtain ⟨_, hClosure⟩ := hClosed
      exact hClosure (φₖ v) v (hMap_φₖ v hv) hv

What this page does not claim

The combining rule is not proved to be a polynomial or analytic. The theorem does not establish the full polynomiality conjecture, which remains open. Continuity on the range does not extend to continuity at the origin for the counterexample rule.

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Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/PolynomialityFromLogic.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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