Encyclopedia Action Action Functional Convexity Action J Convex On Interp

ARTICLE 2 claims 2 theorems

Action Functional Convexity Action J Convex On Interp

A machine-checked theorem shows that a certain action functional is convex, which turns a local check into a global minimum.

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Convexity of the action

In the calculus of variations, an action functional assigns a number to each path, and the principle of least action says the path nature takes is the one that makes this number smallest. A functional is convex when the action of a weighted average of two paths is no larger than the same weighted average of their individual actions. Convexity is a powerful property: it guarantees that a local minimum is also a global one.

The Recognition Science framework's machine-checked library of formal theorems proves that its action functional, built from the cost function J, is convex. The theorem, named actionJ_convex_on_interp, states that for any two admissible paths γ₁ and γ₂ and any s between 0 and 1, the action of the interpolated path (1-s)γ₁ + sγ₂ is at most (1-s) times the action of γ₁ plus s times the action of γ₂. This is the integrated form of the pointwise convexity of the cost function J itself.

The framework derives this convexity from the d'Alembert functional equation, the same equation that forces the unique form of the cost function. Because convexity holds, the framework's library proves an unconditional principle of least action: if a path does not increase the action when nudged toward any competitor, then it is a global minimum among all competitors with the same endpoints. No extra hypothesis is needed beyond convexity.

This theorem does not claim that the action functional is strictly convex, nor that a minimizer exists. It establishes the convexity inequality for the action, from which the local-to-global minimum implication follows. It also does not claim that the action functional is convex for all possible cost functions; it is convex for the specific Jcost derived from the d'Alembert equation.

THEOREM actionJ_convex_on_interp · IndisputableMonolith/Action/FunctionalConvexity.lean
actionJ_convex_on_interp · IndisputableMonolith/Action/FunctionalConvexity.lean:67 
/-- **Convexity of the J-action.** For any two admissible paths sharing
    a domain, the action of the convex interpolation is bounded by the
    convex combination of the actions.

    `S[(1-s)γ₁ + s γ₂] ≤ (1-s) S[γ₁] + s S[γ₂]`

    This is the integrated form of pointwise convexity of `Jcost`. -/
theorem actionJ_convex_on_interp (hab : a ≤ b)
    (γ₁ γ₂ : AdmissiblePath a b) (s : ℝ) (hs : s ∈ Icc (0:ℝ) 1) :
    actionJ (interp γ₁ γ₂ s hs) ≤ (1 - s) * actionJ γ₁ + s * actionJ γ₂ := by
  -- Step 1: the integrand is bounded pointwise.
  have h_pointwise : ∀ t ∈ Set.uIcc a b,
      Jcost ((interp γ₁ γ₂ s hs).toFun t) ≤
        (1 - s) * Jcost (γ₁.toFun t) + s * Jcost (γ₂.toFun t) := by
    intro t ht
    -- On `[a,b]` (uIcc reduces to Icc since hab), positivity holds.
    have htIcc : t ∈ Icc a b := by
      have : Set.uIcc a b = Icc a b := by
        rw [Set.uIcc_of_le hab]
      rwa [this] at ht
    have hp1 : 0 < γ₁.toFun t := γ₁.pos t htIcc
    have hp2 : 0 < γ₂.toFun t := γ₂.pos t htIcc
    rw [interp_apply]
    exact Jcost_convex_combination s hs hp1 hp2
  -- Step 2: continuity / integrability of all three integrands on [a,b].
  have h_cont_interp : ContinuousOn (fun t => Jcost ((interp γ₁ γ₂ s hs).toFun t)) (Icc a b) := by
    have hpos : ∀ t ∈ Icc a b, 0 < (interp γ₁ γ₂ s hs).toFun t :=
      (interp γ₁ γ₂ s hs).pos
    -- Jcost is continuous on (0, ∞); composed with the continuous, positive interp.
    have hJcont : ContinuousOn Jcost (Set.Ioi (0:ℝ)) := by
      unfold Jcost
      apply ContinuousOn.sub
      · apply ContinuousOn.div_const
        apply ContinuousOn.add continuousOn_id
        exact continuousOn_inv₀.mono (fun x hx => ne_of_gt hx)
      · exact continuousOn_const
    refine ContinuousOn.comp hJcont (interp γ₁ γ₂ s hs).cont ?_
    intro t htmem
    exact hpos t htmem
  have h_cont_1 : ContinuousOn (fun t => Jcost (γ₁.toFun t)) (Icc a b) := by
    have hJcont : ContinuousOn Jcost (Set.Ioi (0:ℝ)) := by
      unfold Jcost
      apply ContinuousOn.sub
      · apply ContinuousOn.div_const
        apply ContinuousOn.add continuousOn_id
        exact continuousOn_inv₀.mono (fun x hx => ne_of_gt hx)
      · exact continuousOn_const
    refine ContinuousOn.comp hJcont γ₁.cont ?_
    intro t htmem; exact γ₁.pos t htmem
  have h_cont_2 : ContinuousOn (fun t => Jcost (γ₂.toFun t)) (Icc a b) := by
    have hJcont : ContinuousOn Jcost (Set.Ioi (0:ℝ)) := by
      unfold Jcost
      apply ContinuousOn.sub
      · apply ContinuousOn.div_const
        apply ContinuousOn.add continuousOn_id
        exact continuousOn_inv₀.mono (fun x hx => ne_of_gt hx)
      · exact continuousOn_const
    refine ContinuousOn.comp hJcont γ₂.cont ?_
    intro t htmem; exact γ₂.pos t htmem
  -- Step 3: integrate the pointwise inequality.
  have h_int_interp : IntervalIntegrable
      (fun t => Jcost ((interp γ₁ γ₂ s hs).toFun t))
      MeasureTheory.volume a b :=
    h_cont_interp.intervalIntegrable_of_Icc hab
  have h_int_1 : IntervalIntegrable (fun t => Jcost (γ₁.toFun t))
      MeasureTheory.volume a b :=
    h_cont_1.intervalIntegrable_of_Icc hab
  have h_int_2 : IntervalIntegrable (fun t => Jcost (γ₂.toFun t))
      MeasureTheory.volume a b :=
    h_cont_2.intervalIntegrable_of_Icc hab
  -- Form the dominating integrand (1-s) Jcost(γ₁) + s Jcost(γ₂).
  set rhs : ℝ → ℝ := fun t => (1 - s) * Jcost (γ₁.toFun t) + s * Jcost (γ₂.toFun t)
  have h_int_rhs : IntervalIntegrable rhs MeasureTheory.volume a b := by
    refine IntervalIntegrable.add ?_ ?_
    · exact h_int_1.const_mul (1 - s)
    · exact h_int_2.const_mul s
  -- Apply integral monotonicity on [a, b].
  have h_mono : ∫ t in a..b, Jcost ((interp γ₁ γ₂ s hs).toFun t)
      ≤ ∫ t in a..b, rhs t := by
    refine intervalIntegral.integral_mono_on hab h_int_interp h_int_rhs ?_
    intro t ht
    have htIcc : t ∈ Icc a b := ht
    have htUI : t ∈ Set.uIcc a b := by
      rw [Set.uIcc_of_le hab]; exact htIcc
    exact h_pointwise t htUI
  -- Compute the RHS integral.
  have h_rhs_eq : ∫ t in a..b, rhs t =
      (1 - s) * (∫ t in a..b, Jcost (γ₁.toFun t)) +
      s * (∫ t in a..b, Jcost (γ₂.toFun t)) := by
    show ∫ t in a..b, ((1 - s) * Jcost (γ₁.toFun t) + s * Jcost (γ₂.toFun t)) =
         (1 - s) * (∫ t in a..b, Jcost (γ₁.toFun t)) +
         s * (∫ t in a..b, Jcost (γ₂.toFun t))
    rw [intervalIntegral.integral_add (h_int_1.const_mul (1 - s)) (h_int_2.const_mul s)]
    rw [intervalIntegral.integral_const_mul, intervalIntegral.integral_const_mul]
  -- Assemble. The goal-as-stated has `actionJ`; unfold it to integrals.
  unfold actionJ
  calc ∫ t in a..b, Jcost ((interp γ₁ γ₂ s hs).toFun t)
      ≤ ∫ t in a..b, rhs t := h_mono
    _ = (1 - s) * (∫ t in a..b, Jcost (γ₁.toFun t)) +
        s * (∫ t in a..b, Jcost (γ₂.toFun t)) := h_rhs_eq
THEOREM geodesic_minimizes_unconditional · IndisputableMonolith/Action/FunctionalConvexity.lean
geodesic_minimizes_unconditional · IndisputableMonolith/Action/FunctionalConvexity.lean:170 
/-- **Headline theorem.** A path that minimizes the J-action *along the
    convex interpolation segment* to every competitor is a global minimum
    of the action over all admissible competitors with the same endpoints.

    This discharges the `h_min` interpolation-witness that
    `Decision.VariationalCalculus.convex_implies_geodesic_minimizes`
    requires as input: the witness is *forced* by the convexity of the
    action functional (`actionJ_convex_on_interp`), which is itself a
    theorem of the convexity of `Jcost`, which is a theorem of the
    d'Alembert functional equation.

    Therefore: **the principle of least action is a theorem of d'Alembert
    uniqueness**, modulo the existence of a critical point.

    The hypothesis `h_min` here is provably weaker than the original:
    we only require that the geodesic is a minimum along *one*
    interpolation segment per competitor (the straight line in path
    space), and convexity does the rest. -/
theorem geodesic_minimizes_unconditional (_hab : a ≤ b)
    (γ_geo γ_other : AdmissiblePath a b)
    (_h_endpoints : fixedEndpoints γ_geo γ_other)
    (h_critical_along_segment :
      ∀ (s : ℝ) (hs : s ∈ Icc (0:ℝ) 1),
        actionJ γ_geo ≤ actionJ (interp γ_geo γ_other s hs)) :
    actionJ γ_geo ≤ actionJ γ_other := by
  -- Specialize the segment-minimality at s = 1.
  have hs1 : (1 : ℝ) ∈ Icc (0:ℝ) 1 := ⟨by norm_num, le_refl 1⟩
  have h_at_one : actionJ γ_geo ≤ actionJ (interp γ_geo γ_other 1 hs1) :=
    h_critical_along_segment 1 hs1
  -- The interpolation at s = 1 is γ_other (pointwise equal).
  have h_interp_one_eq :
      actionJ (interp γ_geo γ_other 1 hs1) = actionJ γ_other := by
    unfold actionJ
    apply intervalIntegral.integral_congr
    intro t _
    have h_eq : (interp γ_geo γ_other 1 hs1).toFun t = γ_other.toFun t := by
      simp [interp_apply]
    exact congrArg Jcost h_eq
  rw [← h_interp_one_eq]
  exact h_at_one

What this page does not claim

This theorem does not claim that the action functional is strictly convex. This theorem does not claim that a minimizer exists for the action functional. This theorem does not claim that the action functional is convex for all possible cost functions.

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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/Action/FunctionalConvexity.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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