Action Path Space Fixed Endpoints Trans

The transitivity of fixed endpoints

In the calculus of variations, a path's endpoints are its boundary conditions: the fixed values that a candidate curve must take at the start and end of its interval. The declaration fixedEndpoints_trans records a simple structural fact about these conditions. If path γ₁ shares its endpoints with path γ₂, and γ₂ shares its endpoints with γ₃, then γ₁ shares its endpoints with γ₃. This is the transitivity of the relation "has the same endpoints as".

The declaration lives in the framework's machine-checked library of formal theorems, within a module that sets up the variational stage for the principle of least action. The framework models the action of a path as the integral of its cost function J over the path's interval. The relevant paths are continuous and strictly positive on the interval, and the action is non-negative. The constant path at value 1, where the cost function J reaches its minimum, has action zero.

The transitivity lemma is a building block for the framework's variational argument. It ensures that the relation of sharing endpoints behaves as an equivalence relation, which is a prerequisite for comparing paths that compete in the least-action principle. The lemma is proved directly from the definition of fixed endpoints: two paths share endpoints exactly when their values at the interval's start and end are equal. Transitivity of equality then gives transitivity of the relation.

In Recognition Science, this declaration does not claim that a minimizing path exists, nor that any particular path is a geodesic. It only establishes the algebraic property of the endpoint relation. The framework's library also records that convex interpolation between two admissible paths preserves shared endpoints, and that the action of any admissible path is non-negative. These are separate facts; the transitivity lemma is the one that concerns the relation itself.

lemma fixedEndpoints_trans {a b : ℝ} {γ₁ γ₂ γ₃ : AdmissiblePath a b}
    (h₁ : fixedEndpoints γ₁ γ₂) (h₂ : fixedEndpoints γ₂ γ₃) :
    fixedEndpoints γ₁ γ₃ := ⟨h₁.1.trans h₂.1, h₁.2.trans h₂.2⟩

/-- The J-action functional `S[γ] = ∫_a^b J(γ(t)) dt`.

    This is the central object of the variational principle. Geodesics of
    the Hessian metric `g(x) = J''(x) = 1/x³` minimize this functional
    among admissible paths with fixed endpoints. -/
noncomputable def actionJ {a b : ℝ} (γ : AdmissiblePath a b) : ℝ :=
  ∫ t in a..b, Jcost (γ.toFun t)

/-- The action of the constant path at `1` (the cost-minimum) vanishes. -/
lemma actionJ_const_one {a b : ℝ} :
    actionJ (AdmissiblePath.const a b 1 one_pos) = 0 := by
  unfold actionJ
  simp [AdmissiblePath.const_apply, Jcost_unit0]

/-- Interpolation preserves shared endpoints. -/
lemma interp_fixedEndpoints {a b : ℝ} {γ₁ γ₂ : AdmissiblePath a b}
    (h : fixedEndpoints γ₁ γ₂) (s : ℝ) (hs : s ∈ Icc (0:ℝ) 1) :
    fixedEndpoints γ₁ (interp γ₁ γ₂ s hs) := by
  refine ⟨?_, ?_⟩
  · simp [interp_apply, h.1]; ring
  · simp [interp_apply, h.2]; ring

What this page does not claim

The declaration does not prove that a minimizing path exists for any given boundary conditions. The declaration does not claim that any particular path is a geodesic of the Hessian metric. The declaration does not establish that the action functional is strictly convex.

Verify this page

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

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