Action Path Space Interp One

The interpolation endpoint

In the calculus of variations, a path space is the set of all curves between two fixed points, and a common tool is the straight-line interpolation between two such curves: interpolation (a weighted blend of two functions). For two paths γ₁ and γ₂, the blend at parameter s is (1−s)γ₁ + sγ₂. The declaration interp_one states the simplest endpoint fact about this blend: at s = 1, the interpolation is exactly γ₂, the second path. It is a definitional identity, a reflexivity of the construction, not an existence theorem.

The setting is the Recognition Science framework, which models physical law as the outcome of a forced recognition cost. Within that framework, the path space carries a variational principle: the J-action functional S[γ] = ∫ J(γ(t)) dt, where J(x) = (x + 1/x)/2 − 1 is the unique cost function forced by five plain conditions. The admissible paths are continuous, strictly positive functions on a closed interval, and the interpolation preserves those properties for any s in [0,1]. The endpoint lemma interp_one is the boundary case: it confirms that the family of blends connects the first path to the second path exactly, with no gap at the end.

What this fact does not claim is more interesting than what it does. It does not assert that the interpolated path minimizes the action, nor that any minimizer exists. It does not say that the blend is geodesic, or that the action functional is convex along it (that is a separate lemma, Action.FunctionalConvexity, which uses the closure under interpolation as a hypothesis). The lemma is silent on whether γ₂ is a critical point, a local minimum, or even a reasonable physical trajectory. It is a bookkeeping fact about the parameterization of path space, not a statement about the physics of the action.

The payoff is structural. Because the interpolation is closed under convex combination, the path space is a convex set in the space of functions, and the endpoint lemma pins down the boundary behavior needed for a strict-convexity argument. Without interp_one, the family of blends would be a collection of curves with no guaranteed connection to its own endpoints, and the variational argument would lack a base case. The lemma is the hinge that lets the framework's least-action principle swing from a definition to a theorem.

/-- Interpolation at `s = 1` is the second path. -/
lemma interp_one {a b : ℝ} (γ₁ γ₂ : AdmissiblePath a b) :
    ∀ t, (interp γ₁ γ₂ 1 ⟨by norm_num, le_refl 1⟩).toFun t = γ₂.toFun t := by
  intro t; simp [interp_apply]

/-- The straight-line interpolation between two admissible paths.

    `interp γ₁ γ₂ s = (1 - s) · γ₁ + s · γ₂`.

    The key structural fact is that for `s ∈ [0,1]`, this convex combination
    is again strictly positive and continuous, hence again admissible. -/
def interp {a b : ℝ} (γ₁ γ₂ : AdmissiblePath a b) (s : ℝ)
    (hs : s ∈ Icc (0:ℝ) 1) : AdmissiblePath a b where
  toFun := fun t => (1 - s) * γ₁.toFun t + s * γ₂.toFun t
  cont := by
    have h1 : ContinuousOn (fun t => (1 - s) * γ₁.toFun t) (Icc a b) :=
      γ₁.cont.const_smul (1 - s) |>.congr (fun _ _ => by simp [smul_eq_mul])
    have h2 : ContinuousOn (fun t => s * γ₂.toFun t) (Icc a b) :=
      γ₂.cont.const_smul s |>.congr (fun _ _ => by simp [smul_eq_mul])
    exact h1.add h2
  pos := by
    intro t ht
    have h1s : 0 ≤ 1 - s := by linarith [hs.2]
    have hs' : 0 ≤ s := hs.1
    have hp1 : 0 < γ₁.toFun t := γ₁.pos t ht
    have hp2 : 0 < γ₂.toFun t := γ₂.pos t ht
    -- Either s = 0 (LHS pure γ₁), or s > 0 (RHS strictly positive). Either way > 0.
    rcases lt_or_eq_of_le hs' with hs_pos | hs_zero
    · have := mul_pos hs_pos hp2
      have hnn : 0 ≤ (1 - s) * γ₁.toFun t := mul_nonneg h1s hp1.le
      linarith
    · -- s = 0: the combination is 1 · γ₁ + 0 · γ₂ = γ₁
      simp [← hs_zero, hp1]

/-- 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)

What this page does not claim

The interpolated path minimizes the J-action. The path space is complete or compact. The lemma proves any property of the action functional along the interpolation.

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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