Action Path Space Interp Zero

The path-space interpolation

In the calculus of variations, a path is a function that assigns a value to each moment between a start and an end time. A path space is the collection of all such functions that meet some condition, such as being continuous or staying positive. The Recognition Science framework's ledger, a discrete record of recognition events, gives rise to a cost function J(x) = (x + 1/x)/2 - 1, and the framework builds a variational principle on top of it: the J-action of a path is the integral of J along the path, and paths that minimize this action are the framework's geodesics.

To study these minimizing paths, the framework's machine-checked library of formal theorems defines a straight-line interpolation between two admissible paths. Given two continuous, strictly positive functions γ₁ and γ₂ on an interval, the interpolation at a parameter s is the convex combination (1 - s)·γ₁ + s·γ₂. The declaration interp_zero records the boundary case: when s = 0, the interpolation is exactly the first path, γ₁, at every time t. This is the path-space analogue of a line segment starting at its first endpoint. The companion declaration interp_one records the symmetric fact that at s = 1, the interpolation is the second path.

The structural content of interp_zero is a definitional identity: it follows directly from the definition of the interpolation and the arithmetic of 0 and 1. Its role in the framework is to anchor the convexity argument that shows the J-action functional is strictly convex along such interpolations, which is what guarantees a unique minimizer among paths with fixed endpoints. The declaration also participates in the proof that interpolation preserves shared endpoints: if two paths agree at the start and end of the interval, so does every interpolated path between them.

What interp_zero does not claim is any physical content by itself. It does not assert that any particular path is a geodesic, that a minimizer exists, or that the framework's variational principle matches any measured phenomenon. Those are separate theorems or empirical checks elsewhere in the library. The declaration is a piece of infrastructure: it states one exact fact about the geometry of the path space, and the framework's other results build on it.

/-- Interpolation at `s = 0` is the first path. -/
lemma interp_zero {a b : ℝ} (γ₁ γ₂ : AdmissiblePath a b) :
    ∀ t, (interp γ₁ γ₂ 0 ⟨le_refl 0, by norm_num⟩).toFun t = γ₁.toFun t := by
  intro t; simp [interp_apply]

/-- 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]

/-- 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

It does not claim that any particular path minimizes the J-action, only that the interpolation at s = 0 equals the first path. It does not claim the J-action functional is convex; that is a separate theorem in the library. It does not claim any physical content, such as agreement with measured trajectories.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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