Action Path Space Action J Const One

The zero-action path

In the calculus of variations, an action is a number assigned to a path, typically an integral of a function along it. The Recognition Science framework defines a J-action functional on admissible paths: continuous, strictly positive functions on a closed interval. The functional integrates the cost function J along the path. The constant path at value 1, where the cost function J achieves its minimum, has J-action exactly zero.

The declaration actionJ_const_one is a lemma in the machine-checked library of formal theorems. It states that the action of the constant path at 1 is zero. This is a direct consequence of the cost function's unit normalization: J(1) = 0. The proof is a simple computation, but the statement is structurally significant. It provides the baseline against which all other paths are measured in the variational principle.

In Recognition Science, the J-cost function is forced by five plain conditions, and its minimum occurs at the value 1. The zero-action constant path is the reference point: any other admissible path with the same endpoints has non-negative action, and the constant path at 1 is the unique minimizer. This is the foundation for the principle of least action derived from the J-functional, where geodesics of the Hessian metric g(x) = J''(x) = 1/x³ minimize the action among admissible paths with fixed endpoints.

The lemma does not claim that the constant path at 1 is the only path with zero action, nor does it establish the existence or uniqueness of minimizers for arbitrary boundary conditions. It is a single computational fact, not a variational theorem. The framework's library records it as a building block for the convexity argument that selects the minimizer, but the full variational principle is a separate result.

/-- 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]

/-- 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)

/-- An admissible path on `[a, b]` is a continuous, strictly positive function. -/
structure AdmissiblePath (a b : ℝ) where
  /-- The underlying function. -/
  toFun : ℝ → ℝ
  /-- Continuity on the closed interval. -/
  cont : ContinuousOn toFun (Icc a b)
  /-- Strict positivity on the closed interval. -/
  pos : ∀ t ∈ Icc a b, 0 < toFun t

What this page does not claim

The lemma does not establish existence or uniqueness of minimizers for arbitrary boundary conditions. The lemma does not claim the constant path at 1 is the only path with zero action. The lemma is a computational fact, not a variational theorem.

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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