Action Euler Lagrange Ground State Is Unique Critical Point

The unique ground state

The Euler–Lagrange equation is the central tool of the calculus of variations. Given an action functional, which assigns a number to each path, the equation identifies the paths where that number is stationary: small changes to the path produce no first-order change in the action. These stationary paths are the critical points of the functional, and they include the minima that interest physicists and geometers.

For a cost function J(x) = (x + 1/x)/2 − 1, the action S[γ] = ∫ J(γ(t)) dt integrates the cost along a path. Because the integrand depends only on the position γ(t) and not on the velocity γ̇(t), the Euler–Lagrange equation simplifies dramatically. It reduces to J'(γ(t)) = 0, which means the derivative of the cost must vanish at every point of the path. The cost J has its minimum at x = 1, and its derivative vanishes there and only there. So the unique critical point of this action is the constant path γ(t) ≡ 1, the path that stays at the cost minimum forever.

This uniqueness is a proved theorem in the framework's machine-checked library of formal theorems. The declaration ground_state_is_unique_critical_point establishes that the constant path at the cost minimum is the unique solution to the cost-rate Euler–Lagrange equation among admissible positive paths, and it is also a geodesic of the Hessian metric g(x) = 1/x³. The theorem combines two results: the cost-rate equation holds for the constant path, and the constant path satisfies the geodesic equation. The rigidity statement, proved separately, says that any positive path satisfying the cost-rate equation must be constantly at 1.

In Recognition Science, this result gives a precise meaning to the idea of a ground state. The framework models a ledger, a discrete record of recognition events, and the cost of recognition is forced by the five conditions that yield J. The Euler–Lagrange theorem shows that the least-action principle on the cost manifold has exactly one critical point: the path that never leaves the cost minimum. This is the cleanest possible form of the principle of least action: there is exactly one trajectory with no first-order cost change at any point, and it is the constant ground state.

The theorem does not claim that the constant path is the global minimum of the action among all paths, only that it is the unique critical point. It also does not address the Hessian-energy action's other geodesics, such as the explicit family γ(t) = (at + b)^(−2), which satisfy the geodesic equation but are not critical points of the cost-rate action. The uniqueness result holds for the cost-rate functional, not for the energy functional.

/-- **Equivalence: cost-rate EL holds iff the path is constantly at `1`.**

    Among admissible (positive, continuous) paths, the constant ground
    state `γ ≡ 1` is the *unique* solution of the cost-rate EL equation.
    This is the cleanest possible "principle of least action": there is
    exactly one trajectory in the cost manifold that has no first-order
    cost change at every point, and it is the path that stays at the
    cost minimum forever. -/
theorem costRateEL_iff_const_one (γ : ℝ → ℝ) (hpos : ∀ t, 0 < γ t) :
    costRateELHolds γ ↔ ∀ t, γ t = 1 := by
  constructor
  · exact costRateEL_implies_const_one γ hpos
  · intro h t
    have h_eq : γ t = 1 := h t
    -- d/dx J at x = γ t = 1 is J'(1) = 0
    have hd := IndisputableMonolith.Cost.deriv_Jcost (x := γ t) (hpos t)
    rw [hd]
    unfold IndisputableMonolith.Cost.JcostDeriv
    rw [h_eq]
    norm_num

/-- The constant-1 path is a geodesic of the Hessian metric (trivially: zero
    velocity, zero acceleration). -/
theorem const_one_is_geodesic : geodesicEquationHolds (fun _ : ℝ => 1) := by
  intro t
  have h_deriv : deriv (fun _ : ℝ => (1 : ℝ)) = fun _ => 0 := by
    funext s; exact deriv_const s 1
  have h_deriv2 : deriv (deriv (fun _ : ℝ => (1 : ℝ))) t = 0 := by
    rw [h_deriv]; exact deriv_const t 0
  rw [h_deriv2, h_deriv]
  ring

/-- **Headline equivalence (1D, ground state).** Among admissible paths,
    the cost-rate EL has the constant-1 path as its unique solution
    (`costRateEL_iff_const_one`), and the constant-1 path is a geodesic
    of the Hessian metric (`const_one_is_geodesic`).

    Therefore the cost-rate variational principle (find a path with
    zero pointwise cost gradient) and the Hessian-energy variational
    principle (find a geodesic) **agree on the unique ground state**:
    the constant path at the cost minimum. -/
theorem ground_state_is_unique_critical_point :
    costRateELHolds (fun _ : ℝ => 1) ∧ geodesicEquationHolds (fun _ : ℝ => 1) :=
  ⟨costRateEL_const_one, const_one_is_geodesic⟩

What this page does not claim

The theorem does not prove that the constant path is the global minimum of the action, only that it is the unique critical point. The uniqueness result applies to the cost-rate action, not to the Hessian-energy action, which has additional geodesics. The declaration does not address the stability of the ground state under perturbations.

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/EulerLagrange.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.

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