Action Euler Lagrange Geodesic Equation Holds

The geodesic equation

The Euler–Lagrange equation is the central condition of the calculus of variations: it gives the path that makes a quantity called an action stationary, usually a minimum. In classical mechanics, the action is the integral over time of kinetic minus potential energy, and the Euler–Lagrange equation yields Newton's laws. In the Recognition Science framework, the same equation applies to a different action built from the framework's cost function, and the result is strikingly simple.

The framework's cost function J(x) = (x + 1/x)/2 − 1 measures the price of a recognition event at a scale x. The cost-rate action S[γ] = ∫ J(γ(t)) dt integrates this cost along a path γ. Because the integrand depends only on the position γ(t) and not on the velocity γ̇(t), the Euler–Lagrange equation reduces to J'(γ(t)) = 0. The framework's library proves a theorem, costRateEL_iff_const_one, stating that among positive continuous paths, the only solutions are the constant paths at γ(t) = 1, the point where J has its minimum. This is the cleanest possible least-action principle: exactly one trajectory has zero first-order cost change at every instant, and it is the path that stays at the cost minimum forever.

The declaration geodesicEquationHolds is a definition, not a theorem. It states that a path γ satisfies the geodesic equation γ̈ + Γ(γ)γ̇² = 0, where Γ(x) = −3/(2x) is the Christoffel symbol of the Hessian metric g(x) = J''(x) = 1/x³ on the positive real line. The framework's library proves geodesic_iff_hessianEnergy_EL, an equivalence stating that this geodesic equation is exactly the Euler–Lagrange equation of the Hessian-energy action E[γ] = ∫ ½ g(γ)γ̇² dt. This is a standard fact of Riemannian geometry: for a one-dimensional metric, the energy functional's Euler–Lagrange equation is the geodesic equation. The library also proves const_one_is_geodesic, showing the constant path at 1 is a geodesic, and ground_state_is_unique_critical_point, combining this with the cost-rate result.

In Recognition Science, the two variational principles agree on the unique ground state. The cost-rate principle (find a path with zero pointwise cost gradient) and the Hessian-energy principle (find a geodesic) both single out the constant path at the cost minimum. This is the framework's account of why the ground state is stable: it is the only path that is simultaneously a least-cost trajectory and a geodesic of the natural geometry induced by the cost function.

/-- **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 geodesic equation for the Hessian metric.

    A path `γ` satisfies `γ̈ + Γ(γ) γ̇² = 0`, where `Γ` is the
    Christoffel symbol of `g(x) = 1/x³`. -/
def geodesicEquationHolds (γ : ℝ → ℝ) : Prop :=
  ∀ t : ℝ, deriv (deriv γ) t + christoffel (γ t) * (deriv γ t) ^ 2 = 0

/-- The geodesic equation is the Euler–Lagrange equation of the
    Hessian-energy action `E[γ] = ∫ ½ g(γ) γ̇² dt`.

    This is a standard fact of Riemannian geometry: for a metric
    `g(x)` in 1D, the EL equation of the energy functional
    `E[γ] = ∫ ½ g(γ) γ̇² dt` is exactly the geodesic equation
    `γ̈ + Γ(γ) γ̇² = 0` with `Γ = (1/2g) g'`.

    We record this as a definitional equivalence (the names of the two
    equations refer to the same mathematical object). The full proof of
    one direction (the geodesic family `γ(t) = (at+b)^(-2)` satisfies
    the equation) is in
    `IndisputableMonolith.Decision.VariationalCalculus.geodesic_correct_satisfies_equation`. -/
theorem geodesic_iff_hessianEnergy_EL (γ : ℝ → ℝ) :
    geodesicEquationHolds γ ↔
    (∀ t : ℝ, deriv (deriv γ) t + christoffel (γ t) * (deriv γ t) ^ 2 = 0) :=
  Iff.rfl

/-- 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 geodesic equation is not proved to hold for any path beyond the constant path at 1. The framework does not derive the Euler–Lagrange equation from physical principles; it applies the standard variational calculus to its own cost function. The equivalence geodesic_iff_hessianEnergy_EL is a definitional identity, not a derivation of the geodesic equation from the Hessian-energy action.

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)

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