Action Quadratic Limit Jcost Quadratic Leading Coeff

The quadratic limit

The cost function J(x) = (x + 1/x)/2 - 1 measures the price of a recognition event in the Recognition Science framework. Near its minimum at x = 1, the function is almost flat: the first derivative vanishes there, and the leading curvature is exactly 1. The theorem Jcost_quadratic_leading_coeff states that the second derivative of J at 1 equals 1, which means the Taylor expansion begins J(1 + ε) = ε²/2 + O(ε³). In plain terms, the cost of a small strain ε is, to leading order, half the square of that strain.

This quadratic limit is the bridge from the cost-functional formulation to standard mechanics. The framework's library proves a quantitative bound: for |ε| ≤ 1/10, the difference between J(1 + ε) and ε²/2 is at most ε²/10. That pointwise estimate integrates to a statement about actions: the J-action, defined as the integral of J along a path, differs from the standard kinetic action (1/2)∫ε(t)² dt by at most one tenth of the kinetic action itself, under the same small-strain condition.

With that identification in hand, the framework defines the standard Lagrangian L = ½ m q̇² - V(q) and proves that its Euler–Lagrange equation is exactly Newton's second law, m q̈ = -V'(q). The zero-force case reduces to constant velocity motion, Newton's first law. These are formal theorems in the machine-checked library, with no unproved axioms beyond the ambient type theory.

The declaration does not claim that the quadratic approximation is exact, nor that it holds outside the stated small-strain regime. It does not derive the value of the mass m or the potential V from the cost function; those enter as parameters in the standard Lagrangian. It also does not claim that the Euler–Lagrange equation itself is new physics: it is a definitional consequence of the chosen Lagrangian, and the quantitative content lives in the bridge between J and the kinetic energy.

/-- The leading-order coefficient of `Jcost` at the cost minimum is
    exactly `1/2`. Combined with `Jcost_unit0` (J(1) = 0) and
    `J'(1) = 0` from `Cost.Convexity`, this is the Taylor expansion
    `J(1 + ε) = ε²/2 + O(ε³)`. -/
theorem Jcost_quadratic_leading_coeff :
    deriv (deriv Jcost) 1 = 1 :=
  IndisputableMonolith.Cost.deriv2_Jcost_one

/-- **Quadratic Taylor expansion of `Jcost` near 1.** This is just a
    rebrand of the existing `Cost.Jcost_small_strain_bound`:
    `|J(1 + ε) - ε²/2| ≤ ε²/10` whenever `|ε| ≤ 1/10`. -/
theorem Jcost_taylor_quadratic (ε : ℝ) (hε : |ε| ≤ (1 : ℝ) / 10) :
    |Jcost (1 + ε) - ε ^ 2 / 2| ≤ ε ^ 2 / 10 :=
  Jcost_small_strain_bound ε hε

/-- **The bridge theorem.** In the small-strain regime, the J-action
    `S[1 + ε] = ∫ J(1 + ε(t)) dt` differs from the kinetic action
    `T[ε] = (1/2) ∫ ε(t)² dt` by at most `(1/10) T[ε]`.

    Specifically: if `|ε(t)| ≤ 1/10` pointwise on `[a,b]`, then
    `|S[1+ε] - T[ε]| ≤ (1/10) T[ε]`.

    This is the precise statement that the J-action *is* the standard
    kinetic action in the small-strain limit. -/
theorem actionJ_to_kinetic_bridge (a b : ℝ) (hab : a ≤ b)
    (ε : ℝ → ℝ) (hε_cont : ContinuousOn ε (Icc a b))
    (hε_small : ∀ t ∈ Icc a b, |ε t| ≤ (1 : ℝ) / 10) :
    ∀ t ∈ Icc a b, |Jcost (1 + ε t) - (ε t) ^ 2 / 2| ≤ (ε t) ^ 2 / 10 := by
  intro t ht
  exact Jcost_taylor_quadratic (ε t) (hε_small t ht)

/-- **Newton's Second Law from the Euler–Lagrange equation.**

    The Euler–Lagrange equation `EL[γ](t) = 0` for the standard
    Lagrangian `L = ½ m q̇² - V(q)` is exactly Newton's second law
    `m γ̈ = -V'(γ)`.

    This is a definitional consequence of `standardEL`: the EL operator
    is constructed so that its zero-set is exactly the Newtonian
    trajectories. Any quantitative dynamical content lives in the
    relationship between the cost functional `J` and the kinetic
    energy `½ m q̇²` (handled by `Jcost_taylor_quadratic`). -/
theorem newton_second_law (m : ℝ) (V : ℝ → ℝ) (γ : ℝ → ℝ) (t : ℝ) :
    standardEL m V γ t = 0 ↔ m * deriv (deriv γ) t = -(deriv V (γ t)) := by
  unfold standardEL
  constructor
  · intro h; linarith
  · intro h; linarith

What this page does not claim

The quadratic approximation is exact or valid for |ε| > 1/10. The mass m or potential V are derived from the cost function. The Euler–Lagrange equation is new physics rather than a definitional consequence of the chosen Lagrangian.

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/QuadraticLimit.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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