Action Quadratic Limit Quadratic Limit Status

The quadratic limit status

The declaration quadraticLimit_status is a machine-readable status report. It is a string that names the key results in the Recognition Science library's quadratic limit section and certifies that they were checked with zero unproved axioms. The string reads: "Action.QuadraticLimit: Jcost_taylor_quadratic, kineticAction, newton_second_law (0 sorry, 0 axiom)". In plain terms, it says that the library contains a proof that Newton's second law emerges from a cost functional in the small-strain limit, and that this proof is axiom-clean.

The underlying mathematics is classical. The cost functional J(γ) = ½(γ + γ⁻¹) - 1 has a quadratic Taylor expansion near γ = 1. For small strain ε, where γ = 1 + ε, the theorem Jcost_taylor_quadratic proves the bound |J(1 + ε) - ε²/2| ≤ ε²/10 whenever |ε| ≤ 1/10. This means the cost functional is well approximated by the standard kinetic energy ½ε². The library then defines the kinetic action as the integral of ½ε², and proves that the Euler-Lagrange equation for the standard Lagrangian L = ½m q̇² - V(q) is exactly Newton's second law m q̈ = -V'(q). The bridge theorem actionJ_to_kinetic_bridge shows that the J-action differs from the kinetic action by at most 10% in the small-strain regime.

In Recognition Science, this is the bridge from the cost-functional formulation to standard mechanics. The framework's library proves that the J-action becomes the standard Lagrangian action in the small-strain limit, and its Euler-Lagrange equation becomes Newton's second law. The status string is the library's own summary of this achievement.

The status string does not claim that the quadratic limit is exact, nor that it applies beyond the stated small-strain bound. It does not claim that Newton's second law is derived from first principles without the standard Lagrangian; the theorem newton_second_law is a definitional consequence of the chosen Lagrangian. The status string also does not claim that the identification of the cost functional with kinetic energy is a physical derivation; it is a mathematical approximation result.

def quadraticLimit_status : String :=
  "Action.QuadraticLimit: Jcost_taylor_quadratic, kineticAction, newton_second_law (0 sorry, 0 axiom)"

/-- **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

/-- **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

/-- **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)

What this page does not claim

The quadratic limit is not an exact equality but an approximation with a 10% relative error bound. Newton's second law is not derived from the cost functional alone; it is a consequence of the standard Lagrangian definition. The status string does not claim that the cost functional is physically identified with kinetic energy beyond the small-strain regime.

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)

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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