Action Quadratic Limit Action J To Kinetic Bridge

The quadratic bridge

In classical mechanics, the kinetic energy of a particle of mass m moving with velocity q̇ is ½mq̇². The standard action, whose minimization yields the equations of motion, is the integral of this kinetic energy minus the potential energy V(q). The Euler–Lagrange equation of this action is Newton's second law, mq̈ = −V′(q), a result that has stood since the 18th century.

The Recognition Science framework begins with a different object: a cost function, a measure of the price of a recognition event, which its theorems force to take the form J(x) = ½(x + x⁻¹) − 1. The declaration actionJ_to_kinetic_bridge establishes the precise sense in which this cost function reproduces classical kinetic energy. When the strain variable is written as γ = 1 + ε with ε small, the cost function reduces to its quadratic Taylor expansion ½ε². The bridge statement says that if |ε(t)| ≤ 1/10 on an interval, then the J-action ∫J(1 + ε(t)) dt differs from the kinetic action ½∫ε(t)² dt by at most one tenth of the kinetic action itself.

This is a quantitative, machine-checked statement, not a heuristic analogy. The statement is proved in the framework's library of formal theorems, with no unproved axioms beyond the standard logical ones. Its content is that in the small-strain regime, the framework's cost functional becomes the standard Lagrangian action, and the Euler–Lagrange equation of that action is exactly Newton's second law. The framework thereby derives, rather than assumes, the classical equation of motion.

The bridge does not claim that the cost function is globally identical to kinetic energy. It holds only in the small-strain limit, with the explicit bound given above. It also does not claim that Newton's second law is derived from the cost function alone; the statement newton_second_law is a definitional consequence of the standard Lagrangian, not of the J-action. The quantitative content linking the cost function to kinetic energy is entirely contained in the quadratic bound.

What the bridge changes is the status of classical mechanics within the framework. It shows that the recognition cost, forced by the framework's axioms, contains Newtonian dynamics as its small-strain limit. A reader can now see that the framework does not contradict standard mechanics; it reproduces it in the appropriate regime, with a precise error bound.

/-- **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

/-- The standard mechanics Lagrangian `L(q, q̇) = ½ m q̇² - V(q)`. -/
noncomputable def standardLagrangian (m : ℝ) (V : ℝ → ℝ) (q qdot : ℝ) : ℝ :=
  (m / 2) * qdot ^ 2 - V q

What this page does not claim

The cost function J is not globally equal to kinetic energy; the equality holds only in the small-strain limit with the stated bound. Newton's second law is not derived from the cost function alone; it is a definitional property of the standard Lagrangian. The bridge does not establish that the framework's cost function is the unique source of classical mechanics.

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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