Action Quadratic Limit Newton Second Law

The quadratic limit

Newton's second law, F = ma, is usually taught as a starting point. In the Recognition Science framework, it appears as a consequence of a more basic principle: the cost of recognition. The cost, a measure of how much a recognition event deviates from a perfect match, is forced by the framework's axioms to take a specific form. Near a perfect match, this cost function behaves like a simple quadratic, and it is this quadratic behavior that gives rise to the familiar law of motion.

The framework defines a cost function J(γ) = ½(γ + γ⁻¹) - 1. When the strain γ is close to 1, meaning the system is nearly at rest, the cost is approximately ½ε², where ε is the small deviation. This is the quadratic limit. The framework's library proves that for small strains, the action built from this cost function is nearly identical to the standard kinetic action, ½∫ε(t)² dt. The difference is bounded by a small fraction of the kinetic action itself.

From this bridge, the library constructs the standard Lagrangian L = ½mq̇² - V(q), where m is mass, q is position, and V is potential energy. The Euler-Lagrange equation, which describes the path of least action, is then shown to be exactly Newton's second law: mq̈ = -V'(q). This is a theorem in the framework's machine-checked library of formal theorems. The framework also derives Newton's first law as a special case: when the potential is constant, meaning no force acts, the acceleration is zero and velocity is constant.

In Recognition Science, this is a derivation, not a new physical discovery. The framework does not claim to have found a new law of motion. It shows that the standard law follows from its cost principle in the appropriate limit. The quantitative content of mechanics, such as the exact relationship between the cost function and kinetic energy, lives in the quadratic approximation theorem. The framework's contribution is to show that the structure of classical mechanics, at least in this small-strain regime, is not an independent postulate but a consequence of its foundational cost function.

What the declaration does not claim is equally important. It does not claim to derive the full content of classical mechanics, only the second law for a single particle in a potential. It does not claim that the cost function is a physical energy, only that it behaves like one in the limit. And it does not claim that the framework's axioms are the only way to derive Newton's law; it shows a particular path from its own principles to a familiar result.

/-- **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

/-- **Inertia (Newton's First Law).** When the potential is constant
    (`V' ≡ 0`), the EL equation reduces to `m γ̈ = 0`, i.e., constant
    velocity motion. -/
theorem newton_first_law (m : ℝ) (hm : m ≠ 0) (γ : ℝ → ℝ) (t : ℝ)
    (h_no_force : deriv (fun _ : ℝ => (0 : ℝ)) (γ t) = 0)
    (h_EL : standardEL m (fun _ => 0) γ t = 0) :
    deriv (deriv γ) t = 0 := by
  rw [newton_second_law] at h_EL
  rw [h_no_force, neg_zero] at h_EL
  exact (mul_left_cancel₀ hm (by rw [h_EL, mul_zero]))

/-- **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 full content of classical mechanics, only the second law for a single particle in a potential. That the cost function is a physical energy, only that it behaves like one in the limit. That the framework's axioms are the only way to derive Newton's law.

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.

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