Foundation Pair Kernel Source Variation Action Has Deriv At Line

The exact line derivative

Consider a finite collection of points, each carrying a number called its field value. Pair every two points and assign them a weight. The action, a discrete record of total pair energy, is half the weighted sum of squared differences between field values. Written out, for weights w and field e, it is A(e) = (1/2) ∑_i,j w_ij(e_i − e_j)².

Now move the field along a straight line: replace e by e + tv, where t is a real parameter and v is a fixed direction. The action becomes a quadratic polynomial in t. Its derivative at t = 0, the first variation, is exactly 2 times the polarized Dirichlet form, a symmetric bilinear measure of how e and v co-vary across pairs. The theorem action_hasDerivAt_line proves this fact for any weights and any fields, with no symmetry assumption needed. The coefficient 2 is forced by the half in the action's definition, and it appears exactly, not as a limit.

When the weights are symmetric, meaning w_{ij = wji, the derivative can be rewritten using the graph Laplacian, a weighted sum of differences at each point. In that form the first variation is 2 ∑i vi(Δe)i. This is the standard energy identity of Dirichlet forms, and the theorem provides it without any limiting argument.}

In Recognition Science, this declaration is a mathematical model input. It defines a cost structure and proves its algebraic properties. It does not select a physical source scale, and it does not by itself determine any constants of nature. The framework's library proves the identity in a machine-checked collection of formal theorems, with no unproved assumptions beyond the standard logical axioms.

/-- The derivative of the action along a line has coefficient `2`. -/
theorem action_hasDerivAt_line
    (w : ι → ι → ℝ) (e v : ι → ℝ) :
    HasDerivAt
      (fun t : ℝ => action w (fun i => e i + t * v i))
      (2 * dirichlet w e v)
      0 := by
  have hfun :
      (fun t : ℝ => action w (fun i => e i + t * v i))
        = fun t : ℝ => action w e + 2 * t * dirichlet w e v
            + t ^ 2 * action w v := by
    funext t
    exact action_line_expansion w e v t
  rw [hfun]
  convert
    ((hasDerivAt_const (x := (0 : ℝ)) (action w e)).add
      (((hasDerivAt_id (𝕜 := ℝ) (x := (0 : ℝ))).const_mul 2).mul_const
        (dirichlet w e v))).add
      (((hasDerivAt_id (𝕜 := ℝ) (x := (0 : ℝ))).pow 2).mul_const (action w v))
    using 1
  all_goals norm_num [id]

/-- In Laplacian form, the exact first variation is
`2 * ∑ᵢ vᵢ (Δe)ᵢ`. -/
theorem action_hasDerivAt_line_laplacian
    (w : ι → ι → ℝ) (e v : ι → ℝ)
    (hsymm : ∀ i j, w i j = w j i) :
    HasDerivAt
      (fun t : ℝ => action w (fun i => e i + t * v i))
      (2 * ∑ i, v i * laplacian w e i)
      0 := by
  rw [← dirichlet_eq_sum_mul_laplacian_swapped w e v hsymm]
  exact action_hasDerivAt_line w e v

/-- The symmetric weighted quadratic pair action. -/
def action (w : ι → ι → ℝ) (e : ι → ℝ) : ℝ :=
  (1 / 2 : ℝ) * ∑ i, ∑ j, w i j * (e i - e j) ^ 2

What this page does not claim

The theorem does not require symmetric weights for the basic derivative result. The declaration does not fix any physical constants or source scales. The result does not depend on a limiting argument; it is an exact algebraic identity.

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/Foundation/PairKernelSourceVariation.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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