Foundation Pair Kernel Exact Jevent Interaction Dual Event Source Pairing3 Has D

The pairing's derivative

The declaration eventSourcePairing3_hasDerivAt_line is a theorem in the machine-checked library of formal theorems. It states that a certain real-valued function of a real variable has a derivative at zero. The function is built from a ledger, a discrete record of events, by taking a sum of products between a fixed source and a field that varies linearly with the parameter. The derivative at zero equals the pairing of the source with the direction of variation, a result that follows directly from the linearity of the sum.

In the framework, this lemma is a small piece of a larger construction. The pairing is the linear part of a Legendre functional, which compares a source against a field through an exact cost. The theorem is one of four properties bundled into a certificate for the interaction sector. The other properties are neutrality of the composite source, equivalence of on-shell fields with a nonlinear Gauss equation, and gauge invariance under adding a constant to the field. Together they describe a consistent variational structure before any kinetic scale or coupling is introduced.

The theorem does not claim that the derivative is nonzero, that the critical point exists, or that the on-shell field is unique. It does not select a physical source scale or identify the construction with molecular binding. Those questions remain open. The lemma is a formal statement about a finite sum, and its proof is checked by the kernel. It establishes a local linear fact, nothing more.

theorem eventSourcePairing3_hasDerivAt_line
    (source field variation : DualEventField3) :
    HasDerivAt
      (fun t : ℝ =>
        eventSourcePairing3 source
          (fun i => field i + t * variation i))
      (eventSourcePairing3 source variation) 0 := by
  have haffine :
      (fun t : ℝ =>
        eventSourcePairing3 source
          (fun i => field i + t * variation i)) =
      (fun t : ℝ =>
        eventSourcePairing3 source field +
          t * eventSourcePairing3 source variation) := by
    funext t
    exact eventSourcePairing3_line source field variation t
  rw [haffine]
  convert
    (hasDerivAt_const (x := (0 : ℝ))
      (eventSourcePairing3 source field)).add
      ((hasDerivAt_id (𝕜 := ℝ) (x := (0 : ℝ))).mul_const
        (eventSourcePairing3 source variation))
    using 1 <;> simp

theorem eventSourcePairing3_hasDerivAt_line
    (source field variation : DualEventField3) :
    HasDerivAt
      (fun t : ℝ =>
        eventSourcePairing3 source
          (fun i => field i + t * variation i))
      (eventSourcePairing3 source variation) 0 := by
  have haffine :
      (fun t : ℝ =>
        eventSourcePairing3 source
          (fun i => field i + t * variation i)) =
      (fun t : ℝ =>
        eventSourcePairing3 source field +
          t * eventSourcePairing3 source variation) := by
    funext t
    exact eventSourcePairing3_line source field variation t
  rw [haffine]
  convert
    (hasDerivAt_const (x := (0 : ℝ))
      (eventSourcePairing3 source field)).add
      ((hasDerivAt_id (𝕜 := ℝ) (x := (0 : ℝ))).mul_const
        (eventSourcePairing3 source variation))
    using 1 <;> simp

What this page does not claim

The derivative being zero or nonzero at the critical point. Existence or uniqueness of nonlinear on-shell fields. Any identification with molecular binding or a dimensioned physical source scale. The theorem alone does not establish the full variational structure; it is one of four certificate properties.

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/PairKernelExactJEventInteractionDual.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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