Foundation Pair Kernel Exact Jevent Interaction Dual Exact Jevent Legendre Funct

The derivative along a line

A Legendre functional, in this framework, is a quantity built from a source and a field: it pairs them, then subtracts the cost of the field itself. The declaration exactJEventLegendreFunctional3_hasDerivAt_line is a theorem about how that quantity responds when the field is varied along a straight line in the space of all fields. It states, formally, that the derivative at the starting point equals the source-field pairing of the variation minus the first variation of the cost. In plain terms: if you nudge the field in any direction, the leading change in the functional is exactly that combination, with no higher-order terms at the instant of the nudge.

This is a local statement. It does not say that the functional is minimized, maximized, or stationary at any particular field; it only describes the slope at a point. The theorem holds for every source, field, and variation in the finite-dimensional space of fields on a three-point torus. It is a piece of the framework's library, a machine-checked collection of formal theorems, and it was proved without adding any new axioms or empirical constants.

The result matters because it is the first step toward a critical-point condition. A separate theorem, also in the library, says that a field is on-shell precisely when this derivative vanishes for every variation. That condition is equivalent to a nonlinear Gauss equation, which is the framework's model for how a source produces a field. The derivative theorem is the analytical foundation: it gives the exact expression whose vanishing defines the on-shell condition.

What the declaration does not claim is just as important. It does not assert that on-shell fields exist or are unique; that remains an open problem in the framework. It does not identify the source scale with any physical charge, since the unit source is a convention. And it does not say anything about molecular binding, which the framework's own documentation lists as open. The theorem is a clean, local fact about a finite-dimensional functional, and it is used exactly where it is needed: to define what it means for a field to be stationary.

theorem exactJEventLegendreFunctional3_hasDerivAt_line
    (source field variation : DualEventField3) :
    HasDerivAt
      (fun t : ℝ =>
        exactJEventLegendreFunctional3 source
          (fun i => field i + t * variation i))
      (eventSourcePairing3 source variation -
        exactJFirstVariationPairing
          eventInteractionGraph3 field variation)
      0 := by
  exact
    (eventSourcePairing3_hasDerivAt_line source field variation).sub
      (exactJCostAction_hasDerivAt_line
        eventInteractionGraph3 field variation)

theorem eventOnShell_iff_legendreCritical
    (source field : DualEventField3) :
    IsExactJEventOnShell3 source field ↔
      ∀ variation,
        eventSourcePairing3 source variation -
          exactJFirstVariationPairing
            eventInteractionGraph3 field variation = 0 := by
  unfold IsExactJEventOnShell3 ExactJStationaryAtSource
  constructor
  · intro h variation
    specialize h variation
    simp only [one_mul] at h
    rw [eventSourcePairing3_comm_sum]
    linarith
  · intro h variation
    specialize h variation
    rw [eventSourcePairing3_comm_sum] at h
    simp only [one_mul]
    linarith

theorem eventOnShell_iff_nonlinearGauss
    (source field : DualEventField3) :
    IsExactJEventOnShell3 source field ↔
      ExactJNonlinearGaussEquation
        eventInteractionGraph3 field source 1 :=
  exactJStationaryAtSource_iff_gauss
    eventInteractionGraph3 field source 1

/-- Integer discrete-Gauss source of one posting. -/
def unitPostingSource3
    (event : PostingPair3 3) : DualEventField3 :=
  realifiedElementaryGaussSource event.1 event.2

What this page does not claim

The theorem does not assert existence or uniqueness of on-shell fields. The theorem does not fix a physical scale for the source. The theorem says nothing about molecular binding.

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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