Foundation Pair Kernel Exact Jevent Interaction Dual Event On Shell Iff Nonlinea

The on-shell condition

In classical physics, a field is often said to be "on shell" when it satisfies the equations of motion, the point where a system's action stops changing under small variations. The Recognition Science framework builds an analogous condition for its discrete event ledger, a finite record of posting events. The declaration eventOnShell_iff_nonlinearGauss proves, in the framework's machine-checked library of formal theorems, that a field is on shell exactly when it satisfies a nonlinear Gauss equation, a discrete analog of the classical relation between a field's curvature and its source.

The theorem is an equivalence. It states that for any source and field in the framework's three-event lattice, the predicate IsExactJEventOnShell3, meaning the field is stationary under the exact-J cost action, holds if and only if the field satisfies ExactJNonlinearGaussEquation, the framework's discrete nonlinear Gauss law. A companion theorem makes the content pointwise: when the on-shell condition holds, twice the nonlinear Laplacian of the field equals the source at every lattice point. This is the discrete analog of the classical Gauss law, where the divergence of the field equals its source density.

The framework models this as an interaction sector built before any Born attachment or molecular Hamiltonian. The action is the same finite exact-J cost used elsewhere, and the critical-point law of its Legendre functional yields the nonlinear Gauss equation. The theorem also carries a neutrality property: the composite source of two posting events sums to zero, making the Legendre functional gauge invariant under adding a constant to the field.

What the theorem does not claim is as important as what it proves. The unit source pairing is a q=1 event lattice convention; the exact-J cost does not select a dimensioned physical source scale. Existence and uniqueness of nonlinear on-shell fields, and any identification with molecular binding, remain open targets. The theorem establishes a formal equivalence within the framework's discrete model, not a physical law of nature.

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

theorem eventOnShell_pointwise_nonlinearGauss
    (source field : DualEventField3)
    (h : IsExactJEventOnShell3 source field) :
    ∀ i,
      2 * exactJNonlinearLaplacian
          eventInteractionGraph3 field i =
        source i := by
  have hgauss :=
    (eventOnShell_iff_nonlinearGauss source field).mp h
  intro i
  simpa [ExactJNonlinearGaussEquation] using hgauss i

theorem compositeLegendreFunctional_gauge_invariant
    (cfg : ExactJTwoEventConfig3)
    (field : DualEventField3)
    (c : ℝ) :
    exactJEventLegendreFunctional3 cfg.compositeSource
        (fun i => field i + c) =
      exactJEventLegendreFunctional3
        cfg.compositeSource field :=
  exactJEventLegendreFunctional3_add_const
    cfg.compositeSource field c (compositeSource_sum_zero cfg)

What this page does not claim

The theorem does not establish existence or uniqueness of on-shell fields. The theorem does not identify the framework's interaction with molecular binding. The unit source pairing is a convention, not a derived physical scale.

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: