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

The on-shell condition

In classical physics, a system is said to be on shell when it satisfies the equations of motion, as opposed to being an arbitrary trial configuration. The Recognition Science declaration eventOnShell_iff_legendreCritical proves an equivalence: a field is on shell if and only if it is a critical point of a particular Legendre functional, which is an expression built from a source-field pairing minus an action term. This is a standard variational principle, familiar from classical mechanics where the path taken by a particle is the one that makes the action stationary.

The setting is a discrete one. The framework works with a finite set of points on a torus, and each field assigns a real number to each point. The source is an integer-valued discrete-Gauss profile representing a posting event. The declaration shows that the condition of being on shell, defined through a stationary condition, is exactly equivalent to the field satisfying a nonlinear Gauss equation pointwise. This means the two descriptions, one variational and one differential, pick out the same set of physical fields.

What makes this useful is that it allows two posting sources to be compared through their on-shell dual values without introducing a kinetic scale, a Born coupling, a field normalization, or a molecular Hamiltonian. The framework's library proves this as a theorem, with no axioms beyond the standard three and no empirical constants. The unit source pairing is a convention at the q=1 event lattice, and the exact-J cost does not select a dimensioned physical source scale.

The declaration does not claim that such on-shell fields exist or are unique for a given source, nor does it identify them with molecular binding. Those questions remain open. It also does not claim that the Legendre functional is the only way to characterize on-shell fields, only that this particular equivalence holds in this discrete setting.

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

theorem exactJEventInteractionDual_cert
    (cfg : ExactJTwoEventConfig3) :
    ExactJEventInteractionDualCert3 cfg where
  source_neutral := compositeSource_sum_zero cfg
  line_derivative :=
    exactJEventLegendreFunctional3_hasDerivAt_line cfg.compositeSource
  onShell_iff_gauss :=
    eventOnShell_iff_nonlinearGauss cfg.compositeSource
  gauge_invariant :=
    compositeLegendreFunctional_gauge_invariant cfg

What this page does not claim

Existence or uniqueness of on-shell fields for a given source. Identification of on-shell fields with molecular binding. That the Legendre functional is the only characterization of on-shell fields. Selection of a dimensioned physical source scale by the exact-J cost.

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.

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