Foundation Pair Kernel Exact Jevent Interaction Dual Event On Shell Pointwise No

The on-shell field equation

In Recognition Science, the ledger (a discrete record of events) assigns to each posting event a source profile. The declaration eventOnShell_pointwise_nonlinearGauss proves that if a field is on shell (stationary for the exact-J cost), then at every point of the discrete lattice, twice the nonlinear Laplacian of the field equals the source value. This is a pointwise statement: it holds for each lattice site individually, not merely as an averaged or global relation. The proof is a theorem in the framework's machine-checked library of formal theorems, with no unproved axioms and no empirical constants.

The result is the critical-point law of a Legendre functional, the difference between a source-field pairing and the exact-J cost action. It is a nonlinear Gauss equation because the cost is the exact-J function, not a quadratic form. The theorem connects the abstract variational condition to a concrete differential equation, so that on-shell fields are characterized by this pointwise balance. The framework models this as an interaction sector built before any Born attachment or one-body approximation, using only the same finite exact-J cost that defines the source-first event operator.

What the declaration does not claim is equally precise. It does not select a dimensioned physical source scale: the unit source pairing is a convention at q=1 event lattice, and exact-J does not fix a physical unit. It does not prove existence or uniqueness of nonlinear on-shell fields, nor any identification with molecular binding; those remain open targets. The theorem is a variational and gauge property, not an empirical prediction. It establishes the form of the equation, not the values that solve it.

The practical consequence is that within the framework, two posting sources can be compared through their on-shell dual values without introducing a kinetic scale, Born coupling, field normalization, or molecular Hamiltonian. The pointwise Gauss law is the bridge that makes such comparison possible. It is a structural result: it tells what an on-shell field must satisfy, and it leaves the physical interpretation and the existence of solutions to further work.

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

/-- Exact-J Legendre functional for an integer event source. -/
def exactJEventLegendreFunctional3
    (source field : DualEventField3) : ℝ :=
  eventSourcePairing3 source field -
    exactJCostAction eventInteractionGraph3 field

/-- On-shell field predicate at the explicit q=1 event-lattice convention. -/
def IsExactJEventOnShell3
    (source field : DualEventField3) : Prop :=
  ExactJStationaryAtSource
    eventInteractionGraph3 field source 1

What this page does not claim

The theorem does not prove existence or uniqueness of on-shell fields. The theorem does not identify the framework's interaction with molecular binding. The theorem does not assign a dimensioned physical scale to the unit source.

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)

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Derived articles

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