Foundation Pair Kernel Exact Jevent Interaction Dual

The interaction dual

The foundation pair kernel exact jevent interaction dual is a construction in Recognition Science for comparing two discrete events without first attaching them to a kinetic scale, a Born coupling, a field normalization, or a molecular Hamiltonian. The events are integer discrete-Gauss profiles of realized postings, which are the framework's primitive records of recognition. The construction pairs each event with a field, and the primary action is the same finite exact-J cost used by the source-first event operator.

The object is a Legendre functional, written as the source-field pairing minus the exact-J cost action. Its critical-point law is the exact-J nonlinear Gauss equation. Two posting sources can therefore be compared through on-shell dual values, meaning the field values that make the functional stationary, without any of the usual physical scaffolding. The module proves that this functional is linear in the field along any line, that its derivative at zero is the source-variation pairing, and that an on-shell field is exactly one satisfying the nonlinear Gauss equation pointwise.

The module also proves a neutrality and gauge property. Each unit posting source sums to zero, and the composite source of two realized events also sums to zero. Because of this, adding a constant to the field leaves the full Legendre functional unchanged. The framework models this as gauge invariance: the dual value does not depend on a constant shift of the field, only on the shape of the source.

In Recognition Science, the framework models this as an interaction sector that exists before any Born or one-body attachment. The unit source pairing is a q=1 event lattice convention; exact-J does not select a dimensioned physical source scale. Existence and uniqueness of nonlinear on-shell fields, and any identification with molecular binding, remain open. The module is a machine-checked collection of formal theorems with no sorry, no new axiom, and no empirical constant.

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

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

theorem unitPostingSource3_sum_zero
    (event : PostingPair3 3) :
    ∑ i, unitPostingSource3 event i = 0 := by
  change
    ∑ i, divF (elementaryPosting event.1 event.2) i = 0
  exact elementaryPosting_sum_div_zero event.1 event.2

theorem compositeSource_sum_zero
    (cfg : ExactJTwoEventConfig3) :
    ∑ i, cfg.compositeSource i = 0 := by
  unfold ExactJTwoEventConfig3.compositeSource
  rw [Finset.sum_add_distrib,
    unitPostingSource3_sum_zero,
    unitPostingSource3_sum_zero,
    zero_add]

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

This module does not derive a dimensioned physical source scale. This module does not identify the dual values with molecular binding. This module does not introduce a kinetic scale or coupling constant.

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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