Foundation Pair Kernel Exact Jevent Interaction Dual Composite Legendre Function

A gauge invariance for paired events

In classical physics, a gauge symmetry means that some part of the mathematical description can be changed without changing the physical predictions. The most familiar example is the electric potential: you can add any constant to it everywhere, and all measurable electric fields stay exactly the same. The Recognition Science declaration compositeLegendreFunctional_gauge_invariant establishes an analogous property for a specific interaction between two discrete events, using the framework's own cost-based ledger, a discrete record of realized events and their associated values.

The setting is a finite lattice with three sites. Each event is represented by an integer-valued source profile, and the interaction energy between two such events is defined through a Legendre functional: a pairing between the source and a field, minus the exact J-cost action of that field. The theorem states that if you add the same constant to the field at every site, the value of this functional does not change, provided the total source sums to zero. Since each event's source sums to zero, and the composite of two events also sums to zero, the condition is met. The proof in the machine-checked library of formal theorems chains together two earlier results: one showing the source pairing ignores constant additions, and another showing the cost action does the same.

The practical consequence is that the interaction energy between two posting events is well-defined even though the field itself is only defined up to an additive constant. This is a structural guarantee, not a numerical prediction. It says the framework's chosen interaction sector has the same kind of freedom that electromagnetism has, and that this freedom does not break the energy calculation. The theorem is a variational result: it concerns the critical points of the functional, where the on-shell field satisfies a nonlinear Gauss equation. It does not say anything about what those critical points are, whether they exist uniquely, or what physical meaning they might have.

The declaration's own documentation is explicit about the limits. The unit source pairing is a convention for a q=1 event lattice, and 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 problems. The theorem is a clean piece of mathematics: it proves a symmetry property of a defined object, with no new axioms and no empirical constants. What it does not do is claim that this interaction sector describes any particular physical system, or that the gauge invariance alone forces the form of the interaction.

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)

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 eventSourcePairing3_add_const
    (source field : DualEventField3)
    (c : ℝ)
    (hneutral : ∑ i, source i = 0) :
    eventSourcePairing3 source (fun i => field i + c) =
      eventSourcePairing3 source field := by
  unfold eventSourcePairing3
  simp_rw [mul_add]
  rw [Finset.sum_add_distrib]
  have hconstant :
      (∑ i, source i * c) = 0 := by
    rw [← Finset.sum_mul, hneutral, zero_mul]
  rw [hconstant, add_zero]

/-- Neutral sources make the full Legendre functional gauge invariant. -/
theorem exactJEventLegendreFunctional3_add_const
    (source field : DualEventField3)
    (c : ℝ)
    (hneutral : ∑ i, source i = 0) :
    exactJEventLegendreFunctional3 source
        (fun i => field i + c) =
      exactJEventLegendreFunctional3 source field := by
  unfold exactJEventLegendreFunctional3
  rw [eventSourcePairing3_add_const source field c hneutral,
    exactJCostAction_add_const]

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

What this page does not claim

The theorem does not claim that the interaction sector describes any particular physical system. The theorem does not claim that the on-shell fields exist or are unique for any given source. The theorem does not claim that the gauge invariance alone determines the form of the interaction functional.

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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