Foundation Pair Kernel Exact Jevent Interaction Dual Exact Jevent Interaction Du

The interaction certificate

In Recognition Science, a ledger is a discrete record of events, and each event is a posting: a pair of signed entries. The declaration exactJEventInteractionDual_cert is a machine-checked certificate, a formal theorem in the framework's library, that establishes what can be said about two such postings before any physical interpretation is attached. It proves that two realized posting events, each represented as an integer discrete-Gauss profile, can be compared through a shared cost function, the exact-J cost, without needing a kinetic scale, a Born coupling, a field normalization, or a molecular Hamiltonian.

The certificate assembles four properties into one structure. First, the combined source of the two events sums to zero, a neutrality condition. Second, the Legendre functional, which pairs a source with a field and subtracts the cost action, has a well-defined line derivative. Third, a field is on-shell, meaning it satisfies the critical-point condition, if and only if it obeys the exact-J nonlinear Gauss equation. Fourth, the whole functional is gauge invariant: adding a constant to the field leaves the functional unchanged, provided the source is neutral. These are the four clauses of the certificate, and each is a proved theorem.

The certificate does not claim that this interaction sector corresponds to any known physical force or binding. The docstring is explicit: the unit source pairing is a q=1 event lattice convention, and 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 certificate is a statement about the variational structure of the cost function, not about what that structure means physically.

What the certificate changes is the scope of what can be asserted. Before it, one might have guessed that comparing two events required additional physical input. The certificate shows that the comparison is already determined by the cost function alone, within the discrete event-lattice convention. It is a precise, narrow result: it proves the variational and gauge properties, and it leaves the physical interpretation as a model choice, not a theorem.

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

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

/-- 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]

What this page does not claim

The certificate does not claim that the interaction sector corresponds to any known physical force or binding. It does not claim that exact-J selects a dimensioned physical source scale. It does not claim existence or uniqueness of nonlinear on-shell fields, which remain open.

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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