Foundation Pair Kernel Exact Jnonlinear Gauss S13 Exact Jnonlinear Laplacian Sig

The exact nonlinear Laplacian

The Laplacian is a standard tool that measures how a value at one point differs from the average of its neighbors. In the Recognition Science framework, the ledger (a discrete record of events on a graph) defines a cost for any assignment of numbers to its nodes. The exact nonlinear Laplacian is the object that describes how this cost changes when the numbers at the nodes are varied. It is built from the edge flux, which for an edge with weight w and a field difference d is w times the hyperbolic sine of d. This keeps the full nonlinearity of the cost, unlike a linearized version that would use only d itself.

The central result is an exact identity. For any weighted ledger graph, any field, and any variation of that field, the first variation of the cost equals twice the sum over nodes of the variation times the exact nonlinear Laplacian. This identity is proved in the framework's machine-checked library of formal theorems. The declaration establishes that this identity holds for the signed frame independent construction, meaning the result does not depend on a particular choice of orientation for the graph's edges. The same identity also yields an exact nonlinear Gauss equation, where twice the Laplacian equals a coupling constant times a source term.

The framework also proves that the exact nonlinear Laplacian of a constant field is zero, and that the first variation in a constant direction vanishes. At zero background, the tangent graph (the linearized version) recovers the canonical graph and the standard Green response. With a constant native curvature, the response scale changes by a factor involving the framework's constant hbar. The declaration does not assert that this constant-curvature graph is realized by a single global background, nor does it assert any event action, electromagnetic interpretation, or SI conversion. The result is a mathematical theorem about the discrete structure, not a physical claim.

/-- Ordered-edge symmetry fixes the factor two in the nonlinear Gauss law. -/
theorem exactJFirstVariationPairing_eq_two_sum_mul_laplacian
    {n : ℕ}
    (G : WeightedLedgerGraph n)
    (field variation : Fin n → ℝ) :
    exactJFirstVariationPairing G field variation =
      2 * ∑ i,
        variation i * exactJNonlinearLaplacian G field i := by
  unfold exactJFirstVariationPairing exactJNonlinearLaplacian exactJEdgeFlux
  let L : ℝ :=
    ∑ i, ∑ j,
      G.weight i j *
        Real.sinh (field i - field j) *
        variation i
  let R : ℝ :=
    ∑ i, ∑ j,
      G.weight i j *
        Real.sinh (field i - field j) *
        variation j
  have hR : R = -L := by
    dsimp [R, L]
    exact exactJ_swapped_variation_term G field variation
  calc
    (∑ i, ∑ j,
      G.weight i j *
        Real.sinh (field i - field j) *
        (variation i - variation j))
        = L - R := by
            dsimp [L, R]
            rw [← Finset.sum_sub_distrib]
            apply Finset.sum_congr rfl
            intro i _
            rw [← Finset.sum_sub_distrib]
            apply Finset.sum_congr rfl
            intro j _
            ring
    _ = 2 * L := by rw [hR]; ring
    _ = 2 * ∑ i,
        variation i *
          ∑ j,
            G.weight i j *
              Real.sinh (field i - field j) := by
          congr 1
          dsimp [L]
          apply Finset.sum_congr rfl
          intro i _
          rw [Finset.mul_sum]
          apply Finset.sum_congr rfl
          intro j _
          ring

theorem exactJNonlinearLaplacian_zero_field
    {n : ℕ}
    (G : WeightedLedgerGraph n)
    (i : Fin n) :
    exactJNonlinearLaplacian G (fun _ => 0) i = 0 := by
  simp [exactJNonlinearLaplacian, exactJEdgeFlux]

What this page does not claim

The declaration does not assert that the constant-curvature graph is realized by a physical background. It does not claim any electromagnetic interpretation or SI conversion for the framework's constants. It does not establish the exact nonlinear Laplacian as a physical observable.

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/PairKernelExactJNonlinearGaussS13.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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