Foundation Pair Kernel Exact Jnonlinear Gauss S13 Exact Jfirst Variation Pairing

The exact first variation

In the Recognition Science framework, systems are described by a ledger, a discrete record of events on a graph, where each node holds a value called a field and each edge has a weight. The framework's cost, the exact J cost, is a sum over edges of a function that grows sharply as the field difference across an edge grows. The question of how this cost changes when the field is perturbed is answered by a theorem in the framework's machine-checked library of formal theorems.

The theorem, named exactJFirstVariationPairing_eq_two_sum_mul_laplacian, establishes that the first variation of the exact J cost, when the field is shifted by a small amount, equals two times the sum over all nodes of the variation at that node multiplied by a nonlinear Laplacian of the original field. The nonlinear Laplacian at a node is defined as the sum over its neighbors of the edge weight times the hyperbolic sine of the field difference. This is a direct analogue of the classical discrete Laplacian, but with the linear difference replaced by a hyperbolic sine.

This identity is the key step that connects a variational principle to a field equation. A stationary point of the cost, where the first variation vanishes for all perturbations, is equivalent to the condition that this nonlinear Laplacian equals a source term. The theorem exactJStationaryAtSource_iff_gauss proves this equivalence. This means that finding fields that minimize the cost is the same as solving a nonlinear Gauss law, a discrete version of a fundamental equation in physics.

The theorem also reveals a structural property: the first variation pairing is zero when the variation is a constant function, as shown by exactJFirstVariationPairing_const_direction. This reflects a symmetry of the cost under adding a constant to the field. The framework proves this identity without any unproven assumptions, and it is a foundational piece for understanding how the framework's discrete geometry gives rise to field equations.

What this theorem does not claim is that the nonlinear Laplacian is the only possible one, or that the framework's specific form is uniquely forced. It does not assert that the hyperbolic sine is the only function that could appear. The theorem is a statement about the consequences of the framework's chosen cost function, not a derivation of that cost from first principles. It also does not claim that the framework's discrete equations directly correspond to any particular physical system in the continuous limit.

/-- 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

/-- Exact action stationarity is equivalent to the pointwise nonlinear Gauss
equation. No source scale is selected by the equivalence. -/
theorem exactJStationaryAtSource_iff_gauss
    {n : ℕ}
    (G : WeightedLedgerGraph n)
    (field source : Fin n → ℝ)
    (coupling : ℝ) :
    ExactJStationaryAtSource G field source coupling ↔
      ExactJNonlinearGaussEquation G field source coupling := by
  classical
  constructor
  · intro h i
    have hi := h (fun j => if j = i then (1 : ℝ) else 0)
    rw [exactJFirstVariationPairing_eq_two_sum_mul_laplacian] at hi
    simpa [ExactJNonlinearGaussEquation] using hi
  · intro h variation
    rw [exactJFirstVariationPairing_eq_two_sum_mul_laplacian]
    unfold ExactJNonlinearGaussEquation at h
    calc
      2 * ∑ i,
          variation i * exactJNonlinearLaplacian G field i
          =
        ∑ i,
          variation i *
            (2 * exactJNonlinearLaplacian G field i) := by
              rw [Finset.mul_sum]
              apply Finset.sum_congr rfl
              intro i _
              ring
      _ = ∑ i,
          variation i * (coupling * source i) := by
            apply Finset.sum_congr rfl
            intro i _
            rw [h i]
      _ = coupling * ∑ i, variation i * source i := by
            rw [Finset.mul_sum]
            apply Finset.sum_congr rfl
            intro i _
            ring

theorem exactJFirstVariationPairing_const_direction
    {n : ℕ}
    (G : WeightedLedgerGraph n)
    (field : Fin n → ℝ)
    (c : ℝ) :
    exactJFirstVariationPairing G field (fun _ => c) = 0 := by
  simp [exactJFirstVariationPairing]

What this page does not claim

The theorem does not prove that the hyperbolic sine is the only possible function in the Laplacian. The theorem does not claim the framework's discrete equations directly model any specific physical system. The theorem does not derive the exact J cost function from more basic principles.

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)

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.

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