Foundation Pair Kernel Source Variation Dirichlet Eq Sum Mul Laplacian Swapped

The swapped identity

In the calculus of variations, the Dirichlet form is a standard quadratic energy measuring how much a field field (a function assigning a number to each point) varies from point to point. For a finite set of points with weights weights (numbers saying how strongly each pair is connected), the form is written as half the sum over all pairs of the weight times the squared difference. A classical identity says this energy can be rewritten as the sum over points of the field value times the graph Laplacian (a discrete second-difference operator) of the field.

The declaration dirichlet_eq_sum_mul_laplacian_swapped proves a companion identity: the same energy also equals the sum over points of the second field times the Laplacian of the first field. The proof uses only the symmetry of the weights (w i j = w j i), which makes the two linear terms cancel. This swapped form is what makes exact variation calculations possible: when you perturb one field, the derivative of the energy appears directly as the Laplacian of the other field, with no limiting argument needed.

The identity is a theorem in the machine-checked library of formal theorems, meaning it is verified by a computer from axioms alone. It is part of a family of results: the energy equals the sum of field times Laplacian in both orders, and the quadratic action equals the Dirichlet form on the diagonal. These identities are algebraic facts about symmetric weighted graphs, not physical laws.

In Recognition Science, this identity supports the framework's treatment of variation problems on finite structures. The framework models physical configurations as fields on a discrete ledger, a record of events, and the swapped identity lets it compute exact first variations without approximations. It does not, however, select any physical source scale; the action is a mathematical model input, and the identities hold for any symmetric weights and any fields.

The practical consequence is that calculations involving energy changes become exact algebraic manipulations rather than estimates. For any symmetric weighted graph, the energy and its variations are fully determined by the Laplacian, and the swapped identity shows the two fields play interchangeable roles in the pairing. This is a clean, self-contained result that stands on its own as a piece of discrete mathematics.

/-- The bilinear identity with the variation in the Laplacian slot. -/
theorem dirichlet_eq_sum_mul_laplacian_swapped
    (w : ι → ι → ℝ) (e f : ι → ℝ)
    (hsymm : ∀ i j, w i j = w j i) :
    dirichlet w e f = ∑ i, f i * laplacian w e i := by
  calc
    dirichlet w e f = dirichlet w f e := by
      unfold dirichlet
      congr 1
      apply Finset.sum_congr rfl
      intro i _
      apply Finset.sum_congr rfl
      intro j _
      ring
    _ = ∑ i, f i * laplacian w e i :=
      dirichlet_eq_sum_mul_laplacian w f e hsymm

/-- Swapping the two finite indices turns the second linear term into the
negative of the first. This is the cancellation behind the Dirichlet
identity. -/
private theorem swapped_linear_term
    (w : ι → ι → ℝ) (f e : ι → ℝ)
    (hsymm : ∀ i j, w i j = w j i) :
    (∑ i, ∑ j, w i j * e j * (f i - f j))
      = -(∑ i, ∑ j, w i j * e i * (f i - f j)) := by
  calc
    (∑ i, ∑ j, w i j * e j * (f i - f j))
        = ∑ i, ∑ j, w j i * e i * (f j - f i) := Finset.sum_comm
    _ = ∑ i, ∑ j, w i j * e i * (f j - f i) := by
      apply Finset.sum_congr rfl
      intro i _
      apply Finset.sum_congr rfl
      intro j _
      rw [hsymm j i]
    _ = ∑ i, ∑ j, -(w i j * e i * (f i - f j)) := by
      apply Finset.sum_congr rfl
      intro i _
      apply Finset.sum_congr rfl
      intro j _
      ring
    _ = -(∑ i, ∑ j, w i j * e i * (f i - f j)) := by
      simp only [Finset.sum_neg_distrib]

/-- The quadratic action is the Dirichlet form on the diagonal. -/
theorem action_eq_dirichlet_self (w : ι → ι → ℝ) (e : ι → ℝ) :
    action w e = dirichlet w e e := by
  unfold action dirichlet
  congr 1
  apply Finset.sum_congr rfl
  intro i _
  apply Finset.sum_congr rfl
  intro j _
  ring

What this page does not claim

The identity does not select any physical source scale or coupling constant. The theorem does not apply to non-symmetric weights or infinite vertex sets. The framework's physical interpretation of the ledger is not established by this algebraic identity.

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

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