Foundation Pair Kernel Source Variation

The pair action and its forced identities

In Recognition Science, the framework's account of physical structure, the ledger (a discrete record of events) assigns a real number to each vertex of a finite graph. A pair action is a quadratic energy over that ledger: for weights w_ij and field values e_i, it is defined as (1/2)∑_i,j w_ij (e_i − e_j)². This is a mathematical model input, not a derived law; the framework takes it as given.

The classical content here is the calculus of such quadratic forms. The action is a weighted Dirichlet form, and its associated graph Laplacian is Δe_i = ∑_j w_ij (e_i − e_j). When the weights are symmetric (w_ij = w_ji), the action equals the inner product of the field with its Laplacian: A(e) = ∑_i e_i (Δe)_i. That identity, the Dirichlet bilinear form, is the bridge between the energy and the operator.

The central result is the exact line variation. For a one-parameter family e + t v, the action expands as A(e + t v) = A(e) + 2t D(e,v) + t² A(v), where D is the polarized Dirichlet form. The linear coefficient is exactly 2, not an approximation; the derivative at t = 0 is 2∑_i v_i (Δe)_i. This coefficient is forced by the quadratic symmetry alone, with no additional physical source scale selected.

The framework also proves the unit dipole laws. A unit dipole is a field that is +1 at one vertex and −1 at another. Pairing any field with a unit dipole gives the potential drop: ∑_i e_i (dipole)_i = e_a − e_b. If the Laplacian of a field is exactly a unit dipole, then the action equals the potential drop; if twice the Laplacian is a unit dipole, the action is half the drop. These are exact identities.

In Recognition Science, this framework isolates the algebra that a symmetric quadratic pair action forces, before any physical interpretation. The identities are established in a machine-checked library of formal theorems, with no unproved assumptions and no new axioms. The framework does not claim that the action itself is derived from its core forcing chain; it is a model input whose consequences are then rigorously established.

/-- Kernel-clean energy identity: `A(e) = ∑ᵢ eᵢ (Δe)ᵢ`. -/
theorem action_eq_sum_mul_laplacian
    (w : ι → ι → ℝ) (e : ι → ℝ)
    (hsymm : ∀ i j, w i j = w j i) :
    action w e = ∑ i, e i * laplacian w e i := by
  rw [action_eq_dirichlet_self]
  exact dirichlet_eq_sum_mul_laplacian w e e hsymm

/-- Exact line expansion of the quadratic action. The linear coefficient is
exactly twice the polarized Dirichlet form. -/
theorem action_line_expansion
    (w : ι → ι → ℝ) (e v : ι → ℝ) (t : ℝ) :
    action w (fun i => e i + t * v i)
      = action w e + 2 * t * dirichlet w e v + t ^ 2 * action w v := by
  unfold action dirichlet
  have hpoint : ∀ i j,
      w i j * ((e i + t * v i) - (e j + t * v j)) ^ 2
        = w i j * (e i - e j) ^ 2
          + 2 * t * (w i j * (e i - e j) * (v i - v j))
          + t ^ 2 * (w i j * (v i - v j) ^ 2) := by
    intro i j
    ring
  simp_rw [hpoint]
  simp only [Finset.sum_add_distrib, ← Finset.mul_sum]
  ring

/-- Pairing any field with a unit dipole gives its potential drop. -/
theorem sum_mul_dipole [DecidableEq ι] (e : ι → ℝ) (a b : ι) :
    (∑ i, e i * dipole a b i) = e a - e b := by
  unfold dipole
  simp [mul_sub, Finset.sum_sub_distrib]

/-- If the Laplacian is a unit dipole, the energy is the potential drop. -/
theorem action_eq_potential_drop_of_laplacian_eq_dipole
    [DecidableEq ι]
    (w : ι → ι → ℝ) (e : ι → ℝ) (a b : ι)
    (hsymm : ∀ i j, w i j = w j i)
    (hsource : ∀ i, laplacian w e i = dipole a b i) :
    action w e = e a - e b := by
  rw [action_eq_sum_mul_laplacian w e hsymm]
  calc
    (∑ i, e i * laplacian w e i) = ∑ i, e i * dipole a b i := by
      apply Finset.sum_congr rfl
      intro i _
      rw [hsource i]
    _ = e a - e b := sum_mul_dipole e a b

What this page does not claim

The pair action itself is not derived from the framework's forcing chain; it is a model input. The framework does not select a physical source scale or coupling strength. The identities hold only for finite vertex types and symmetric weights, not for general graphs or non-symmetric kernels.

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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