Foundation Pair Kernel Exact Jnonlinear Gauss S13 Exact Jtangent Laplacian Signe

The tangent picture

The declaration exactJTangentLaplacian_signed_frame_independent belongs to a family of results about how a discrete field responds to a small push. In the framework's picture, a field is a list of numbers attached to the vertices of a weighted graph, and the graph's edges carry weights that record how strongly two vertices interact. The exact J cost function, which the framework derives from its five axioms, assigns an energy to each field configuration. The declaration concerns the tangent picture: take a background field, nudge it by a tiny perturbation, and ask how the energy changes to first order. That first-order change is governed by a linear operator called the tangent Laplacian, built from the background field's local differences.

What the declaration establishes is a symmetry property of that tangent Laplacian. For any two perturbations, the pairing defined by the tangent Laplacian is symmetric: the response to one perturbation in the direction of another equals the response in the reversed order. This symmetry is not automatic for a general nonlinear operator; it is a structural fact that the exact J cost function inherits from its form. The declaration also shows that the tangent Laplacian is positive semidefinite, meaning its action on any perturbation is nonnegative, and that constant perturbations lie in its kernel, so shifting the whole field by a constant costs nothing. These properties make the tangent Laplacian a well-behaved object for linear analysis around a background.

Within the framework, this result supports a specific interpretation. The tangent Laplacian at a background field is the graph whose edge weights are the original weights multiplied by the hyperbolic cosine of the background field difference across that edge. At zero background, this tangent graph reduces to the original graph, recovering the simpler linear response. The framework uses this to model a constant native curvature: a background field whose differences are all equal to a fixed drop, which scales the tangent graph uniformly. The declaration does not assert that such a constant-drop background exists for a given graph; that realization is a separate predicate. It also does not assert that the tangent Laplacian is the full nonlinear Laplacian, nor that the linear response it describes is the whole story of the field's behavior.

In plain terms, the declaration pins down the local, linearized behavior of the framework's cost function around any background. It gives the framework a clean object, the tangent Laplacian, with the symmetry and positivity needed for stable analysis. What it does not do is claim that this linear picture is complete, that the constant-curvature background is always realizable, or that the framework's constants, such as the event action hbar, have any electromagnetic or SI interpretation. Those remain separate questions, and the declaration is silent on them.

/-- Bilinear Hessian pairing of the exact action. -/
def exactJHessianPairing
    {n : ℕ}
    (G : WeightedLedgerGraph n)
    (background left right : Fin n → ℝ) : ℝ :=
  ∑ i, ∑ j,
    G.weight i j *
      Real.cosh (background i - background j) *
      (left i - left j) *
      (right i - right j)

theorem exactJHessianAction_nonneg
    {n : ℕ}
    (G : WeightedLedgerGraph n)
    (background perturbation : Fin n → ℝ) :
    0 ≤ exactJHessianAction G background perturbation := by
  unfold exactJHessianAction action
  apply mul_nonneg
  · norm_num
  · apply Finset.sum_nonneg
    intro i _
    apply Finset.sum_nonneg
    intro j _
    exact mul_nonneg
      ((exactJTangentGraph G background).weight_nonneg i j)
      (sq_nonneg _)

theorem exactJTangentLaplacian_const_zero
    {n : ℕ}
    (G : WeightedLedgerGraph n)
    (background : Fin n → ℝ)
    (c : ℝ)
    (i : Fin n) :
    exactJTangentLaplacian G background (fun _ => c) i = 0 := by
  simp [exactJTangentLaplacian, laplacian]

theorem exactJTangentGraph_zero_weight
    {n : ℕ}
    (G : WeightedLedgerGraph n)
    (i j : Fin n) :
    (exactJTangentGraph G (fun _ => 0)).weight i j =
      G.weight i j := by
  simp [exactJTangentGraph]

theorem nativeCurvatureTangentGreenScale :
    realGreenScaleFromPostingMagnitude
        (nativeOrderedExactJSource /
          (1 + Constants.hbar)) =
      nativeExactJConjugateSource /
        (1 + Constants.hbar) := by
  rw [tangentGreenScale_eq
    (1 + Constants.hbar) nativeOrderedExactJSource
    (ne_of_gt nativeCurvature_pos)]
  unfold nativeOrderedExactJSource
  field_simp [ne_of_gt nativeCurvature_pos]

What this page does not claim

The declaration does not assert that a constant-curvature background is realizable for every graph. The declaration does not claim that the tangent Laplacian captures the full nonlinear response of the field. The declaration does not assign any electromagnetic or SI meaning to the framework's internal constants.

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.

Derived articles

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