Foundation Pair Kernel Exact Jnonlinear Gauss S13 Constant Curvature Signed Post

A Green function for curved space

A Green function is a standard tool in physics and engineering: it answers the question, if you poke a system at one point, how does the effect spread to every other point? On a flat, regular lattice the answer is a familiar, symmetric pattern. The Recognition Science framework asks what happens to that pattern when the underlying geometry is curved instead of flat, and its machine-checked library of formal theorems contains a result, constantCurvatureSignedPostingAttachment3_realGreen, that settles the question for one specific family of curved spaces.

The framework models reality as a ledger, a discrete record of events and the connections between them. Each connection has a weight, and the cost of a configuration is the sum over connections of a nonlinear function of the field difference, written as cosh(field_i - field_j) - 1. The first variation of this cost gives a nonlinear version of the discrete Laplacian, the operator that measures how a field spreads. The theorem at issue concerns a constant-curvature version of this setup, where every connection carries the same geometric factor, and it proves that a Green function built from the standard flat-space formula, with the source strength divided by that curvature, still satisfies the nonlinear Gauss law.

Concretely, the theorem states that for any positive curvature, any source strength q, and any primitive posting event on a torus of size at least 2, the field defined by the scaled Green function satisfies the exact nonlinear Gauss equation. The proof is a direct computation: it verifies that the nonlinear Laplacian of the proposed field equals the source term divided by two, with the curvature factor absorbed into the field's definition. The result is not an approximation; it is an exact identity in the formal library, checked line by line by the Lean 4 kernel with no unproven assumptions.

What the theorem does not claim is just as important. It does not assert that such a constant-curvature geometry actually arises from the framework's dynamics. The constant-curvature graph is a tangent model, a linearized approximation around a background, and realizing it by one global background field remains a separate, unproven predicate. The theorem also does not assign physical meaning to the curvature, the source, or the field: it is a statement about a mathematical structure, not about electromagnetism or any specific force. Finally, it does not claim that the Green function is unique or that it minimizes the action; it only shows that this particular field satisfies the defining equation.

The consequence is that the framework's nonlinear generalization of Gauss's law is not a dead end. Even in curved geometry, the familiar Green-function machinery survives, with a simple rescaling by the curvature. This opens the door to asking whether such constant-curvature configurations can be realized dynamically, a question the framework leaves explicitly open.

theorem constantCurvatureSignedPostingAttachment3_realGreen
    {N : ℕ} [NeZero N]
    (hN : 2 ≤ N)
    (curvature : ℝ)
    (hcurvature : 0 < curvature)
    (q : ℝ)
    (event : PostingPair3 N)
    (hevent : event ∈ realizedPrimitivePostingPairs3 N)
    (σ : Equiv.Perm (Fin 3)) :
    ConstantCurvatureSignedPostingAttachment3
      curvature hcurvature.le q event σ
      (constantCurvatureSignedGreenField3 curvature q event) := by
  constructor
  · exact hevent
  · rw [postingMagnitudeActionLaw_iff_scaledSourceEquation
      q
      (constantCurvatureGraph curvature hcurvature.le
        (signedRecognitionProductionGraph3 N σ)).weight
      (constantCurvatureSignedGreenField3 curvature q event)
      event.1 event.2
      (constantCurvatureGraph curvature hcurvature.le
        (signedRecognitionProductionGraph3 N σ)).weight_symm]
    exact
      constantCurvatureSignedGreenField3_scaledSourceEquation
        hN curvature hcurvature q event σ

/-- 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

/-- Explicit constant-curvature tangent model. Background realization is kept
as the separate predicate above. -/
def constantCurvatureGraph
    {n : ℕ}
    (curvature : ℝ)
    (hcurvature : 0 ≤ curvature)
    (G : WeightedLedgerGraph n) :
    WeightedLedgerGraph n :=
  nonnegativeScaleGraph curvature hcurvature G

What this page does not claim

The theorem does not claim that constant-curvature geometry is dynamically realized. The theorem does not assign physical meaning to the curvature, source, or field. The theorem does not claim uniqueness or minimality of the Green function.

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: