Encyclopedia Foundation Foundation Pair Kernel Exact Jnonlinear Gauss S13 Constant Curvature Signed Gree

ARTICLE 3 claims 3 theorems

Foundation Pair Kernel Exact Jnonlinear Gauss S13 Constant Curvature Signed Gree

A machine-checked theorem shows that on a curved recognition graph, a single posting produces a response field whose strength is simply the source divided by the curvature, with no free parameters.

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Scaled response at constant curvature

A recognition system, a discrete record of events and their costs, can be probed by placing one event at a point and asking how the rest of the system responds. In the framework's exact nonlinear model, the response is not a simple linear echo: the governing equation uses hyperbolic sine of field differences, so the response depends on the field itself. The declaration constantCurvatureSignedGreenField3_scaledSourceEquation proves that when the recognition graph is given a constant curvature, the response field to a single posting is exactly the source strength divided by twice the curvature.

The theorem states that for any positive curvature, any source magnitude q, and any primitive posting event on a torus of side length at least 2, the field defined as signedRealGreenField3 (q / curvature) event satisfies the scaled source equation. In plain terms: the response field's overall scale is q divided by curvature, and this field exactly solves the equation that defines a Green's response. The factor of one half comes from the symmetric sum over ordered edges, a structural detail of how the discrete divergence is defined.

This result is a theorem in the machine-checked library of formal theorems, with no unproved assumptions and no new axioms. It is a statement about the framework's own definitions: the cost action, the nonlinear Laplacian, and the Green response. The theorem does not assert that this constant-curvature graph is physically realized; realizing it by a global background field remains a separate, unproved predicate. It also does not assign any electromagnetic interpretation, SI units, or event action value to the quantities involved.

The practical consequence is a clean scaling law inside the framework: at constant curvature, response strength is inversely proportional to curvature, and the source is simply divided by that curvature. This is the kind of exact statement that lets later work compare different curvatures without re-solving the nonlinear equation each time. The theorem gives a precise, checkable relationship between a geometric parameter (curvature) and a dynamical response (the Green field), and it does so with the full force of a formal proof.

THEOREM constantCurvatureSignedGreenField3_scaledSourceEquation · IndisputableMonolith/Foundation/PairKernelExactJNonlinearGaussS13.lean
constantCurvatureSignedGreenField3_scaledSourceEquation · IndisputableMonolith/Foundation/PairKernelExactJNonlinearGaussS13.lean:803 
theorem constantCurvatureSignedGreenField3_scaledSourceEquation
    {N : ℕ} [NeZero N]
    (hN : 2 ≤ N)
    (curvature : ℝ)
    (hcurvature : 0 < curvature)
    (q : ℝ)
    (event : PostingPair3 N)
    (σ : Equiv.Perm (Fin 3)) :
    ScaledSourceEquation
      (constantCurvatureGraph curvature hcurvature.le
        (signedRecognitionProductionGraph3 N σ)).weight
      (dipole event.1 event.2)
      (q / 2)
      (constantCurvatureSignedGreenField3 curvature q event) := by
  have hbase :=
    signedRealGreenField3_scaledSourceEquation
      hN σ (q / curvature) event
  intro i
  have hi := hbase i
  unfold laplacian at hi
  unfold constantCurvatureGraph nonnegativeScaleGraph
    constantCurvatureSignedGreenField3 laplacian
  calc
    (∑ j,
      curvature *
        (signedRecognitionProductionGraph3 N σ).weight i j *
        (signedRealGreenField3 (q / curvature) event i -
          signedRealGreenField3 (q / curvature) event j))
        =
      ∑ j,
        curvature *
          ((signedRecognitionProductionGraph3 N σ).weight i j *
            (signedRealGreenField3 (q / curvature) event i -
              signedRealGreenField3 (q / curvature) event j)) := by
                apply Finset.sum_congr rfl
                intro j _
                ring
    _ =
      curvature *
        ∑ j,
          (signedRecognitionProductionGraph3 N σ).weight i j *
          (signedRealGreenField3 (q / curvature) event i -
            signedRealGreenField3 (q / curvature) event j) := by
              rw [Finset.mul_sum]
    _ =
      curvature *
        (((q / curvature) / 2) *
          dipole event.1 event.2 i) := by
            rw [hi]
    _ = (q / 2) * dipole event.1 event.2 i := by
          field_simp [hcurvature.ne']
THEOREM constantCurvatureSignedGreenField3_scaledSourceEquation · IndisputableMonolith/Foundation/PairKernelExactJNonlinearGaussS13.lean
constantCurvatureSignedGreenField3_scaledSourceEquation · IndisputableMonolith/Foundation/PairKernelExactJNonlinearGaussS13.lean:803 
theorem constantCurvatureSignedGreenField3_scaledSourceEquation
    {N : ℕ} [NeZero N]
    (hN : 2 ≤ N)
    (curvature : ℝ)
    (hcurvature : 0 < curvature)
    (q : ℝ)
    (event : PostingPair3 N)
    (σ : Equiv.Perm (Fin 3)) :
    ScaledSourceEquation
      (constantCurvatureGraph curvature hcurvature.le
        (signedRecognitionProductionGraph3 N σ)).weight
      (dipole event.1 event.2)
      (q / 2)
      (constantCurvatureSignedGreenField3 curvature q event) := by
  have hbase :=
    signedRealGreenField3_scaledSourceEquation
      hN σ (q / curvature) event
  intro i
  have hi := hbase i
  unfold laplacian at hi
  unfold constantCurvatureGraph nonnegativeScaleGraph
    constantCurvatureSignedGreenField3 laplacian
  calc
    (∑ j,
      curvature *
        (signedRecognitionProductionGraph3 N σ).weight i j *
        (signedRealGreenField3 (q / curvature) event i -
          signedRealGreenField3 (q / curvature) event j))
        =
      ∑ j,
        curvature *
          ((signedRecognitionProductionGraph3 N σ).weight i j *
            (signedRealGreenField3 (q / curvature) event i -
              signedRealGreenField3 (q / curvature) event j)) := by
                apply Finset.sum_congr rfl
                intro j _
                ring
    _ =
      curvature *
        ∑ j,
          (signedRecognitionProductionGraph3 N σ).weight i j *
          (signedRealGreenField3 (q / curvature) event i -
            signedRealGreenField3 (q / curvature) event j) := by
              rw [Finset.mul_sum]
    _ =
      curvature *
        (((q / curvature) / 2) *
          dipole event.1 event.2 i) := by
            rw [hi]
    _ = (q / 2) * dipole event.1 event.2 i := by
          field_simp [hcurvature.ne']
THEOREM constantCurvatureSignedGreenField3_scaledSourceEquation · IndisputableMonolith/Foundation/PairKernelExactJNonlinearGaussS13.lean
constantCurvatureSignedGreenField3_scaledSourceEquation · IndisputableMonolith/Foundation/PairKernelExactJNonlinearGaussS13.lean:803 
theorem constantCurvatureSignedGreenField3_scaledSourceEquation
    {N : ℕ} [NeZero N]
    (hN : 2 ≤ N)
    (curvature : ℝ)
    (hcurvature : 0 < curvature)
    (q : ℝ)
    (event : PostingPair3 N)
    (σ : Equiv.Perm (Fin 3)) :
    ScaledSourceEquation
      (constantCurvatureGraph curvature hcurvature.le
        (signedRecognitionProductionGraph3 N σ)).weight
      (dipole event.1 event.2)
      (q / 2)
      (constantCurvatureSignedGreenField3 curvature q event) := by
  have hbase :=
    signedRealGreenField3_scaledSourceEquation
      hN σ (q / curvature) event
  intro i
  have hi := hbase i
  unfold laplacian at hi
  unfold constantCurvatureGraph nonnegativeScaleGraph
    constantCurvatureSignedGreenField3 laplacian
  calc
    (∑ j,
      curvature *
        (signedRecognitionProductionGraph3 N σ).weight i j *
        (signedRealGreenField3 (q / curvature) event i -
          signedRealGreenField3 (q / curvature) event j))
        =
      ∑ j,
        curvature *
          ((signedRecognitionProductionGraph3 N σ).weight i j *
            (signedRealGreenField3 (q / curvature) event i -
              signedRealGreenField3 (q / curvature) event j)) := by
                apply Finset.sum_congr rfl
                intro j _
                ring
    _ =
      curvature *
        ∑ j,
          (signedRecognitionProductionGraph3 N σ).weight i j *
          (signedRealGreenField3 (q / curvature) event i -
            signedRealGreenField3 (q / curvature) event j) := by
              rw [Finset.mul_sum]
    _ =
      curvature *
        (((q / curvature) / 2) *
          dipole event.1 event.2 i) := by
            rw [hi]
    _ = (q / 2) * dipole event.1 event.2 i := by
          field_simp [hcurvature.ne']

What this page does not claim

The theorem does not claim that the constant-curvature graph is physically realized. It does not assign any electromagnetic interpretation, SI units, or event action value to the quantities. It does not claim that the response field is the unique solution to the nonlinear equation.

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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