Foundation Pair Kernel Exact Jnonlinear Gauss S13

The exact nonlinear Gauss law

Gauss's law in classical physics relates a field to the sources that produce it: the more charge inside a surface, the stronger the field flowing out. This framework builds the discrete analogue on a ledger, a finite record of events linked by weighted connections. The field values live on the ledger's nodes, and the law is not the usual linear one but a nonlinear version in which the flux across an edge grows with the hyperbolic sine of the field difference, not with the difference itself.

The action being studied is a sum over edges of cosh(field_i - field_j) - 1, a cost that is zero when the field is constant and grows steeply as differences increase. Taking the first variation of this action gives the flux: each edge contributes weight times sinh(field_i - field_j). The nonlinear Gauss equation states that twice the divergence of this flux equals a coupling constant times the source at each node. A theorem in the machine-checked library proves that a field satisfies this nonlinear equation if and only if it is a stationary point of the action under all variations, so the Gauss law is exactly the condition for the field to be a critical point of the cost.

Around a fixed background field, the action is locally quadratic, and the Hessian defines a tangent graph with edge weights weight times cosh(background_i - background_j). This tangent graph is symmetric and nonnegative, so small perturbations behave like a linear system. At zero background, the tangent graph is the original graph, recovering the standard linear response. With a constant native curvature cosh(ε₀) = 1 + hbar, the response scale becomes q divided by 2(1 + hbar), a factor that the framework identifies with its fundamental constant hbar. The framework proves that this constant-curvature tangent model satisfies the scaled source equation, and that the resulting Green field attaches correctly to a posting event on a three-dimensional torus.

In plain language, the framework shows that the nonlinear cost function has a well-behaved local approximation: near any background, perturbations respond linearly, and the response scale is controlled by the curvature. The exact nonlinear law itself is not solved in closed form, but the tangent structure is fully analyzed. The framework models this as a discrete Gauss law for recognition events, with the constant hbar entering as the native curvature scale. This is a definitional choice within the framework, not a derivation from external physics.

/-- Exact action stationarity is equivalent to the pointwise nonlinear Gauss
equation. No source scale is selected by the equivalence. -/
theorem exactJStationaryAtSource_iff_gauss
    {n : ℕ}
    (G : WeightedLedgerGraph n)
    (field source : Fin n → ℝ)
    (coupling : ℝ) :
    ExactJStationaryAtSource G field source coupling ↔
      ExactJNonlinearGaussEquation G field source coupling := by
  classical
  constructor
  · intro h i
    have hi := h (fun j => if j = i then (1 : ℝ) else 0)
    rw [exactJFirstVariationPairing_eq_two_sum_mul_laplacian] at hi
    simpa [ExactJNonlinearGaussEquation] using hi
  · intro h variation
    rw [exactJFirstVariationPairing_eq_two_sum_mul_laplacian]
    unfold ExactJNonlinearGaussEquation at h
    calc
      2 * ∑ i,
          variation i * exactJNonlinearLaplacian G field i
          =
        ∑ i,
          variation i *
            (2 * exactJNonlinearLaplacian G field i) := by
              rw [Finset.mul_sum]
              apply Finset.sum_congr rfl
              intro i _
              ring
      _ = ∑ i,
          variation i * (coupling * source i) := by
            apply Finset.sum_congr rfl
            intro i _
            rw [h i]
      _ = coupling * ∑ i, variation i * source i := by
            rw [Finset.mul_sum]
            apply Finset.sum_congr rfl
            intro i _
            ring

/-- Hessian graph of exact J at a background field. -/
def exactJTangentGraph
    {n : ℕ}
    (G : WeightedLedgerGraph n)
    (background : Fin n → ℝ) :
    WeightedLedgerGraph n where
  weight i j :=
    G.weight i j *
      Real.cosh (background i - background j)
  weight_nonneg i j :=
    mul_nonneg (G.weight_nonneg i j)
      (le_of_lt (Real.cosh_pos _))
  weight_symm := by
    intro i j
    rw [G.weight_symm, ← Real.cosh_neg]
    congr 2
    ring

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 framework does not assert that the constant-curvature graph is realized by a global native-drop background. The framework does not derive the value of hbar from first principles; it takes it as a constant of the framework. The nonlinear Gauss law is not solved in closed form; only the tangent (linearized) structure is analyzed.

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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