Gravity Analysis Recognition Mesh Exact Jbridge4 D Exact Jsecond Diff Independen

The amplitude independence

In numerical analysis, a second difference measures how a quantity curves as its input changes. The declaration exactJSecondDiff_independent_of_amplitude concerns a specific action, a rule that assigns a number to a field configuration, defined on a discrete mesh, a finite grid of points that approximates a continuous space. The theorem states that for any nonzero amplitude, the second difference of this action is identical. Concretely, if the action is evaluated at amplitudes ε₁ and ε₂, both nonzero, the second difference is the same number in both cases. The proof is a short algebraic identity: the second difference simplifies to a quantity called the mesh Hessian, which does not depend on the amplitude at all.

This is a theorem in the framework's machine-checked library of formal theorems, meaning a computer has verified the proof. The result is not a statement about the physical world directly; it is a property of a mathematical object the framework has chosen to study. The action is defined as a quadratic form, half the square of the amplitude times a Hessian, and the second difference of a pure quadratic form is always its Hessian. The theorem confirms that this holds in the framework's specific construction, where the Hessian is the exact midpoint Bloch symbol, a term from the study of waves on periodic lattices.

What the theorem does not claim is equally important. It does not say that the action itself is independent of amplitude; the action grows with the square of the amplitude. It does not claim that the mesh Hessian equals the Hessian of a continuum gravitational action; that identification is a separate, open problem. The theorem also does not assert that the second difference is the actual second derivative of a physical action; it is a discrete object, and its limit as the mesh becomes infinitely fine is a different question, addressed by other theorems in the library.

In the framework's broader program, this theorem is a small but necessary step. It shows that a certain discrete construction has a well-defined curvature-like quantity that does not depend on how strongly the field is excited. This independence is what allows the framework to later ask whether that quantity converges to a continuum limit. Without it, the limit question would not be well-posed. The theorem, therefore, is a piece of groundwork: it makes a later, larger claim possible, but it does not itself establish that claim.

/-- The action is quadratic (not the previous definitional `0` shell):
its amplitude second difference is independent of `ε` for `ε ≠ 0`. -/
theorem exactJSecondDiff_independent_of_amplitude
    (M : RecognitionFreudenthalMesh4D) (m : IntMode4) (E : Mat4)
    {ε₁ ε₂ : ℝ} (h₁ : ε₁ ≠ 0) (h₂ : ε₂ ≠ 0) :
    exactJSecondDiff M m E ε₁ = exactJSecondDiff M m E ε₂ := by
  rw [exactJSecondDiff_eq_meshHessian M m E h₁,
    exactJSecondDiff_eq_meshHessian M m E h₂]

/-- **THEOREM:** for `ε ≠ 0` the amplitude second difference equals the
mesh true-Regge Hessian exactly (pure quadratic action). -/
theorem exactJSecondDiff_eq_meshHessian
    (M : RecognitionFreudenthalMesh4D) (m : IntMode4) (E : Mat4)
    {ε : ℝ} (hε : ε ≠ 0) :
    exactJSecondDiff M m E ε = meshTrueReggeQuadraticHessian M m E := by
  unfold exactJSecondDiff exactJActionOnMesh
  have hε2 : ε ^ 2 ≠ 0 := pow_ne_zero 2 hε
  field_simp [hε2]
  ring

/-- Recognition exact-J action on the mesh at amplitude `ε`.

MODEL: identified with the true-weight Regge quadratic Hessian on the
same edge-class perturbation `ε • E` (homogeneous of degree two in the
fold).  Not an ArbitraryPullback / TestVariationPullback substitute.
Schläfli elevation remains OPEN. -/
def exactJActionOnMesh (M : RecognitionFreudenthalMesh4D)
    (m : IntMode4) (E : Mat4) (ε : ℝ) : ℝ :=
  (1 / 2) * ε ^ 2 * meshTrueReggeQuadraticHessian M m E

What this page does not claim

The action itself is independent of amplitude; it grows with the square of the amplitude. The mesh Hessian equals the Hessian of a continuum gravitational action; that identification is open. The second difference is the second derivative of a physical action in the continuum.

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/Gravity/Analysis/RecognitionMeshExactJBridge4D.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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