Constants Curvature Cost Form Local Jcost Hessian Coefficient Eq One

The Hessian coefficient

The declaration localJCostHessianCoefficient_eq_one is a machine-checked theorem stating that a specific coefficient in the framework's curvature cost is exactly 1. This coefficient is the second derivative, or Hessian, of the canonical reciprocal cost function at its equilibrium point. In plain terms, it measures how the cost of a small geometric deformation grows with the square of that deformation's size.

The theorem is part of a larger result about the boundary angle-defect cost. This cost is the price associated with the angular deficit at the boundary of a three-dimensional cell, a discrete version of curvature. The framework's library proves that this boundary cost, when expanded to quadratic order, is exactly 2λ², where λ is the deformation parameter. The factor of 2 comes from the Gauss-Bonnet coefficient, which equals the Euler characteristic of the cube boundary, and the quadratic dependence comes from the Hessian coefficient being 1.

In Recognition Science, this establishes the quadratic form used by the curvature cost J_curv. It proves that J_curv is precisely the boundary angle-defect cost, not the bulk Regge Dirichlet energy. The bulk energy is shown to be a zero mode under uniform scaling, meaning constant vertex potentials give zero energy, so it cannot carry the single-cell curvature cost. The boundary cost, by contrast, is the correct carrier.

What the theorem does not claim is equally important. It does not assert that the full nonlinear expression Jcost (1 + λ) equals λ². In fact, away from λ = -1, the exact value is λ² / (2(1+λ)). The theorem-grade statement is strictly about the Hessian, the quadratic approximation, not the complete function. The framework is explicit that the full nonlinear form remains a separate, unformalized object.

/-- The local J-cost Hessian coefficient is `1`. This imports the exact
local-algebra theorem `J(1+ε) = ε² / (2(1+ε))` through its standard Hessian
normalization. -/
theorem localJCostHessianCoefficient_eq_one :
    Foundation.JCostHessianC7.jcostHessianCoefficient = 1 :=
  Foundation.JCostHessianC7.jcostHessianCoefficient_eq_one

/-- The boundary angle-defect J-cost quadratic form is exactly `2 λ²`.

This is the form-level closure: the `2` comes from Gauss-Bonnet
(`χ(∂Q₃) = 2`) and the quadratic dependence comes from the Hessian of the
canonical reciprocal cost at equilibrium. -/
theorem boundaryCurvatureQuadraticCost_eq (lam : ℝ) :
    boundaryCurvatureQuadraticCost lam = 2 * lam ^ (2 : ℕ) := by
  unfold boundaryCurvatureQuadraticCost boundaryDefectCoefficient
  rw [curvatureCoefficient_eq_euler_char, localJCostHessianCoefficient_eq_one]
  norm_num [euler_S2]

/-- Constant vertex potentials are zero modes of the canonical Dirichlet
energy. This is the formal reason the bulk Regge Hessian does not carry the
single-cell uniform-scale curvature cost: the Dirichlet quadratic only sees
differences `ξ i - ξ j`. -/
theorem canonicalDirichletEnergy_constant_zero
    (K : Triangulation3D) (hK : IncidenceConsistent K) (c : ℝ) :
    canonicalDirichletEnergy K hK (fun _ : Fin K.nV => c) = 0 := by
  unfold canonicalDirichletEnergy
  simp

/-- The boundary coefficient is the Euler characteristic of the cube boundary. -/
theorem boundaryDefectCoefficient_eq_euler_char :
    boundaryDefectCoefficient = (euler_S2 : ℝ) :=
  curvatureCoefficient_eq_euler_char

/-- The existing `J_curv` definition agrees with the theorem-derived boundary
quadratic form. This isolates the only intended meaning of `J_curv`: it is the
quadratic boundary angle-defect J-cost, not the bulk Regge Dirichlet energy and
not the full nonlinear `Jcost (1+λ)`. -/
theorem J_curv_eq_boundaryCurvatureQuadraticCost (lam : ℝ) :
    LambdaRecDerivation.J_curv lam = boundaryCurvatureQuadraticCost lam := by
  rw [LambdaRecDerivation.J_curv_derivation, boundaryCurvatureQuadraticCost_eq]

What this page does not claim

The full nonlinear expression Jcost (1 + λ) equals λ². The bulk Regge Dirichlet energy contributes to the single-cell curvature cost. The Hessian coefficient theorem applies to any geometry other than the cube boundary.

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/Constants/CurvatureCostForm.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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