Constants Curvature Cost Form J Curv Eq Boundary Curvature Quadratic Cost

The curvature cost form

The declaration J_curv_eq_boundaryCurvatureQuadraticCost establishes that the framework's curvature cost, written J_curv(λ), is exactly the quadratic form 2λ². Here λ is a small deformation parameter measuring how far a cell's geometry departs from its equilibrium. The theorem is a form-level closure: it identifies the boundary angle-defect cost, which is the cost associated with the total angular defect around a cell's boundary, as the source of J_curv, and it fixes the coefficient 2 through two independent facts.

The first fact is the Gauss-Bonnet coefficient. The boundary defect coefficient, which is the total angular defect in units of one full turn, equals the Euler characteristic of the cube boundary, which is 2. The second fact is the local J-cost Hessian coefficient, which is 1. The Hessian is the second derivative of the cost at equilibrium, and its value 1 comes from the exact local-algebra theorem J(1+ε) = ε² / (2(1+ε)) under standard normalization. Multiplying these gives the boundary curvature quadratic cost as 2λ², and the theorem proves that J_curv(λ) equals this expression.

The theorem also records a negative result about the bulk. The bulk Regge/Dirichlet quadratic energy, which is a standard discretization of curvature on a triangulated 3D manifold, cannot be the source of the one-cell curvature cost under a uniform conformal scale. Constant vertex potentials are zero modes of the canonical Dirichlet energy, meaning the Dirichlet quadratic only sees differences between neighboring vertices and vanishes for uniform scaling. This is the formal reason the bulk Hessian does not carry the single-cell uniform-scale curvature cost.

The declaration does not claim that the full nonlinear expression Jcost(1+λ) is exactly λ². In fact, Jcost(1+λ) = λ² / (2(1+λ)) away from λ = -1. The theorem-grade statement is only the Hessian/quadratic-form statement. The certificate structure CurvatureCostFormCert packages all five facts: the bulk zero mode, the boundary cost equality, the closed form 2λ², the Euler characteristic coefficient, and the Hessian coefficient being 1. 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 expression.

/-- 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]

/-- 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]

/-- 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 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

/-- 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 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]

What this page does not claim

The full nonlinear expression Jcost(1+λ) is exactly λ² for all λ. The bulk Regge Dirichlet energy is the source of the one-cell curvature cost under uniform conformal scale. The theorem applies to the full nonlinear expression Jcost(1+λ) away from λ = -1.

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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