Constants Curvature Cost Form Boundary Curvature Quadratic Cost Eq

The boundary curvature cost

In the Recognition Science framework, the curvature of a discrete geometric object is not a free quantity but a cost. The declaration boundaryCurvatureQuadraticCost_eq establishes the precise quadratic form of that cost for a boundary angle defect. It states that the boundary curvature cost for a small angular deviation λ is exactly 2λ². The factor of 2 comes from the Gauss-Bonnet theorem applied to the boundary of a three-dimensional cell, whose Euler characteristic is 2. The quadratic dependence on λ comes from the Hessian, or local curvature, of the fundamental reciprocal cost function at its equilibrium point.

This theorem is a form-level closure. It confirms that the existing definition J_curv, used elsewhere in the framework, is precisely this quadratic boundary angle-defect cost and nothing else. The theorem does not claim the full nonlinear expression Jcost(1 + λ) is exactly λ². In fact, away from λ = -1, that expression equals λ² / (2(1+λ)). The theorem-grade statement is limited to the Hessian and quadratic form. It also explicitly excludes the bulk Regge Dirichlet energy, which is shown to be a zero mode under uniform scaling, meaning it cannot carry this curvature cost.

What this establishes in practice is a clean, provable foundation for one piece of the framework's curvature cost. It gives a concrete, checkable formula for a boundary term. It does not, however, claim to have derived the full nonlinear curvature cost, nor does it claim the bulk energy is irrelevant in all contexts. The theorem is a precise, narrow statement about a quadratic approximation at the boundary.

/-- 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

/-- 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

This theorem does not claim the full nonlinear expression Jcost(1 + λ) is exactly λ². This theorem does not claim the bulk Regge Dirichlet energy is irrelevant in all contexts. This theorem does not claim to have derived the complete curvature cost for all geometric configurations.

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)

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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