Constants Curvature Cost Form Boundary Curvature Quadratic Cost

The boundary curvature cost

In the Recognition Science framework, the curvature of a discrete three-dimensional cell is assigned a cost. The declaration boundaryCurvatureQuadraticCost establishes the precise quadratic form of this cost: for a boundary angle-defect parameter λ, the cost is exactly 2λ². The factor of 2 comes from the Gauss-Bonnet theorem for the boundary of a cube, whose Euler characteristic is 2. The quadratic dependence on λ comes from the Hessian, or second-order approximation, of the framework's canonical reciprocal cost function at its equilibrium point, where that Hessian coefficient is exactly 1.

The declaration also clarifies what the curvature cost is not. It is not the bulk Regge or Dirichlet energy, which measures differences between neighboring vertex potentials and therefore vanishes for a constant potential, a fact proved in the same module. The quadratic form is specifically a boundary effect, tied to the angle defect on the surface of the cell. In addition, the declaration explicitly states that it does not claim the full nonlinear expression Jcost(1+λ) is exactly λ². In fact, away from λ = -1, the full expression is λ² / (2(1+λ)). The theorem-grade statement is confined to the quadratic form, which is the Hessian-level approximation.

The practical consequence is that the existing J_curv definition, used elsewhere in the framework, is now isolated to mean exactly this quadratic boundary angle-defect cost. This is a closure result: it ties a previously defined quantity to a theorem-derived form, ensuring that when J_curv appears, its meaning is unambiguous and its provenance is a machine-checked proof, not a definitional choice.

/-- 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 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) is not proved to equal λ²; the theorem covers only the quadratic form. The boundary curvature cost is not the bulk Regge or Dirichlet energy. This result does not derive the fine-structure constant or any other specific physical coupling.

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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