Constants Curvature Cost Form

The curvature cost form

In the Recognition Science framework, the curvature cost form is the precise quadratic expression for the cost of a small angular defect at the boundary of a single cell. The framework establishes that this cost is exactly 2λ², where λ is the angular defect measured in units of one full turn. The number 2 comes from the Euler characteristic of a cube's boundary, which is 2, and the quadratic dependence comes from the Hessian of the canonical reciprocal cost at equilibrium.

The framework's ledger, a discrete record of recognition events, assigns costs to geometric configurations. The framework first shows that the bulk Regge/Dirichlet quadratic energy cannot be the source of this curvature cost: constant vertex potentials are zero modes of the graph Laplacian, meaning the Dirichlet quadratic only sees differences between vertices, not uniform scaling. This is the formal reason the bulk term does not carry the single-cell uniform-scale curvature cost.

The boundary angle-defect cost, however, has a theorem-tier quadratic form. The local J-cost Hessian coefficient is 1, and the Gauss-Bonnet defect coefficient is χ(∂Q₃) = 2, so the quadratic boundary cost is 2λ². This is a proved theorem in the machine-checked library of formal theorems, stated as boundaryCurvatureQuadraticCost_eq.

The framework also proves that the existing J_curv definition agrees exactly with this 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). The honest boundary is that the theorem closes the quadratic form, not the full nonlinear expression, which is λ² / (2(1 + λ)) away from λ = -1.

What this means in plain language: the framework has a precise, machine-checked answer for the cost of a small angular defect at a cell boundary. The cost is quadratic in the defect, with a coefficient fixed by topology (the Euler characteristic) and by the Hessian of the canonical cost. This is a form-level closure that pins down the quadratic behavior without overclaiming the full nonlinear expression.

/-- 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 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

The full nonlinear expression Jcost (1 + λ) is exactly λ²; the theorem only closes the quadratic form. The bulk Regge Dirichlet energy contributes to the single-cell uniform-scale curvature cost; it is a zero mode. The curvature cost form is derived from the forcing chain or the golden ratio; it is a separate module.

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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