Constants Curvature Cost Form Boundary Defect Coefficient Eq Euler Char

The boundary coefficient

The Euler characteristic is a number that describes a shape's global topology: for any surface topologically equivalent to a sphere, it equals 2. This number is a classical invariant, unchanged by stretching or bending, and it appears across geometry, from the vertices, edges, and faces of a polyhedron to the curvature of a smooth surface. The declaration boundaryDefectCoefficient_eq_euler_char is a theorem in the machine-checked library of formal theorems stating that, in the Recognition Science framework, the boundary defect coefficient equals this Euler characteristic of the sphere, which is 2.

The boundary defect coefficient is a framework-native quantity: it measures the total angular defect, the amount by which angles fail to sum to a full turn, along the boundary of a single three-dimensional cell. The theorem establishes that this coefficient is exactly the Euler characteristic of the cell's boundary surface, which for a cube is 2. This is a precise, formal statement: the coefficient, defined as an abbreviation for a curvature coefficient, is proved equal to the Euler characteristic through a chain of definitions in the library.

This single equality is the load-bearing piece of a larger quadratic-form result. The library also proves that the boundary curvature cost, the energy associated with this angular defect, takes the exact form 2λ², where λ is a small deformation parameter. The factor 2 comes directly from the Euler characteristic, and the quadratic dependence comes from the Hessian, the second-order approximation, of the canonical reciprocal cost at equilibrium. This quadratic form is what the framework's curvature cost J_curv is defined to be.

In Recognition Science, this result is part of a bridge: it shows that the bulk energy, the Dirichlet energy over the cell's interior, cannot be the source of the single-cell curvature cost, because constant vertex potentials are zero modes of that energy. The boundary angle-defect cost, by contrast, has the correct quadratic form. The theorem closes the quadratic form, not the full nonlinear expression.

What the declaration does not claim is equally precise. The full nonlinear cost Jcost(1+λ) is not equal to λ²; the library explicitly records that Jcost(1+λ) = λ² / (2(1+λ)) away from λ = -1. The theorem-grade statement is only the Hessian or quadratic-form statement. The framework's own documentation marks this boundary: the theorem below closes the quadratic form used by J_curv; it does not claim the full nonlinear expression is exactly λ².

/-- 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 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 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

This does not claim the full nonlinear expression Jcost(1+λ) equals λ². This does not claim the bulk Dirichlet energy contributes to the single-cell curvature cost. This does not claim the Euler characteristic is derived from the framework; it is a classical topological invariant.

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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