Constants Curvature Cost Form Canonical Dirichlet Energy Constant Zero

The flat direction

The theorem canonicalDirichletEnergy_constant_zero concerns a ledger, a discrete record of events, where each event carries a number called a potential. The energy in question, the canonical Dirichlet energy, measures how much neighboring potentials differ. The theorem states that if every potential in the ledger is the same constant c, then the energy is exactly zero. This holds for any triangulated three-dimensional structure that satisfies the framework's incidence consistency condition.

The reason is plain: the Dirichlet quadratic only sees differences between neighboring potentials. If all potentials are equal, every difference is zero, so the sum of squared differences is zero. This is not a special property of the framework's geometry; it is a general fact about any graph Laplacian. The theorem gives it a formal name and a machine-checked proof inside the framework's library of formal theorems.

The consequence is that the bulk Regge quadratic, a classical discretized gravity action, cannot be the source of the one-cell curvature cost under a uniform conformal scale. If you multiply all lengths by the same factor, the constant potential mode remains a zero mode, so the bulk energy does not respond to that uniform scaling. The curvature cost must come from somewhere else.

In Recognition Science, the framework models the boundary angle-defect cost as the true curvature cost. The boundary cost has a quadratic form: its coefficient is the Euler characteristic of the cube boundary, which is 2, and its Hessian coefficient from the canonical reciprocal cost is 1. The theorem proves that this boundary quadratic cost equals exactly 2λ², where λ is the scale perturbation. This is the form-level closure that the existing J_curv definition agrees with.

The theorem does not claim that the full nonlinear expression Jcost(1+λ) equals λ². In fact, away from λ = -1, the exact expression is λ² / (2(1+λ)). The theorem-grade statement is only about the Hessian, the quadratic approximation near equilibrium. It also does not claim that the bulk Regge energy is irrelevant to all curvature questions; it only shows that uniform scaling is a zero mode for it.

/-- 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 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) equals λ². The bulk Regge energy is irrelevant to all curvature questions, only that uniform scaling is a zero mode. The theorem derives the fine-structure constant or any other specific physical constant.

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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