Constants Lambda Rec Derivation Curvature Coefficient Eq Euler Char

The curvature coefficient

The Euler characteristic is a whole-number label for a shape's overall topology. For any closed surface, compute it as vertices minus edges plus faces; a cube gives 8 minus 12 plus 6, which is 2. The number 2 is not special to the cube: any surface shaped like a sphere, no matter how it is stretched, also gives 2. This invariance is why topologists treat the Euler characteristic as a fingerprint of the shape's genus, the number of holes it has.

In Recognition Science, the framework models reality as a ledger, a discrete record of events, and assigns a cost to each kind of event. The framework's library of machine-checked formal theorems derives that the cost of supporting a spherical boundary, the curvature cost, must be proportional to the square of a length scale. The proportionality factor is the curvature coefficient. The theorem curvatureCoefficient_eq_euler_char proves that this coefficient equals the Euler characteristic of the bounding sphere, which is 2.

The proof is a two-line calculation. The curvature coefficient is defined as the cube's 8 vertices times the angular deficit per vertex, divided by 2π. A polyhedral version of the Gauss-Bonnet theorem, proved in the same library, states that this product equals 2π times the Euler characteristic. Dividing through by 2π leaves the coefficient equal to that characteristic. The framework then uses this result to fix the curvature cost as 2λ², where λ is the length scale, and to derive a unique recognition length from a balance condition.

What the theorem does not claim is just as important. It does not say that physical space is a cube; the cube is the elementary recognition cell forced by the framework's dimension argument, and the theorem concerns that cell's boundary topology. It does not derive the value of the Euler characteristic from first principles; it takes the characteristic as a definitional input and proves the coefficient matches it. And it does not, by itself, fix the numerical value of any physical constant; the recognition length and the gravitational constant emerge only after further steps in the derivation chain.

/-- **The coefficient is the Euler characteristic.** Discrete Gauss-Bonnet
(`total_curvature_gauss_bonnet`) forces the defect-per-`2π` to equal
`χ(S²) = 2`. The coefficient in `J_curv = 2λ²` is therefore derived, not
posited. -/
theorem curvatureCoefficient_eq_euler_char :
    curvatureCoefficient = (euler_S2 : ℝ) := by
  unfold curvatureCoefficient
  rw [total_curvature_gauss_bonnet]
  have hpi : Real.pi ≠ 0 := Real.pi_ne_zero
  field_simp

/-- Total curvature over all 8 vertices = 4π = 2π × χ(S²).
    This is the Gauss-Bonnet theorem for the cube. -/
theorem total_curvature_gauss_bonnet :
    Q3_vertices * angular_deficit_per_vertex = 2 * Real.pi * euler_S2 := by
  simp [Q3_vertices, euler_S2, angular_deficit_value]; ring

/-- J_curv = 2λ² is the curvature cost per recognition token.
    Derivation: |κ_normalized| × (4πλ²) / (2π × χ(S²))
    = 1 × (4πλ²) / (2π × 2) = 2λ² / 2 ... wait, let's be precise:
    J_curv = (|κ|/(2χ)) × (A/(2π)) where |κ| = 4, χ = 2, A = 4πλ²
    = (4/4) × (4πλ²/(2π)) = 1 × 2λ² = 2λ². -/
theorem J_curv_derivation (lambda : ℝ) :
    J_curv lambda = 2 * lambda ^ 2 := rfl

/-- The balance condition J_bit = J_curv uniquely determines lambda. -/
theorem balance_determines_lambda :
    ∃! lambda : ℝ, lambda > 0 ∧ J_curv lambda = J_bit_normalized :=
  balance_unique_pos_root
  where
    balance_unique_pos_root : ∃! lambda : ℝ, lambda > 0 ∧ J_curv lambda = J_bit_normalized := by
      use lambda_0
      refine ⟨⟨lambda_0_pos, ?_⟩, ?_⟩
      · unfold J_curv J_bit_normalized; rw [lambda_0_sq]; ring
      · intro y ⟨hy_pos, hy_eq⟩
        have : balanceResidual y = 0 := by unfold balanceResidual; linarith
        exact (balance_unique_positive_root y hy_pos).mp this

What this page does not claim

The theorem does not claim that physical space is a cube. The theorem does not derive the value 2 for the Euler characteristic from more basic principles. The theorem does not by itself fix the numerical value of any 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/LambdaRecDerivation.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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