Constants Lambda Rec Derivation J Curv Eq Coefficient Mul Sq

The curvature cost formula

The declaration J_curv_eq_coefficient_mul_sq is a theorem in the Recognition Science framework's machine-checked library of formal theorems. It states that the curvature cost, written J_curv(λ), equals the curvature coefficient times the square of the recognition length λ. The curvature coefficient is itself proved to equal the Euler characteristic of a sphere, which is 2. So the theorem's content is that J_curv(λ) = 2λ².

This is not a free choice. The framework derives it from a chain of forced steps. The cost functional J(x) = (x + 1/x)/2 - 1 is forced by the recognition composition law. Minimizing that cost on a discrete ledger forces three dimensions and the cube Q₃ as the elementary cell. Polyhedral Gauss-Bonnet on the cube's surface, which is topologically a sphere, fixes the total curvature at 4π. The curvature cost then takes the form 2λ², with the coefficient 2 coming from the Euler characteristic, not from any fitted parameter.

The theorem's role is to set up a balance condition. The framework equates the curvature cost J_curv(λ) to the bit cost J_bit, which is normalized to 1. Solving 2λ² = 1 gives the unique positive solution λ = 1/√2. This is the recognition length in cost units. The theorem J_curv_eq_coefficient_mul_sq is the step that makes that balance equation well-defined, because it supplies the exact form of the curvature cost.

What the theorem does not claim is just as precise. It does not claim that the curvature coefficient is a free parameter; the coefficient is forced by the Euler characteristic. It does not claim that G, the gravitational constant, appears anywhere in the derivation; the chain runs from the composition law to the balance condition with no G, ℏ, or c. It does not claim that the recognition length is a measured quantity; it is a derived root of a forced equation. The theorem is a formal statement about the curvature cost's shape, not an empirical claim about the universe.

In Recognition Science, this theorem is a load-bearing step in the derivation of the recognition length λ_rec. It shows that the curvature cost has a specific quadratic form, which then allows the balance condition to have a unique positive solution. The theorem's contribution is to pin down the exact coefficient in that quadratic form, making the subsequent derivation of λ_rec non-circular.

/-- `J_curv(λ) = curvatureCoefficient · λ²`: the curvature cost is the
Gauss-Bonnet coefficient times the squared embedding scale. -/
theorem J_curv_eq_coefficient_mul_sq (lambda : ℝ) :
    J_curv lambda = curvatureCoefficient * lambda ^ 2 := by
  unfold J_curv
  rw [curvatureCoefficient_eq_euler_char]
  norm_num [euler_S2]

/-- **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

/-- 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

/-- lambda_0 is the unique positive root of the balance residual. -/
theorem balance_unique_positive_root (lambda : ℝ) (hlambda : lambda > 0) :
    balanceResidual lambda = 0 ↔ lambda = lambda_0 := by
  unfold balanceResidual J_curv J_bit_normalized lambda_0
  constructor
  · intro h
    have hsq : lambda ^ 2 = 1 / 2 := by linarith
    have hlam_sqrt : lambda = Real.sqrt (1 / 2) := by
      rw [← Real.sqrt_sq (le_of_lt hlambda), hsq]
    rw [hlam_sqrt, Real.sqrt_div (by norm_num : (0:ℝ) ≤ 1), Real.sqrt_one]
  · intro h
    rw [h, div_pow, Real.sq_sqrt (by norm_num : (0:ℝ) ≤ 2)]
    ring

What this page does not claim

The theorem does not claim that the curvature coefficient is a free parameter; it is forced by the Euler characteristic. The theorem does not claim that G, ℏ, or c appear anywhere in the derivation of J_curv. The theorem does not claim that the recognition length is a measured quantity; it is a derived root of a forced equation.

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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