CCFU Proof 2 — The Lorentzian Hyperboloid Identity

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CCFU Proof 2 — The Lorentzian Hyperboloid Identity

05/14/2026

Given.
The second-order linear recurrence:

    C₂:  x(n+2) = x(n+1) + x(n)

Assume x(n) ≠ 0.  Let R = x(n+1)/x(n) ∈ ℝ.

Define.

    U₁ = (2R − 1) / √5
    U₂ = (R + 2) / √5
    V  = R

Claim.

    U₁² + U₂² − V² = 1

Proof.

    U₁² + U₂² = ((2R−1)² + (R+2)²) / 5
              = (4R²−4R+1 + R²+4R+4) / 5
              = (5R² + 5) / 5
              = R² + 1

    U₁² + U₂² − V² = R² + 1 − R² = 1  ∎

Note.
This is a one-parameter Lorentzian hyperboloid identity, not a full H² embedding with metric and curvature. For every real R, the point (U₁, U₂, V) lies on the unit hyperboloid U₁² + U₂² − V² = 1 in signature (2,1). In the positive C₂ ratio dynamics, R > 0. The normalization √5 is the C₂ discriminant scale. No free parameters are chosen.

Corollary — C₂ defines a null ruling.

Write (U₁, U₂, V) = P + R·D, where

    P = (−1/√5, 2/√5, 0)
    D = (2/√5, 1/√5, 1)

Under the Lorentzian form ⟨u,v⟩ = u₁v₁ + u₂v₂ − u₃v₃:

    ⟨P,P⟩ = 1
    ⟨D,D⟩ = 0
    ⟨P,D⟩ = 0

Therefore ⟨P+RD, P+RD⟩ = 1 for all R ∈ ℝ. The direction D is null; the C₂ ratio curve is a null ruling on the Lorentzian hyperboloid.  ∎