CCFU Proof 18 — Uniqueness of C₂ Among ±1 Recurrences

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CCFU Proof 18 — Uniqueness of C₂ Among ±1 Recurrences

05/17/2026

Given.
All second-order recurrences x(n+1) = a·x(n) + b·x(n-1)
with a, b ∈ {+1, −1}.

Claim.
C₂ (i.e., a=b=1) is, up to isomorphism, the unique such
recurrence with real hyperbolic roots and a reciprocal
magnitude pair.

Proof. There are exactly four cases.

Case 1: (a,b) = (1,1).
    λ² − λ − 1 = 0,  Δ = 5,  roots: φ, −1/φ.
    Real roots, reciprocal magnitudes {φ, 1/φ}. ✓

Case 2: (a,b) = (−1,1).
    λ² + λ − 1 = 0,  Δ = 5,  roots: 1/φ, −φ.
    Isomorphic to Case 1 via x_n → (−1)ⁿ x_n.
    Same magnitudes, same chain. ✓

Case 3: (a,b) = (1,−1).
    λ² − λ + 1 = 0,  Δ = −3,  complex roots, |λ| = 1.
    No real reciprocal pair. Chain breaks at step 1. ✗

Case 4: (a,b) = (−1,−1).
    λ² + λ + 1 = 0,  Δ = −3,  complex roots, |λ| = 1.
    No real reciprocal pair. Chain breaks at step 1. ✗

Conclusion.
Only Cases 1 and 2 yield Δ = 5, real hyperbolic roots,
and a reciprocal magnitude pair. These two are isomorphic.
Cases 3 and 4 give Δ = −3 with complex roots on the unit
circle; no growth/decay split, no real reciprocal pair,
no factor completion.

Therefore C₂ is, up to isomorphism, the unique minimal
parameter-free second-order recurrence whose spectral chain
reaches sig(3,2).  ∎

[No dependencies. Self-contained.]