CCFU Proof 19 — Explicit A₄: Companion Matrix from C₂

2605195700865

CCFU Proof 19 — Explicit A₄: Companion Matrix from C₂

05/19/2026

Given.
spec(A₂) = {φ, −1/φ}, where φ = (1+√5)/2.

By Proof 17 (factor completion), the canonical four-mode spectrum is:
    {φ, 1/φ, +1, −1}

Step 1 — Characteristic polynomial.

    p(λ) = (λ − φ)(λ − 1/φ)(λ − 1)(λ + 1)
         = (λ² − √5·λ + 1)(λ² − 1)
         = λ⁴ − √5·λ³ + √5·λ − 1  ∎

Step 2 — Companion matrix.

    A₄ = [[0, 0, 0, 1],
          [1, 0, 0, −√5],
          [0, 1, 0, 0],
          [0, 0, 1, √5]]

This is the companion matrix of p(λ).
By construction, spec(A₄) = {φ, 1/φ, +1, −1}.  ∎

Step 3 — Invariant form.
Let D = diag(φ, 1/φ, +1, −1) and define:

    G̃ = [[0, 1, 0, 0],
         [1, 0, 0, 0],
         [0, 0, 1, 0],
         [0, 0, 0, 1]]

    sig(G̃) = (3,1).

Every nonzero entry G̃ᵢⱼ satisfies λᵢλⱼ = 1, so DᵀG̃D = G̃.
Let P be the eigenvector matrix (A₄ = PDP⁻¹, invertible
since the four eigenvalues are distinct).
Define G₄ = P⁻ᵀG̃P⁻¹.
Then A₄ᵀG₄A₄ = G₄.  ∎

Step 4 — Determinant.

    det(A₄) = φ · (1/φ) · 1 · (−1) = −1.

For later use.
Since A₄ᵀG₄A₄ = G₄, also (A₄²)ᵀG₄A₄² = G₄.
Moreover det(A₄²) = 1
and spec(A₄²) = {φ², 1/φ², 1, 1}.
Thus A₄² is a G₄-orthogonal, determinant-one map with
positive hyperbolic spectrum. This is the boost component
used in the translation-length proof (Proof 12).

Conclusion.
A₄ is an explicit real 4×4 companion matrix,
constructed from spec(A₂) alone via the factor completion
of Proof 17. It preserves G₄ with sig(3,1) and has det = −1.
No free numerical input is introduced; the √5 scale is
forced by Δ(C₂) = 5.

This closes the external dependency of Proof 4 on Theory #15c.

[Dependency: Proof 17. No external references.]