CCFU Proof 4 — sig(G₄) = (3,1)

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CCFU Proof 4 — sig(G₄) = (3,1)

05/15/2026

Given.
spec(A₄) = {φ, 1/φ, +1, −1}

This spectrum is derived from C₂ by separating
spec(A₂) = {φ, −1/φ} into signs and magnitudes.
[Dependency: Theory #15c / CCFU Paper I]

Construction.  Define G̃ in the eigenbasis by the reciprocal pairing rule:

    G̃[i,j] = 1 if λᵢ · λⱼ = 1, else 0.

Reciprocal pair (φ, 1/φ):
    φ · (1/φ) = 1  →  off-diagonal block [[0,1],[1,0]]
    Eigenvalues of this block: +1, −1
    Contribution to signature: (1,1)

Self-reciprocal modes (+1 and −1):
    (+1)·(+1) = 1  →  diagonal entry +1
    (−1)·(−1) = 1  →  diagonal entry +1
    Contribution to signature: (2,0)

Total signature:
    (1,1) + (2,0) = (3,1)

Therefore sig(G₄) = (3,1).  ∎

Verification of invariance.
In the eigenbasis, D = diag(φ, 1/φ, +1, −1).
For every nonzero entry G̃ᵢⱼ, we have λᵢλⱼ = 1, hence
    (DᵀG̃D)ᵢⱼ = λᵢλⱼ G̃ᵢⱼ = G̃ᵢⱼ.
All zero entries remain zero. Therefore DᵀG̃D = G̃.

Standard basis.
If A₄ = PDP⁻¹, define G₄ = P⁻ᵀG̃P⁻¹. Then:
    A₄ᵀG₄A₄ = (PDP⁻¹)ᵀ(P⁻ᵀG̃P⁻¹)(PDP⁻¹)
             = P⁻ᵀDᵀG̃DP⁻¹
             = P⁻ᵀG̃P⁻¹
             = G₄.  ∎