CCFU Proof 17 — Two Canonical Four-Mode Completions of C₂

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CCFU Proof 17 — Two Canonical Four-Mode Completions of C₂

05/17/2026

Given.
The C₂ companion matrix and its spectrum:

    A₂ = [[1, 1],
          [1, 0]]

    λ² − λ − 1 = 0

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

Completion 1 — Factor completion (sign/magnitude).

Every nonzero real eigenvalue has a unique factorization
λ = sign(λ) · |λ|:

    φ    = (+1) · φ
    −1/φ = (−1) · (1/φ)

Extract: signs {+1, −1}, magnitudes {φ, 1/φ}.
Union:

    spec₁ = {φ, 1/φ, +1, −1}

Reciprocal pairing (λᵢλⱼ = 1):
    (φ, 1/φ): sig(1,1)
    +1 self-reciprocal: sig(1,0)
    −1 self-reciprocal: sig(1,0)

Total:
    sig₁ = (1,1) + (1,0) + (1,0) = (3,1)  ∎

Completion 2 — Symmetry completion (±-closure).

Close the magnitudes {φ, 1/φ} under negation (λ → −λ):

    spec₂ = {φ, −φ, 1/φ, −1/φ}

Reciprocal pairing:
    (φ, 1/φ): sig(1,1)
    (−φ, −1/φ): sig(1,1)

Total:
    sig₂ = (1,1) + (1,1) = (2,2)  ∎

Note on terminology.
Completion 1 is a unique real factorization, not a group closure.
Completion 2 is a group-theoretic closure under negation.
Both are canonically determined by spec(A₂) with no free choices.

Minimal parent.

Completion 1 gives sig(3,1). Completion 2 gives sig(2,2).

To contain sig(3,1): p ≥ 3, q ≥ 1.
To contain sig(2,2): p ≥ 2, q ≥ 2.
Together: p ≥ 3, q ≥ 2.
Minimal: sig(3,2), dim = 5.
Sufficiency: sig(3,2) contains sig(3,1) by dropping one negative
direction, and sig(2,2) by dropping one positive direction.
The lower bound is attained.

This recovers the Minimal Parent Theorem (Proof 5) with both
signatures now derived from spec(A₂) alone.  ∎

[No dependencies. Self-contained. Recovers Proofs 4 and 5.]