CCFU Proof 3 — Conformal Preservation J(0)ᵀGJ(0) = βG

2605155663445

CCFU Proof 3 — Conformal Preservation J(0)ᵀGJ(0) = βG

05/15/2026

Given.
The nonlinear memory map on C² = R⁴:

    F(z,w) = (z² + βw, z),  β ∈ ℝ, β ≠ 0.

Write z = x₁ + ix₂, w = x₃ + ix₄.

Jacobian at the origin (z=0, w=0):

    J(0) = [[0, 0, β, 0],
            [0, 0, 0, β],
            [1, 0, 0, 0],
            [0, 1, 0, 0]]

Bilinear form.  Define Q = x₁x₃ + x₂x₄, represented by:

    G = [[0, 0, 1, 0],
         [0, 0, 0, 1],
         [1, 0, 0, 0],
         [0, 1, 0, 0]]

    sig(G) = (2,2).

Note: under the convention vᵀGv, this matrix represents 2Q.
The scalar factor does not affect signature or conformal preservation.

Claim.
    J(0)ᵀ G J(0) = β·G  for all β ∈ ℝ, β ≠ 0.

Proof.

    J(0)ᵀ = [[0, 0, 1, 0],
              [0, 0, 0, 1],
              [β, 0, 0, 0],
              [0, β, 0, 0]]

    J(0)ᵀ · G = [[1, 0, 0, 0],
                  [0, 1, 0, 0],
                  [0, 0, β, 0],
                  [0, 0, 0, β]]

    (J(0)ᵀ · G) · J(0) = [[0, 0, β, 0],
                            [0, 0, 0, β],
                            [β, 0, 0, 0],
                            [0, β, 0, 0]]

                         = β · G  ∎

Signature.
Eigenvalues of G: +1, +1, −1, −1.  sig(G) = (2,2).
Since sig(2,2) is symmetric under sign reversal, β < 0 preserves the signature class.

Limitations.
β = 0 gives 0·G (degenerate). Result holds for β ≠ 0.
This is conformal preservation at the fixed point.
The full nonlinear map does not preserve Q globally:

    Q(F(z,w)) = |z|²Re(z) + βQ(z,w) ≠ βQ(z,w) in general.

Status.

[PROVEN]
J(0)ᵀGJ(0) = βG, β ∈ ℝ, β ≠ 0.

[DISPROVEN FOR THIS Q]
Global nonlinear preservation Q(F(z,w)) = βQ(z,w).

[RESOLVED LATER — Proof 15]
The differential signature is globally preserved:
sig(J(z)ᵀGJ(z)) = (2,2) for all z, β ≠ 0.

[OPEN]
Whether a modified global form-level invariant exists.