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Status: exact contraction theorem; original visual contains a decomposition error documented below.

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First-principles formalism

Set

$$ A=K^\dagger K. $$

Because $K$ is Hermitian,

$$ A=K^2,\qquad \ker A=\ker K=E_{47}. $$

For

$$ 0<\varepsilon<\frac{2}{\|A\|}, $$

define

$$ T=I-\varepsilon A. $$

Then

$$ T^n\longrightarrow P_{47}, $$

and

$$ x_{n+1}=(I-\varepsilon K^\dagger K)x_n\longrightarrow P_{47}x_0. $$

Logical deductions

• Kernel components are fixed.
• Complement components decay geometrically.
• The map preserves diagonal $SU(2)$ symmetry because $K$ is a polynomial in $C$.
• Exact projection discards information in the $78$-dimensional complement.