The typed calculation behind the ten-generator claim

Furey participation: closure certificate

This page exposes the assumptions, proof chain, certificate files and authority boundary behind the chapter claim.

Claim under audit

On the declared four-real-dimensional response carrier, the three typed generators \(\mathcal A\), \(\mathcal H\) and \(\mathcal C\) generate all of \(\mathfrak{sp}(4,\mathbb R)\). The Furey-shaped source product exponentiates to \(\mathcal C\) on the aggregate channel.

Start with the picture

Think of three control knobs acting on the same two-channel instrument. One turns phase, one stretches the two channels in opposite ways, and one continuously mixes the channels. The certificate checks that switching these controls in different orders produces ten independent motions—the full set allowed by the declared symplectic carrier.

Objects and assumptions

Carrier	\(\mathbb R^2_{\mathrm{phase}}\otimes\mathbb R^2_{\mathrm{channel}}\), with every generator written in the same ordered basis.
Preserved form	\(\Omega=J\otimes I\), where \(J\) is the two-dimensional quarter-turn matrix.
Typed generators	\(\mathcal A=J\otimes e_-\), \(\mathcal H=K\otimes D\) and \(\mathcal C=I\otimes\varepsilon\), with \(e_-=(I-D)/2\).
Meaning of closure	Real linear span after repeated matrix commutators \([X,Y]=XY-YX\).

Certificate chain

Commuting \(\mathcal C\) with \(\mathcal A\) supplies \(J\otimes X\); commuting once more supplies \(J\otimes D\). Combining \(J\otimes D\) with \(\mathcal A\) supplies \(J\otimes I\).
Commuting \(\mathcal C\) with \(\mathcal H\) supplies \(K\otimes X\).
The phase turn \(J\otimes I\) rotates \(K\) into \(X\), supplying \(X\otimes D\) and \(X\otimes X\).
Further commutators supply \(K\otimes I\) and \(X\otimes I\). Together with the original terms, these form \(I\otimes\varepsilon\) plus \(\{J,X,K\}\otimes\{I,X,D\}\).
Those ten matrices are linearly independent and preserve \(\Omega\). Since \(\dim\mathfrak{sp}(4,\mathbb R)=10\), the generated algebra is exactly \(\mathfrak{sp}(4,\mathbb R)\).

Necessity check

Input set	Closure dimension	Reading
Remove \(\mathcal A\)	3	The Jordan translation is necessary.
Remove \(\mathcal H\)	4	The split dilation is necessary.
Remove \(\mathcal C\)	4	The continuous channel rotation is necessary.
Keep all three	10	Full \(\mathfrak{sp}(4,\mathbb R)\) closure.

The source flow reaches the channel generator

For the source-space product \(G_{\mathrm{src}}=F_Cp\), the certificate checks \(G_{\mathrm{src}}^{\mathsf T}=-G_{\mathrm{src}}\) and \(G_{\mathrm{src}}^2=-I\), hence \(\exp(tG_{\mathrm{src}})=\cos(t)I+\sin(t)G_{\mathrm{src}}\). The aggregate map \(A\) obeys \(AG_{\mathrm{src}}=\varepsilon A\), so \(A\exp(tG_{\mathrm{src}})=\exp(t\varepsilon)A\). Realification is precisely the channel generator \(\mathcal C\).