Theorem: Sourced semantic constructor

On ${\mathcal{D}}_{\mathrm{src}}$, the rule

is well defined. For a fixed declared frame, its deficit and homogeneous affine matrix data are Lipschitz in the input state. In particular,

Substitution into Theorem thm:persistence returns the complete new geometry and certificate, evaluated at $s_c=-1/2$, with internal matter unchanged.

If $I_f(\varrho )=0$, $\kappa =0$, or $\theta =0$, the geometric output is the reference family at ${\mu }_0$.

Proof. The interval $I$ is convex and contains ${\mu }_0$. Thus ${\mu }_{\theta }(\varrho )\in I$ whenever $\varrho \in {\mathcal{D}}_{\mathrm{src}}$. Trace duality proves (eq:source-lipschitz).

For the spectral norm,

With tildes denoting homogeneous affine matrices,

The remaining fixed packet components do not vary.

The geometric construction was proved independently for every scalar $\mu \in I$, so substitution is not circular. The stated controls set the selected deficit to ${\mu }_0$.